The spacetime Penrose inequality with charge and original-data rigidity

Proof. Let D be the exterior region defining \Gamma. Choose r so that \Gamma lies strictly inside S_r and a neighborhood of S_r lies in the manifold interior of D. The truncated region D_r is a compact smooth manifold with intrinsic boundary \Gamma\sqcup S_r. Its boundary orientation on \Gamma is opposite to the chosen end orientation. Stokes’ Theorem gives 0=\int_{D_r}d\beta=\int_{S_r}\beta-\int_\Gamma\beta. The same argument on the truncation of \Omega proves the other equality. Contact between \Gamma and S occurs on the smooth intrinsic boundary of D_r; it neither deletes a boundary term nor creates another face. Every boundary component is included.□

Proof. Choose a point of S. Its boundary collar, connectedness of \Omega, and the single coordinate end give an embedded proper ray starting transversely at that point and eventually equal to a straight coordinate ray. Indeed, join the collar to the end by a path, perturb its compact part to an embedded arc in dimension three, shorten at its last crossing of a large coordinate sphere, and attach the outward straight ray. The two joins can be smoothed in disjoint coordinate balls.

The normal disk bundle of the ray is trivial because its base is an interval. The tubular-neighborhood construction on its compact portion, followed by the straight tail, gives an embedded tube

whose initial disk lies in S, whose positive-parameter part lies in \operatorname{int}\Omega, and whose tail has a fixed positive coordinate radius. Let \omega be a smooth two-form compactly supported in the interior of the transverse disk, oriented so that \int_{\mathbb D^2}\omega=1. Define \beta on the tube by \Phi^*\beta=\operatorname{pr}_{\mathbb D^2}^*\omega, and extend by zero. The transverse compact support makes this extension smooth at the lateral boundary; it is smooth up to S and closed.

Its comass is bounded on the compact portion and on the straight tail, the latter because the coordinate radius is fixed and g_{ij}\to\delta_{ij}. Thus C_\beta=\sup_\Omega\|\beta\|_{*,g}<\infty. A sufficiently distant sphere meets the supporting tail in a positively oriented graph over the entire support of \omega and has flux one. Lemma 3.1 gives the same flux on every full cut. Hence

Finally S itself is an admissible cut.□

Proof. Let C_t be the compact region inside S_t, including the filling. On a sufficiently distant end the smooth unit field Z=\nabla r/|\nabla r|_g satisfies

Increase R_0 so this holds for r>R_0 and F\subset\operatorname{int}C_{R_0}.

For any bounded finite-perimeter set M, the slicing and Gauss–Green formulas give, for almost every t>R_0,

where b(t)=\int_{S_t}\mathbf1_M\,dA_g uses the common almost-everywhere trace. In Equation 3.5, cut off Z beyond the bounded set M if necessary. Since |Z|_g=1,

Clipping strictly decreases perimeter if the discarded volume is positive.

Fix R_1>R_0 and clip each member of a minimizing sequence at a good radius in (R_0,R_1). The resulting sets lie in C_{R_1}, contain F, and have bounded perimeter and volume. Compactness in the space BV of functions of bounded variation gives an L^1 convergent subsequence, and lower semicontinuity gives a minimizer. The obstacle condition is closed under L^1 convergence. These compactness and lower-semicontinuity results apply in finitely many coordinate charts on a larger compact neighborhood [1, Theorem 3.23 and Proposition 3.6]. Equation 3.7 equates this confined infimum with the unrestricted bounded-set infimum. Applying it to a minimizer at good radii decreasing to R_0 proves that the minimizer has zero volume outside C_{R_0}. Its perimeter support therefore lies strictly inside C_{R_1}.

Uniform end bounds make the error in Equation 3.4 uniform, so the same construction works for the stated family. Only boundedness and the closedness in L^1 of the containment condition were used for the obstacle; a large compact truncation is always a finite-perimeter competitor. This proves the last assertion.□

Proof. Let M minimize and let M' be a bounded finite-perimeter local competitor, without an obstacle restriction. The set M'\cup F is admissible. Perimeter submodularity gives

Choose a smooth compactly supported vector field V with |V|_g\leq1 which agrees with the outward unit normal of F on \partial F. The signed-distance collar and a cutoff provide such a field. Gauss–Green gives

Consequently, for \Lambda=\|\operatorname{div}_gV\|_\infty,

The unchanged perimeters outside the variation neighborhood cancel, so this is also a local inequality.

The exact obstacle regularity result we use is Huisken–Ilmanen’s Regularity Theorem 1.3(iii): in a smooth Riemannian domain of ambient dimension less than eight, a pure perimeter minimizer respecting a C^2 obstacle has C^{1,1} boundary, smooth off contact [12]. Here M minimizes against every compactly supported competitor containing the open obstacle \operatorname{int}F; the metric and obstacle are smooth, the dimension is three, and the bulk term is zero. Lemma 3.3 keeps its frontier away from the auxiliary truncation boundary. The regular representative is the closure of its measure-theoretic interior, whose boundary equals the perimeter support. Away from F, both signs of every compact normal variation are allowed, so this smooth boundary has mean curvature zero. The stated Sobolev regularity follows from C^{1,1}.

A compact embedded frontier has finitely many components: it has a finite cover by connected graph neighborhoods. If its complement had a bounded open component, that component’s boundary would be a nonempty union of frontier components. Adding it to M removes their positive area without adding a new frontier and contradicts minimality. There is only one unbounded complementary component, because \widehat\Omega has one end and M is bounded.

Given an admissible smooth exterior D with full boundary \Gamma, the bounded set M_D=\widehat\Omega\setminus\operatorname{int}D contains F and has boundary exactly \Gamma. Therefore P_g(M_D)=\operatorname{Area}_g(\Gamma), including every coincident portion on S. In particular D=\Omega gives M_D=F. This proves p_g(F)\leq a_g(S).

Conversely, flow a minimizing set M for small time s>0 by a compactly supported smooth field pointing strictly outward on S. The image M_s contains F in its interior. Indeed it contains the flowed obstacle, whose boundary in the signed-distance collar has moved strictly outside S. The flow’s surface Jacobians converge uniformly to one, so P_g(M_s)\to P_g(M).

A compact C^1 embedded boundary has a C^1 defining function with nonzero differential near its boundary. Approximate this function in C^1 by smooth functions; the implicit function theorem yields smooth embedded boundaries converging in C^1, with the same enclosed side. Apply this to \partial M_s. Its positive distance from F preserves enclosure under sufficiently close approximation, and surface Jacobians give area convergence. Fill bounded complementary components of each approximation. This deletes components and does not increase area. The closure of the remaining exterior is connected, contains the whole distant end, and has smooth compact two-sided intrinsic boundary in \operatorname{int}\Omega. It is an admissible full cut. Letting the approximation error and then s tend to zero proves a_g(S)\leq p_g(F).□

Proof. At contact, write frontier and obstacle as graphs u\geq\psi, with u\in C^{1,1} and \psi smooth. On the contact set Du=D\psi and D^2u=D^2\psi almost everywhere. The second statement follows from the fact that the weak derivative of a Sobolev function vanishes almost everywhere on a level set, applied to the Lipschitz functions D_i(u-\psi). Off contact u solves the minimal graph equation; at almost every contact point its second derivatives equal those of the minimal graph \psi. Thus it solves the same equation weakly throughout the patch. The bounded gradient makes this a uniformly elliptic equation, so interior elliptic regularity and bootstrapping give smoothness. The strong maximum principle for the difference of the two smooth minimal graph equations gives local coincidence at each contact point. Contact is also closed, so connectedness extends the coincidence over the obstacle component.□

Proof. Two-dimensional area elements satisfy dA_{g_j}\geq c_jdA_g, so a_{g_j}(S)\geq c_ja_g(S). For the upper bound, test with any fixed smooth cut of g-area less than a_g(S)+\varepsilon, use compact convergence on that cut, and then let \varepsilon\downarrow0.□

Proof. Confinement supplies one compact set C containing every minimizer. Uniform metric comparison on C gives \varepsilon_j\downarrow0 such that for every set M' whose perimeter is supported there,

Taking infima, using confinement for both metrics, proves p_{g_j}(F)\to p_g(F), and hence P_g(M_j)\to p_g(F). Compactness in BV and lower semicontinuity give a subsequential L^1 limit M containing F and supported in C, with P_g(M)\leq p_g(F). Therefore it minimizes, proving Equation 3.11.

The vector-measure continuity needed here can also be proved directly. Use the fixed metric g, and write

where \nu_j,\nu are outward unit normals. Gauss–Green and L^1 convergence give weak convergence of these vector measures. Their total variation masses converge by Equation 3.11. Lower semicontinuity with nonnegative continuous weights and convergence of the total masses also imply |\lambda_j|\rightharpoonup|\lambda|.

Approximate the measurable unit normal \nu in L^2(|\lambda|) by a continuous vector field V on C with |V|_g\leq1. One obtains such approximations in finitely many bundle charts, joins them by a partition of unity, and projects onto the unit ball. They make \int(1-g(V,\nu))\,d|\lambda| arbitrarily small. For each such V,

A continuous function of a unit normal extends continuously to the closed unit-ball bundle and is uniformly continuous on that compact bundle. The displayed estimate, Cauchy–Schwarz, and weak convergence of |\lambda_j| give convergence of its integrals, first at a fixed approximation V, then as the approximation error tends to zero. Apply this to the even function \Psi(x,\nu^{\perp_g}) to obtain Equation 3.12. This is the needed case of Reshetnyak’s Continuity Theorem [1, Theorem 2.39].

The constant metric sequence gives compactness of \mathcal M_g(F). For a continuous symmetric tensor h, the function \Psi(x,T)=\tfrac12\operatorname{tr}_T h is continuous, giving the last assertion.□

Proof. The determinant formula for a two-plane area gives, uniformly over planes in the compact truncation,

For every finite-perimeter competitor M' in that truncation,

with the same remainder modulus. By Proposition 3.7, the minimum of L_h exists.

Testing a(s) with any M\in\mathcal M_g(F) gives

In the other direction, take a minimizer M_s for g_s. Its g-perimeter is bounded and at least a(0), so Equation 3.15 gives

Along a sequence realizing the lower limit, strict compactness gives a subsequence converging to a member of \mathcal M_g(F); continuity of L_h yields the opposite inequality. No differentiable choice of minimizer is needed. Since F\subset M, the frontier does not enter \operatorname{int}F, which proves the final assertion.□

Proof. If M_1,M_2 minimize, their union and intersection contain F. Submodularity gives

Thus both also minimize. If their frontier planes were distinct at an intersection, the tangent half-spaces would meet transversely. The boundary of their union or intersection would have a corner: its tangent cone would contain two distinct half-planes. This contradicts its C^1 regularity from Proposition 3.4.

Here is the required closedness argument. On the common compact truncation, Equation 3.10 has a fixed constant. It yields uniform density bounds at every frontier point, for sufficiently small coordinate balls centered there:

For clarity, compare with M\setminus B_r and M\cup B_r. For either phase volume v(r), the isoperimetric inequality for its intersection with the ball and these comparisons give

The coarea formula identifies v' with the slice area up to fixed coordinate constants. Since v(r)\leq Cr^3, the last term is absorbed for small r. Integrating (v^{1/3})'\geq c gives v(r)\geq cr^3; both phases are nonempty at every scale at a perimeter-support point. Relative isoperimetry gives the perimeter lower bound; the same competitors and the coordinate-sphere area bound give the upper bound. Smooth compact geometry makes these constants uniform.

Suppose now M_j\to M in \mathcal M_g(F) and x_j\in\partial M_j tend to x. The two volume bounds pass to every fixed small ball about x under L^1 convergence; hence x\in\partial M. Huisken–Ilmanen’s Regularity Theorem 1.3(ii) gives uniform local C^{1,\alpha} estimates for a fixed 0<\alpha\leq1/2 [12]. Its dependencies are the distance to the ambient domain boundary, obstacle C^{1,\alpha} bounds, metric C^1 bounds and positive definiteness, and the bulk bound, here zero. All are fixed in this application. Arzelà–Ascoli in local graph charts gives subsequential C^{1,\beta} convergence, 0<\beta<\alpha. The density bounds and the L^1 limit identify the limiting sheet with \partial M. Thus

The incidence set \{(M,x,T_x\partial M):M\in\mathcal M_g(F),\,x\in\partial M\} is therefore compact. Its spatial projection \mathcal Z is compact, and the previously proved uniqueness of the plane at each limit point makes T continuous.

By Proposition 3.7, the surface measures depend weakly continuously on M, so their average defines the finite Borel measure \eta. Each surface uses the same plane T(x) at its points. Integrating this identity proves Equation 3.16.□

Proof. The scalar-curvature formula in dimension three is R_{u^4g}=u^{-5}(-8\Delta_g u+R_gu). Both \tau^2 and |K|^2 acquire the factor u^{-4}. This proves the first identity. Write \pi=K-\tau g and f=\log u. The transformed trace reversal is \pi'=u^2\pi. In the divergence of this tensor use

Contracting the two covariant tensor slots gives u^2\operatorname{div}_{g'}(u^2\pi) =\operatorname{div}_g\pi+4K(\nabla f,\cdot); the terms containing \tau\,df cancel. This proves the momentum identity. Since dV_{g'}=u^6dV_g, fixed flux forms give X_i'=u^{-6}X_i. In an orthonormal frame their components therefore scale by u^{-4}, whereas u^{-2}Y has the same orthonormal components as Y. The electromagnetic identity and Equation 4.3 follow. Its last term is nonnegative for u\geq1, and K(\nabla u,Y)\geq-|K||du| proves Equation 4.4. Finally H'=u^{-2}(H+4\partial_\nu\log u) and \operatorname{tr}'_S K'=u^{-2}\operatorname{tr}_S K.□

Proof. Indices in the following calculation are raised with \eta. Define

It is skew in l,a. Expanding its divergence and collecting the two Hessians, wave operator, and trace terms of the linearized Ricci tensor gives

The last estimate follows because the difference between exact and linearized Einstein curvature is bounded by C(|h||\partial^2h|+|\partial h|^2).

In coordinates adapted to a background plane x^0=0, direct substitution in Equation 4.7 gives

Here j is spatial. The last expression for K^{\mathrm{lin}} follows by expanding the future unit normal and its connection; in Gaussian normal coordinates it reduces to K^{\mathrm{lin}}=\tfrac12\partial_0h. For fixed k, the extra vector in the middle line has identically zero spatial divergence. Its flux through any exterior coordinate sphere is zero: extend h_{0i} smoothly across the ball and integrate that identity. The value of the extension cannot affect the flux. The nonlinear errors in K and in its trace reversal are O(r^{-1-2q_0}) and hence have integrated flux O(R^{1-2q_0}). Consequently the plane fluxes \int\mathcal F_{i0b}n^i\,dA/(16\pi) equal (E,P_i).

For a fixed translation vector a, regard \tfrac12\mathcal F_{lab}a^b dx^l\wedge dx^a as a two-form and take its background Lorentzian Hodge dual. With orientation dx^0\wedge dx^1\wedge dx^2\wedge dx^3, its integral on an outward plane sphere is the flux just computed, with the common orientation chosen to give positive Schwarzschild energy. Equation 4.8 bounds its exterior derivative. Two large plane sections can be joined along a cylinder in r\asymp R with absolute three-dimensional coordinate volume O(R^3). Stokes’ Theorem bounds the difference of the integrals by

Replacing the sections by spheres or uniformly comparable ellipsoids in their planes has the same estimate, using the intervening annular three-region. The limiting flux is thus independent of the plane for fixed a. Tensor covariance under constant Lorentz changes proves Equation 4.6, including the plus sign of its spatial components. Substitution of b=(E/m,-P/m) gives m.

For the last assertions, write the stationary metric in a rest coordinate system as

The orbit metric is h_s=\gamma+W^{-1}\beta\otimes\beta, so h_s-\gamma=O_1(r^{-2q_0}). Its extra ADM flux is O(R^{1-2q_0}) and vanishes. Finally let \gamma=\delta+k with k=O_2(r^{-q_0}), and express its pullback under y=x+\zeta(x). Taylor expansion with one derivative gives

The linear ADM integrand of 2\operatorname{sym}D\zeta is \Delta\zeta_i-\partial_i\operatorname{div}\zeta =\partial_j(\partial_j\zeta_i-\partial_i\zeta_j). Its flux is zero by a smooth interior extension and skew symmetry. The remainder flux is O(R^{1-2q_1}+R^{1-q_0-q_1})\to0. This proves coordinate invariance in the precise class used here. A change of stationary time coordinate changes the threading one-form but leaves h_s itself unchanged.□

Proof. Choose smooth majorants and positive interpolations with the stated tails; the bound on K permits a constant radial majorant Cr^{-1-q} outside a compact set. When K is compactly supported, choose b with compact support instead. On \Omega_L=\{r\leq L\}, including the compact core, impose the additional boundary condition \phi=0 at r=L. Continue from t=0 to t=1 in

The linearization has principal part -\Delta_g and drift -t b\nabla\phi/\sqrt{|d\phi|^2+a^2} of norm at most b. The inner Neumann and outer Dirichlet boundaries are disjoint smooth components. The homogeneous mixed problem has zero kernel: a nonconstant positive maximum or negative minimum is excluded by the strong maximum principle and the boundary point lemma, and a constant is excluded by the Dirichlet condition. The mixed elliptic Fredholm index is zero, by continuation from the mixed Laplacian, so the linearization is invertible.

Here are bounds closing this continuation. On a fixed \Omega_L, mixed W^{2,p} estimates, p>3, and \sqrt{|d\phi|^2+a^2}\leq |d\phi|+a give

The second inequality is gradient interpolation, with its small parameter chosen after C_L. If the supremum norms were unbounded, divide a sequence by its supremum norm. Compactness in C^1 gives a function of supremum one solving -\Delta z=t b|dz|, with zero mixed boundary data. This is a homogeneous equation with bounded drift, by taking the drift to be t b\nabla z/|dz| where dz\ne0 and zero elsewhere. The same maximum principles give a contradiction. The resulting W^{2,p} bounds, followed by elliptic regularity, close the continuation. The solution is nonnegative: an interior negative minimum contradicts -\Delta\phi>0, and a negative minimum at S contradicts the boundary point lemma, since the outward domain derivative is -\partial_\nu\phi=1.

The bounds can be made independent of L on each compact subset. To see this without a bounded-domain constant depending on L, put f(r)=r^{-1}(1-r^{-\delta_0}) on a sufficiently distant tail. Direct differentiation gives

The same estimate holds with b in place of |K|. Since ba=O(r^{-3-q}) and \delta_0<q, a sufficiently large multiple A f satisfies -\Delta_g(Af)\geq b\sqrt{|d(Af)|^2+a^2}+w. Comparison on the exterior annulus, choosing its inner boundary value to dominate \phi, gives

with fixed L_*. If the core supremum diverged along an exhaustion, normalize by it and apply the preceding local estimates, including the inner boundary estimates. A subsequence converges locally in C^1 to a nonzero nonnegative solution of -\Delta z=b|dz|, with \partial_\nu z=0 and z=O(r^{-1}). The tail estimate forces its positive maximum to occur in a compact set; the strong and boundary maximum principles again exclude it. Thus exhaustion gives a positive smooth solution of Equation 4.10.

Rescale each exterior annulus to unit size. The metric has uniformly bounded C^2 coefficients, the scaled drift has norm O(r^{-q}), and the already proved bound is \phi=O(r^{-1}). Interior W^{2,p} estimates and then Schauder estimates give d\phi=O(r^{-2}), d^2\phi=O(r^{-3}). In the compact-b case the equation is the linear Poisson equation on a smooth symbol end; differentiating its rescaled form gives the estimates at every order. Its right side is integrable in either case. The divergence theorem therefore gives a finite limit of \int_{r=L}\partial_{\nu_L}\phi\,dA_g. Its difference from the Euclidean gradient flux is O(L^{-q}), so that limit also exists. For u=1+s\phi, -\Delta u-|K||du|\geq s w. Apply Equations 4.4 and 4.5. The scalar-curvature conclusion follows from its conformal formula. The energy increment is finite and tends to zero with s, because its ADM flux contribution is -(s/2\pi)\lim\int\partial_{\nu_L}\phi\,dA_g.□

Proof. Choose 0<\delta_0<\min(q,1) and f as in Equation 4.11. Far enough out f'<0 and -\Delta_g f-|K||df|\geq0. Let \chi(r) be a smooth nondecreasing cutoff from zero to one supported in this valid range. Define

there, and extend it constantly over the compact core. It is bounded, positive, smooth, and equals f far out. The radial differentiation formula shows

For each s\geq0 set u_s=1+s\phi and use (u_s^4g,u_s^2K) with the original forms. Charged DEC and all boundary signs persist. Completeness persists, since u_s\geq1, and the full-cut infimum can only increase. The data still have decay exponent greater than 1/2. The new constraint terms have orders r^{-3-\delta_0} and r^{-3-q}, so their absolute integrability follows from that of the old sources and Equation 4.2.

The asymptotic expansion u_s=1+s/r+o_2(r^{-1}) gives, directly in the original chart,

Indeed the leading new metric term is 4s\delta_{ij}/r and its energy flux is 32\pi s; the factor 1/(16\pi) yields 2s. The trace reversal is multiplied by u_s^2, whose additional momentum flux is O(r^{-q}) and tends to zero.

The assumed timelike numerical conclusion implies r_{A,s}\leq2\sqrt{(E+2s)^2-|P|^2}. By Proposition 3.2, A_{\min,g}(S)>0, while r_{A,s}\geq\sqrt{A_{\min,g}(S)/(4\pi)}. If E\leq|P|, take s\downarrow(|P|-E)/2 from above. The right side tends to zero and contradicts this fixed positive lower bound. Thus E>|P|. Applying the assumed timelike result to the original data finishes the proof. No common limiting boost is used.□

Proof. We first specify the ordinary divergence inverse used in the proof. For a mean-zero smooth compactly supported source in a ball, the support-preserving Bogovskiĭ operator gives a smooth compactly supported vector field, with divergence equal to the source and a gain of one derivative in W^{\ell,p}. The regularized integral operators and their support and Sobolev mapping properties are given in [10, Theorems 3.2 and 4.9, Corollary 4.11]. To use balls on the annulus, cover the prescribed source support by finitely many balls whose closures lie inside the annulus and join them by a finite connected family of such balls. Split a source by a partition of unity. Along a spanning tree of overlapping balls transfer the integral of each leaf piece to its parent using a fixed unit-integral bump in the overlap. Removing leaves successively leaves each piece with integral zero, since the total integral is zero. Applying the ball operators and adding the results gives a divergence inverse with fixed compact support. The finite cutoffs and bumps preserve all the stated Sobolev estimates. Use successively larger fixed compact supports for successive applications below.

For a scalar f with zero affine moments solve \partial_iV_i=f. Compact support and integration by parts imply \int V_i=-\int z_i f=0. Solve \partial_j B_{ij}=V_i componentwise. Then the symmetric part P=(B+B^{\mathsf T})/2 satisfies \partial_i\partial_jP_{ij}=f. In dimension three the inverse of trace reversal is

This gives the scalar solution and gains two derivatives.

For momentum, zero translation moments permit a componentwise solve \partial_jB_{ij}=V_i. Put S_{ij}=(B_{ij}-B_{ji})/2. The rotation moments give

Solve \partial_kD_{ijk}=-S_{ij}, choosing D skew in i,j, and define

Directly using that skew symmetry gives C_{ijk}=-C_{ikj} and (C_{ijk}-C_{jik})/2=D_{ijk}. It follows that P_{ij}=B_{ij}+\partial_kC_{ijk} is symmetric and \partial_jP_{ij}=V_i. Set k=P-\tfrac12\operatorname{tr}P\,\delta. There is one net derivative of gain: the second divergence solve gains a derivative and the displayed correction differentiates it once. All supports remain compact in the prescribed larger subannulus. Integration against the adjoint kernels also proves the necessity of the stated moment conditions.□

Proof. Use z=x/R on A=\{1<|z|<4\} and regard the scaled fields as g(Rz) and RK(Rz). The constraints scale by R^2. Write H=g(Rz)-g_*(Rz) and k=R(K(Rz)-K_*(Rz)). These have C^2\times C^1 norm O(R^{-\beta}). If \lambda equals one near the inner boundary, zero near the outer boundary, and takes values in [0,1], blend the metric and second form using \lambda. The linearized constraint error beyond the blend of the two linear sources is

If P(H)=H-\operatorname{tr}H\,\delta, the scalar commutator is

Thus \|f_R\|_{W^{1,p}}+\|V_R\|_{W^{1,p}}=O(R^{-\beta}) for every fixed finite p. Only the second derivatives of the original metric and first derivatives of its second form are used.

We next estimate all adjoint-kernel moments. Put h=g-g_*, \kappa=K-K_* in the physical x coordinates, and s=\mathcal L_xh, V=\mathcal D_x\kappa. They belong to L^1. In fact the scalar linear source for either datum equals its 2\mu plus an error bounded by

and the momentum linear source equals J plus an error bounded by

These errors are O(r^{-2-2q}) for the original data and O(r^{-4}) for the reference. They are integrable because q>1/2. For any integrable source F on the end,

Indeed splitting at a fixed M and dividing by R bounds the outer contribution by 4\int_{r>M}|F|, while the inner contribution tends to zero; then let M\to\infty.

For an affine scalar p define its linear constraint boundary flux

For a Euclidean Killing field Z, define B_Z(t;\kappa)=\int_{r=t}P(\kappa)_{ij}Z_i n^j\,dA. Integration by parts gives, for either type of test, the difference of the two boundary fluxes as the integral of the test times the linear source; the adjoint-kernel term is zero. For p=1 and constant Z, the boundary flux differences tend to zero as t\to\infty, since the two ADM vectors agree. For momentum the difference between Euclidean and geometric trace reversal has integrated error O(t^{1-2q})+O(t^{-1})\to0. For homogeneous linear p or rotational Z, integration from a fixed sphere and Equation 4.17 instead give

No limit for these first-moment fluxes is required.

For clarity, the scaling factors in the commutator moments are as follows. A constant test gives

A homogeneous linear test p(z) gives

For momentum the identical calculation uses R\kappa(Rz); translation tests have factor R^{-1}, rotation tests factor R^{-2}. Equations 4.17 and 4.18 therefore give

for every fixed affine p and every fixed Killing field Z.

Choose a fixed nonnegative bump \chi supported in the annulus and positive on a ball. The Gram matrix of the affine basis with weight \chi is positive definite. Subtract the linear combination of the smooth functions \chi p having the same affine moments as f_R. Likewise use the Gram matrix of the Killing-field basis and the vector bumps \chi Z for V_R. Equation 4.19 says that the coefficients of all these bumps are o(R^{-1}). Their norms of every fixed order therefore have this size. The remaining sources have zero moments and W^{1,p} norm O(R^{-\beta}). Apply Lemma 4.5 with the negative signs to cancel them. The resulting compact corrections have norms

Taking p>3 controls their C^2\times C^1 norms and ensures positivity of the corrected metric for large R.

The nonlinear constraint map about (\delta,0) has quadratic remainder bounded in C^0 by C R^{-2\beta} on these scaled data. Its sources consequently differ from the convex blend of the original and vacuum sources by o(R^{-1})+O(R^{-2\beta}) in C^0. The original scaled momentum is O(R^{-\beta}); changing its norm from the original metric to the corrected metric costs only O(R^{-2\beta}). Convexity of the cone \{(\mu,J):\mu\geq|J|\} therefore proves a scaled DEC deficit of size o(R^{-1})+O(R^{-2\beta})=o(R^{-1}). Undoing the scaling multiplies sources by R^{-2} and gives Equation 4.15. All corrections are supported away from the two annular boundaries, so the asserted exact matching and smoothness follow.□

Proof. The reference plane. Since m>0, use the exterior Schwarzschild metric

at sufficiently large radius. Set v=P/E, so |v|<1, and A=(I-vv^{\mathsf T})^{-1/2}. Parametrize the plane T=v\cdot y by y=Ax. Its induced background Minkowski metric is exactly |dx|^2. The unit normal and an adapted spatial frame are related to the horizontal ones by the Lorentz boost with velocity v. By Lemma 4.2, the plane data have

They are vacuum, with g_*-\delta=O_\ell(r_x^{-1}) and K_*=O_\ell(r_x^{-2}) at every order. Identify their x coordinates with the original end coordinates and apply Lemma 4.6.

Bending the reference end. Beyond r_x=4R replace the plane by the graph

where \psi is smooth, is one on (-\infty,0], is zero on [1,\infty), and has values in [0,1]. Choose L sufficiently large that |v|(1+\|\psi'\|_\infty/L)<1; then choose R_b to be a sufficiently large fixed multiple of R. The bend starts in the exact reference portion. Its background gradient has norm at most the preceding strict bound, so it is uniformly spacelike in G_* once R is large. It becomes a horizontal Schwarzschild slice for r_y\geq R_b e^L. Denote the resulting data before conformal repair by (g_R,K_R).

For fixed L, the graph has |dT|\leq c<1, |d^2T|\leq C/r_y, and derivatives of the background Schwarzschild coefficients are O(r_y^{-2}). The graph formulas thus give uniform comparability of g_R with \delta_y, |\partial g_R|\leq C/r_y, and |K_R|\leq C/r_y throughout the bend. The annular corrections have stronger corresponding bounds. It follows that, throughout these large-radius ranges,

Here and below constants may depend on L and v, but not on R.

Retain the original forms on the same manifold during the replacement and bend. Since y=Ax is fixed and all these metrics are uniformly comparable, their norms are O(r_y^{-2}) and their electromagnetic ball term is O(r_y^{-4}). The bend is vacuum for the total constraints. Thus Lemma 4.6, together with I_{g_R}(Y)=O(r_y^{-4}), implies the following estimate for the charged deficit. With \xi=r_y/R there are fixed 0<a<\xi_0<b such that the data are original for \xi\leq\xi_0 and horizontal Schwarzschild for \xi\geq b, and

The constant radial formulas used next are extended across the core.

Repair of the charged deficit. Choose a smooth \omega\geq0 vanishing near a and at least one on [\xi_0,b+1]. On [a,b+1] solve

and put z_R=0 below a. The constant C_0 will be chosen large using Equation 4.23. On the horizontal tail let

Starting at b, smoothly continue its derivative to t^{-2} over [b,b+1], using a convex interpolation of the positive derivative already prescribed and t^{-2}. Thus

The quantities A_R(\infty) and F_R(\xi)=\int_\xi^\infty z_R(t)\,dt are uniformly bounded for fixed L. Define

It is smooth, at least one, and constant on the compact region r_y\leq aR. Direct differentiation, with F_R'=-z_R, gives

On [a,b] this is at least e_RR^{-3}(c z_R'-C z_R). Choose C_0 sufficiently large to make it nonnegative everywhere there and at least c'e_RR^{-3} on [\xi_0,b]. On the horizontal tail K_R=0. Writing w_*=1+m/(2r_y), the radial conformal Laplacian formula is \Delta_{w_*^4\delta}u=w_*^{-6}r_y^{-2} \partial_{r_y}(r_y^2w_*^2\partial_{r_y}u). It yields the exact identity

Choose e_R\to0 with e_R/(\eta_R+R^{-1})\to\infty. For fixed L, Equations 4.24, 4.26, and 4.27, followed by Equation 4.4, show that (g_0,K_0)=(u_R^4g_R,u_R^2K_R) satisfies charged DEC. All boundary signs persist, since u_R is constant near S.

By Equation 4.25,

for each fixed R,L. Its ADM energy and momentum are therefore

The second form is compactly supported and the scalar curvature has O_\ell(r_y^{-4}) decay on the tail. Both assertions follow from the exact horizontal form and Equation 4.27.

Strictification and the limiting comparisons. Fix this repaired exterior and apply the compact-K part of Lemma 4.3, with a fixed 0<\delta_0<1. A further conformal change by 1+s\phi, s>0, gives the strict inequalities and every stated decay and integrability property. Its energy tends to E_0 as s\downarrow0, its momentum remains zero, and it can be made arbitrarily small in any prescribed finite collection of compact smooth norms.

It remains to check the area comparison, including cuts that escape into the bend. On the Schwarzschild graph, the spatial Schwarzschild metric dominates |dy|^2 and its lapse squared is at most one. Moreover Equation 4.22 gives

Since 0\leq\psi\leq1, the tensor square of this expression is bounded above by (v\cdot dy)^2+C L^{-1}|dy|^2. Consequently, on the plane and bend,

On the replacement annulus g_R=\delta_x+O(R^{-\beta}), while the original metric has g=\delta_x+O(R^{-q}) throughout the distant end. Inside the replacement radius the metrics agree. Both conformal changes increase the metric. Therefore the final metric has a global lower bound g_{R,L,s}\geq(1-C'/L-o_R(1))g.

Choose L_j\to\infty first. For each L_j choose R_j so large that all the preceding constructions apply, their error bounds are at most 1/j, and the energy increment in Equation 4.28 is at most 1/j. The choices are possible because for fixed L the repair constants and A_R(\infty) are bounded independently of R. Then choose s_j>0 small enough that the strictification energy increment and the errors in the first j compact smooth seminorms are at most 1/j. The replacement and bend move to infinity; u_R\to1 uniformly and is constant on every fixed compact set eventually. Thus g_j,K_j converge locally smoothly to g,K. The field convergence follows from the unchanged forms and the smooth convergence of the volume forms. This proves the energy, momentum, and local convergence assertions. The global lower metric bound proves completeness and gives, for every full cut \Gamma,

Taking infima yields the required lower limit for the areas. For the upper limit choose, for any \eta>0, a fixed smooth full cut \Gamma_\eta with \operatorname{Area}_g(\Gamma_\eta)\leq A_{\min,g}(S)+\eta. It is compact, so local convergence gives

Let \eta\downarrow0. This proves Equation 4.20 for precisely the original full cut definition. If one uses the perimeter formulation, Proposition 3.4 identifies the same infimum, including the coincident obstacle frontier. At every stage the charge fluxes were preserved as integrals of the two original closed forms.□

Proof. Every connected component of the boundary is an orientable closed surface, hence is diffeomorphic to the boundary of a handlebody of its genus. Attach a separate such handlebody on each component. The collar theorem gives signed collar coordinates, increasing toward the original exterior. Extend the metric and tensor smoothly a short distance inward. The strict expansion inequality persists on a shorter collar by continuity.

On the next fixed collar, interpolate the metric to a product metric. The metrics in this interpolation are positive definite and form a fixed compact family; their cross-sectional mean curvatures are therefore bounded. Before making this interpolation, subtract \sigma_i Lg from K, with L\ge0 increasing smoothly from zero. This changes \sigma_i\operatorname{tr}_{\rm tan}K by -2L. Start the change within the region where the expansion is already strict; after L has become large, it dominates the bounded mean curvature and the bounded tensor terms in the interpolation. One may then interpolate the remaining tensor to the pure trace -\sigma_iL_0g, still retaining strictness. Finish with an exactly product segment. This gives the asserted fixed transition.

Choose a smooth metric on each handlebody extending its product boundary collar, and put K=-\sigma_iL_0g on that cap. Insert between the transition and cap as many copies of the fixed product segment as are needed. A length bounded by a fixed multiple of 1+N is available, with the multiple enlarged when the height barriers are chosen. There are finitely many fixed transition and cap pieces. Product translation identifies all local neck charts, so all asserted local bounds and the local injectivity radius are uniform. Only the cylinder volume grows with its length. The original end is unaltered, proving completeness and the one-end assertion. Extending r as a positive constant along the inserted pieces proves the weight assertion. This construction concerns the metric and tensor only; all subsequent uses of the flux forms take place on the original exterior, where they remain closed with their prescribed two fluxes.□

Proof. First obtain the scalar identity before conformal multiplication. On \Omega_N\times\mathbb R consider the stationary Lorentzian metric

Written in the coordinates (x,z), it has spatial metric g, lapse U=l\sqrt D, and shift \gamma=l^2\nabla f=Uw. Its future unit normal to the constant-z slices is \mathbf n=U^{-1}(\partial_z-\gamma)=U^{-1}\partial_z-w. With the convention for K in Definition 2.1, stationarity gives

Here \operatorname{sym} means the average of a two-tensor and its transpose. The contracted Gauss equation and normal variation of the second fundamental form give, with this sign convention,

Since H_b is independent of z, \mathbf n(H_b)=-w(H_b). Substituting and using \operatorname{div}(Uw)=-UH_b yields

In the coordinates z-f(x) the same metric is the warped static metric -l^2d(z-f)^2+\bar g. Its scalar curvature is \mathbf R=R_{\bar g}-2l^{-1}\Delta_{\bar g}l. The inverse and determinant of \bar g are A_d and D\det g, respectively. Hence

For the second identity, divide by U and use d\log U=p(1+v)dt+H^f(w,\cdot) and d\log l=pdt; the resulting difference is H^f(w,\cdot)+p\{vdt+w(t)w\}=-b(w,\cdot). Equations 5.13 and 5.14 now give

For clarity, the insertion of the original constraint tensor is an exact cancellation. Namely,

where symmetry of \mathsf P_K permits use of Equation 5.12. Moreover, \mu=R_g/2-\mathsf B(K,K)/2 and \mathsf Q=K-b. Expanding \mathsf B(\mathsf Q,\mathsf Q) in Equation 5.15 therefore proves

In dimension three the conformal scalar law is

The same determinant calculation as above gives \Delta_{\bar g}t =U^{-1}\operatorname{div}(UA_d\nabla t)-p|dt|_{A_d}^2. Set W=\mathsf P_{\mathsf Q}w+2A_d\nabla t. Since u=e^tU,

It follows that

Differentiating Equation 5.4 gives the useful exact one-form identity

In the block notation, \operatorname{tr}T=F-\chi j, and \mathsf P_{\mathsf Q}w has transverse component a_wM+pv y and axial component -a_w(F-\chi j). Thus both sides of the first identity below have transverse component u(qy+a_wM) and axial component u(2dx+\chi a_wj-a_wF):

For the second identity, use u^{-1}\operatorname{div}(uw)=-H_b+w(t) =\operatorname{tr}H^f+(2p+1)w(t). Replacing the flux in Equation 5.17 and differentiating uFw gives Equation 5.10 with Equation 5.9.

To check the full expansion, put z_T=\operatorname{tr}T=F-\chi j. The transverse, mixed, and axial blocks of \mathsf Q are T, M+pa_wy, and j+2pa_wx. Therefore

and

Combining the terms in M,y completes the square |M+s_pa_wy|^2-v|y|^2/4. The remaining linear terms in x equal 2s_pa_wx(F-z_T)=2s_p\chi ja_wx; the remaining terms in j,F equal -z_Tj+F(F+(1-\chi)j)=\chi j^2-\chi Fj+F^2. These are exactly all the terms in Equation 5.11.□

Proof. Completing first the x square and then the F square in Equation 5.11 gives the exact expression

Indeed, the residual coefficient of j^2 is

Every displayed square has a positive coefficient, and 2s_p-v/4\ge3s_p/2\ge3/4. The last square implies dj^2\le3(N+2)^2\mathcal T/8 and \chi^2j^2\le3\mathcal T/2. The F square then gives F^2\le3\mathcal T; hence |T|^2=|T^0|^2+(F-\chi j)^2/2\le13\mathcal T/2. The transverse square gives |y|^2\le4\mathcal T/3. Undoing the x shift gives

Likewise,

Adding these bounds proves Equation 5.20. In particular the coercivity loses no exponential factor from l or d.

When dt=0, Equation 5.11 reduces to

The matrix of the last two variables (F,\sqrt\chi j) is

Its eigenvalues are 1 and (3-\chi)/4\in[1/2,3/4]. This proves Equation 5.21.□

Proof. Use the graph of f in the Riemannian warped product g+l^2dz^2. At dt=0 its second fundamental form, up to its irrelevant overall normal sign, is H^f; its induced metric is \bar g and its inverse is A_d. At that point l^{-1}\nabla^2l=p\nabla^2t. The horizontal Ricci tensor of the ambient metric is \operatorname{Ric}_g-p\nabla^2t, its unit vertical Ricci component is -p\Delta_gt, and its mixed Ricci components vanish. Its scalar curvature is R_g-2p\Delta_gt. The graph unit normal has horizontal part -w and vertical part of length \sqrt d. The contracted graph Gauss equation therefore gives

At a stationary point the conformal law contributes exactly -2\operatorname{tr}_{A_d}\nabla^2t. This proves Equation 5.24, including all the Hessian terms.

To prove the inequality, first use C=H^f+K. Its transverse trace is F-\chi j, so direct contraction gives

Subtracting this from the stationary expression for \mathcal T gives the exact formula

Since \chi\le d, we have b_*\ge\chi(1+d/2)\ge\chi and

Young’s inequality, in the form d|Fj|\le b_*j^2/2+d^2F^2/(2b_*), thus proves

For the remaining polarization, let X=A_d^{1/2}CA_d^{1/2} and Y=A_d^{1/2}KA_d^{1/2}, regarding these as symmetric matrices in a g-orthonormal frame. Then

Here d^2/\chi=dq/2\le(N+2)/2 controls the only axial loss. The trace-reversal map on symmetric three-by-three matrices has operator norm two, whence

Also \mathcal S(K)\le|K|^2. As \mathcal S(C-K)=\mathcal S(C) -\{(\operatorname{tr}X)(\operatorname{tr}Y)-\langle X,Y\rangle\} +\mathcal S(K), Young’s inequality with coefficient 1/12 of \mathcal E_0 and Equation 5.27 imply

Use \mathcal E_0\ge\mathcal T and 0\le p\le N to obtain Equation 5.25.□

Proof. At a positive interior maximum of f, the left side of Equation 5.31 is nonpositive and the right side is strictly positive. A negative minimum is excluded in the same way. The zero boundary value therefore gives f=0. The defining formulas now give t=Z, \mathcal T=2s_p|dZ|^2, u=e^Zl(Z), and V=2e^Zl(Z)\nabla Z. Differentiating this flux proves Equation 5.33. For b=0, define \Psi(z)=\int_0^z e^sl(s)\,ds. The equation is \Delta_g\Psi(Z)=0 with zero boundary value, so \Psi(Z)=0 by the maximum principle. Since \Psi'>0, Z=0. This identifies the starting solution for continuation without assuming existence at any other parameter value.□

Proof. At an interior maximum of h, its gradient vanishes, and hence D=1, A_\chi=I. The trace equation there reads

At an interior minimum the reverse inequality holds. The boundary height is zero and |\operatorname{tr}_gK_a|\le C uniformly on the filled manifolds. This proves the first assertion. It also proves h=0 when K_a=0.

Choose r_0 outside the support of K and large enough for the following radial calculation. At a contact point with H_R=C_0(r^{-b_1}-R^{-b_1}), the slope axis of h is the radial axis. The transverse eigenvalues of A_\chi are one and its axial eigenvalue is 0<\chi\le1. The asymptotic bounds for the prepared metric imply, uniformly for this entire range of \chi,

Indeed, in the Euclidean metric the radial and transverse Hessian eigenvalues are respectively C_0b_1(1+b_1)r^{-b_1-2} and -C_0b_1r^{-b_1-2}; the metric and connection errors are o(r^{-b_1-2}). Since 1-b_1>0, one choice of r_0 makes the inequality strict for every slope and every N. Choose C_0 so that H_R\ge C at r=r_0 for every R\ge2r_0. If h-H_R had a positive interior maximum, the matching gradients would determine the same t, l, D, and A_\chi in evaluating the trace at that point, while \nabla^2h\le\nabla^2H_R. The trace equation, K=0, and (6.5) would give 0<h\le(l/(\eta\sqrt D)) \operatorname{tr}_{A_\chi}\nabla^2H_R<0. Applying the same argument to -h proves the end bound. The two barriers vanish at r=R; their one-sided derivatives there, together with the zero tangential derivative of h, imply (6.4).

For a fixed boundary slope the function

has derivative 1+pv/2\ge1. At t=-\epsilon its value is at most

This is negative for sufficiently large R at fixed N; for example it suffices to require C^2\ell^{-1}R^{-2b_1-2}<e^{4\epsilon}-1. Since its value at the actual boundary t is Z=0, strict monotonicity gives t>-\epsilon.□

Proof. The outer boundary is excluded by Lemma 6.1. Consider first b=1 and an interior minimum with t=-\epsilon. Write e for the slope axis, arbitrary at zero slope. The stationary-point calculation of Lemma 5.4 gives

where \mathcal S(B)=\tfrac12((\operatorname{tr}_{A_d}B)^2 -|B|^2_{A_d\otimes A_d}). The signs in this inequality use p,v\ge0, A_d>0, and \nabla^2t\ge0. The same lemma supplies a constant c_1>0, independent of N, and the algebraic estimate

Both formulas concern the actual tensor K_a; no energy condition on the filling or on the scaled tensor is used.

By the scalar identity (5.10) and F=\eta f, w(F)=G, so comparison with (6.6) yields

with B_N\le C(1+N), using the coefficient 6N+25 in Lemma 5.4. To justify the last bound throughout the noncompact range, observe that \mu_a-R_g/2=((\operatorname{tr}K_a)^2-|K_a|^2)/2, and that this term, J_a, K_a, and F\operatorname{tr}K_a are supported in the enlarged core, where they are bounded and \rho_0 is bounded below. On the end, |F|^2=|h|^2\le C\rho_1 by Lemma 6.1, while |\operatorname{Ric}_g|\le C r^{-3}\le C\rho_1. The same estimates on a fixed connecting annulus are absorbed into the constant. This proves the displayed bound with constants independent of R and a.

At the assumed minimum m_0=1. Equating the last lower bound with (6.2) gives

Choose 0<\delta_0<\frac12\min(c_1,1), and then choose the polynomial C_N\ge B_N+c_*+1. The right side of (6.8) is positive because \rho_0>0, a contradiction.

In the remaining homotopy segment a=0, Lemma 6.1 gives f=0. Consequently t=Z, v=0, A_\chi=I, u=e^t l(t), and \mathcal T=(2p+1)|dt|^2. At an interior minimum t=-\epsilon, the equation is

Its left side is nonnegative, whereas its right side is negative if b>0. If b=0, set \Phi_0(t)=\int_0^t e^s l(s)\,ds. Then \Delta_g\Phi_0(t)=0 with zero boundary data, hence \Phi_0(t)=0 and t=0, since \Phi_0 is strictly increasing. This proves the assertion on the whole homotopy.□

Proof. Only the support of the penalty needs consideration for (6.10). There -\epsilon\le t<-\epsilon+1/N\le1, and therefore, when Z\ge2,

The radius weights are bounded above, so the negative penalty has magnitude at most C C_N. Thus one may take

Increasing this expression to a polynomial proves (6.10).

Choose a nondecreasing Lipschitz function q which is zero on (-\infty,Z_0], one on [Z_0+1,\infty), and has 0\le q'\le2. It vanishes on the outer boundary. Testing the divergence equation with q(Z) gives

The last inequality is the scalar estimate -2x^2+yx\le y^2/8, and A\le I. The last integrand is supported in \mathcal C_N, where u=l e^{2Z-t}\le e^{1+\epsilon+2(Z_0+1)} on the strip. Its integral is consequently at most \exp(P_N). All terms on the left are nonnegative by (6.10), and q=1 above Z_0+1, so

The same argument applies to any fixed enlargement of the core, since \rho has a positive lower bound there.

Here is a pointwise conversion of this estimate to the claimed moment. Since l\ge\ell\ge\eta and D\ge1,

The final inequality is \sqrt D=e^{2(Z-t)}\ge e^{Z-t}. Thus (6.13), \rho\ge c on the core, and \eta^{-1}=e^{3N\epsilon/2} bound the integral in (6.11) above Z_0+1. On the complementary set, \sqrt D=e^{2(Z-t)}\le e^{2(Z_0+1+\epsilon)} and e^Z\le e^{Z_0+1}; multiplication by the core volume C(1+N) gives the same bound there.□

Proof. Use the graph of f in the product metric g+dz^2 on a larger base patch times the entire real line. Put

Its measure is d\nu_1=\sqrt{E_f}\,dV_g and its inverse metric on base covectors is A_1. Equations (6.9) give

The axial eigenvalues of A_1,A_d,A are respectively E_f^{-1},d,\chi, and 2d/(2+N)\le\chi\le d. Hence the measures and inverse metrics are pairwise comparable with factors bounded by C(2+N)\ell^{-2}\le\exp(P_N).

The scalar mean curvature of this graph in g+dz^2 is

Coercivity, Proposition 5.3, controls the transverse block of H^f+K_a and the axial block multiplied by \sqrt d by P_N\sqrt{\mathcal T}. Since E_f^{-1}\le e^2d\le e^2\sqrt d and |K_a|\le C, it follows that

This estimate uses weighted axial contraction, rather than an unweighted bound for the full Hessian.

Each larger base patch can be smoothly isometrically embedded in a fixed Euclidean space. Only finitely many compact geometric models are needed: the fixed exterior and transition patches, the fixed caps, and translates of product-neck patches. One can obtain such embeddings by extending each model to a compact smooth Riemannian manifold and applying the smooth isometric embedding theorem [16, Theorem 2, p. 59]. The Euclidean dimensions and the second fundamental forms on the smaller patches therefore have fixed bounds. Taking the product of these embeddings with the identity of the real line gives an isometric embedding of the ambient product. The Euclidean mean curvature vector of the graph has norm at most |H_1|+C, because the trace of the ambient second fundamental form over its three unit tangent vectors is bounded by a fixed constant.

The Euclidean submanifold Sobolev inequality of Michael and Simon [15], in the form proved in [19, Chapter 4, Section 6, Theorem 6.7], applies to this smooth three-dimensional immersed graph and compactly supported test functions: if \zeta\ge0, then

There is no boundary term because the support is in the interior of the larger patch. Apply it to \zeta=|\varphi|^4 and use Cauchy–Schwarz on 4|\varphi|^3|d\varphi| and |\mathbf H||\varphi|^4 to obtain

The zero function is harmless; otherwise this follows by dividing the intermediate inequality by (\int|\varphi|^6\,d\nu_1)^{1/2} and squaring. The already established metric, measure, and mean curvature comparisons yield (6.14).□

Proof. Put B=e^t l, \kappa=K_a(w,\cdot) and \gamma=d\log B=(1+p)dt. Coercivity and A\le A_d show that

with B_N polynomial. The boundedness of the weights in the penalty gives, globally, \Xi\ge\delta_0\mathcal T-L_N for a polynomial L_N. Test \operatorname{div}V=u\Xi with the nonnegative function \psi^2e^{2sZ}/B. As u/B=\sqrt D, exact differentiation gives

For clarity, set X=|dZ|_A and Y=|d\psi|_A\le|d\psi|_g. Young’s inequality and (6.16) give the following pointwise bounds for the terms on the right:

Choose the polynomial threshold s_0\ge1+C\delta_0^{-1}B_N^2, with its constant large enough that the first X^2 error is at most s\psi^2X^2. Substituting these five estimates into (6.17) leaves at least \delta_0\mathcal T/2+sX^2 on its left and at most P_N(1+s)(\psi^2+Y^2) on its right, all multiplied by e^{2sZ}. Absorbing the fixed factor \delta_0/2 proves (6.15).□

Proof. First take b=1 and write Y=e^Z. Choose nested core patches U_r\Subset U_R with 1/2\le r<R\le1, in uniform local coordinates, and cutoffs with |d\psi|_g\le C/(R-r). Combining Lemmas 6.4 and 6.5, applied to \varphi=\psi e^{sZ}, gives

Indeed, differentiation of \psi e^{sZ} contributes at most 2e^{2sZ}|d\psi|_A^2+2s^2\psi^2e^{2sZ}|dZ|_A^2; the latter integral is bounded by 2s times (6.15). This accounts for the power (1+s)^2 in (6.19).

Set s_i=3^i s_0, take successive spatial gaps proportional to b2^{-i-1}, where 0<b\le1 is the total gap, and iterate (6.19). The product of constants has logarithm at most

The identities

therefore give

where D_N\le C(1+\log S_N+\log E_N+\log(1+s_0))\le P_N. The limiting norm is the supremum because each individual solution and its positive smooth measure are smooth on the closure of the larger patch. No uniform bound for that individual supremum has been used to obtain (6.20).

We next lower the initial integrability exponent explicitly. By Lemma 6.3, \int_{U_R}Y\,d\nu\le e^{L_N} with L_N polynomial. Put M(r)=\max(1,\sup_{U_r}Y) and \theta=1-1/(2s_0). The inequality Y^{2s_0}\le M(R)^{2s_0-1}Y and (6.20) yield

Choose increasing radii r_j whose successive gaps are c2^{-j-1} and whose limit lies strictly inside the larger patch. Iterating (6.21) m times gives

The first term tends to zero: the supremum on that fixed larger patch is finite for the particular smooth solution. Moreover,

Consequently

This computation is the required control of the parameter dependence: both iteration steps enlarge only a polynomial in the logarithm of the supremum. Uniform patches cover the entire enlarged core and the support of K, so (6.22) bounds Z there with one polynomial independent of the number of patches.

Outside this covered region K=0. If Z attained a larger maximum, exceeding also Z_0, the maximum would be interior since Z=0 on the outer sphere. At such a point V=2uA\nabla Z and

by (6.10), a contradiction. This proves the global upper bound on the first segment.

On the second segment a=0, the trace equation again gives f=0 and t=Z. If b>0, a positive interior maximum with Z>\max(0,-\epsilon+1/N) has m_0=0, dt=0, and hence \operatorname{div}V\le0<b u\rho. Together with zero boundary values this is already a uniform upper bound. If b=0, the transformed harmonic equation used in Lemma 6.2 gives Z=0.

Finally Lemma 6.2 makes the lower bound strict, (6.9) gives the warp bounds, and D=e^{4(Z-t)}\le\exp(P_N) gives |df|_g=\sqrt{D-1}/l\le\exp(P_N). Lemma 6.1 and \eta^{-1}=e^{3N\epsilon/2} give |f|\le\exp(P_N). This proves (6.18).□

Proof. We give the compactness and improvement arguments, including the treatment of the face. The identities involving third derivatives are distributional. For a C^2 function and metric their trace-reversed form follows by C^2 smoothing: the fluxes converge locally uniformly, the Hessian quadratics converge in L^1, and the Ricci terms converge locally uniformly. The original axial equation is substituted only after this identity has been established; neither \chi nor G is differentiated or mollified. The same argument on each side of a face retains the normal flux jump. Thus the proof applies in particular to the C^{2,\alpha} functions arising in continuation below.

A Hessian estimate above exponent two. Normalize the metric at the center of a small coordinate ball. The coordinate principal matrix a in Equation 7.1 satisfies

after shrinking the ball by an amount determined by the metric bounds. For a compactly supported function Y on Euclidean space, the Fourier transform gives \|D^2Y\|_2=\|\Delta Y\|_2, where the matrix norm is Frobenius. The Calderón–Zygmund estimate at any fixed exponent larger than two [11, Theorem 9.9], interpolated with this identity, implies that the norm of D^2\Delta^{-1}:L^s\longrightarrow L^s tends to one as s\downarrow2. Choose s_0>2 sufficiently close to two that this norm times 1-c/2 is less than one. Writing \Delta Y=(I-a):D^2Y+a:D^2Y and absorbing gives the global estimate. Apply it to \zeta Y, and use

Using intermediate balls and absorbing successively gives, for 2\le s\le s_0,

For clarity, the successive absorption uses radii tending geometrically from r/2 to r: choose the interpolation coefficient smaller than the inverse of the geometric loss to the power 2s. The resulting series converges. This proves Equation 7.3 without a derivative of a.

The trace-reversed identity. In Euclidean coordinates put H=D^2z, q=1-\chi, and W=|Dz|^2/2. Since He= D W/|Dz| wherever Dz\ne0, direct multiplication gives the identity, also valid when Dz=0,

Moreover \operatorname{tr}H=qH(e,e)+G. Choose (1-c)^2<\theta<1. Young’s inequality then gives

If |Dz|\le1 and S=1-|Dz|^2, we consequently have

This is a divergence inequality with measurable coefficients; it has no D\chi term. Intrinsically the last member of Equation 7.4 gains \operatorname{Ric}_g(\nabla z,\nabla z). Passing to coordinates with the volume density and replacing covariant by ordinary derivatives adds bounded flux and scalar errors. On rescaled balls whose metric tends to the Euclidean metric in C^1, whose smooth-side curvature tends to zero, and whose normalized forcing tends to zero, these errors have the form

with fixed ellipticity constants and \varepsilon\to0.

A gradient drop or entry into an affine regime. Fix 0<r_*<1/32. For every b_0>0 there exist k_*\in(r_*,1) and an error tolerance \varepsilon_*>0 such that a normalized solution on B_1, with |\nabla z|\le1 and errors at most \varepsilon_*, satisfies one of

for an affine L. To prove this, suppose the assertion fails, let the errors tend to zero, and let the gradient supremum on B_{r_*} tend to one. The scalar weak Harnack inequality [11, Theorem 8.18], applied to the nonnegative supersolution S in Equation 7.6, gives

for a fixed p>0. Equation 7.3 gives a uniform local L^2 bound for DS. Here is the extra truncation step needed to retain the negative Hessian term. Testing the supersolution inequality by \zeta^2(b-S)_+ yields

Indeed the derivative of (b-S)_+ contributes the negative of the elliptic energy on \{S<b\}; the cutoff and B terms are absorbed by Young’s inequality. Since S\to0 in measure, split DS into the sets \{S<b\} and \{S\ge b\}. The first has limiting L^1 norm at most Cb by Equation 7.8; the second has vanishing L^1 norm by the uniform L^2 bound and its vanishing measure. Thus DS\to0 locally in L^1. Testing Equation 7.6 with a fixed nonnegative cutoff now gives D^2z\to0 locally in L^2. After subtraction of a constant, the gradient bound gives uniform compactness, and the limit is affine. Its slope has length one because S\to0 in measure. This contradicts failure of the second alternative in Equation 7.7.

Improvement near a nonzero affine function. Suppose on B_1 that |z-L|\le b\le b_0, 1/4\le|DL|\le4, and the gradient is bounded by a fixed constant. Suppose also that metric oscillation, connection errors, and forcing are at most b\delta. For Y=(z-L)/b, Equation 7.3 gives a uniform W^{2,s_0} bound on smaller balls; its source is bounded by C\delta. Put e_0=DL/|DL|. The directions satisfy

Hence they converge in measure to the fixed axis as b_0,\delta\to0. Hölder’s inequality, with exponent 2s_0/(s_0-2) for this bounded coefficient error, proves

Only the axis is frozen; \chi remains an arbitrary measurable function of position.

There is a zero-Dirichlet correction U\in W^{2,2}(B_{3/4})\cap W^{1,2}_0(B_{3/4}) with

Indeed solve U=\Delta_D^{-1}((1-\chi)U_{e_0e_0}+R) by contraction in the Hessian norm. On a convex ball integration by parts gives

the difference is the boundary mean curvature times (\partial_\nu U)^2, which is nonnegative. Thus the contraction factor is at most 1-c. Poincaré’s inequality controls the lower derivatives. In dimension three the Sobolev embedding W^{2,2}\hookrightarrow C^{0,1/2} gives \|U\|_\infty\to0. This is the dimension-specific uniform correction required here.

Write Y_0=Y-U, and rotate so that e_0 is the third coordinate axis. Then v=\partial_3Y_0\in W^{1,2} satisfies

distributionally: it is exactly the distributional derivative of \Delta_{e_0^\perp}Y_0+\chi\partial_3^2Y_0=0. The scalar divergence-form Hölder estimate [11, Chapter 8] and its energy inequality therefore give, on smaller balls,

for some \alpha_1\in(0,1). Since \Delta Y_0=(1-\chi)\partial_3v=:F_0, Cauchy–Schwarz gives

This also controls all the first derivatives of Y_0. To see the point explicitly, localize F_0 to a slightly larger ball and take its Newton potential P. In the difference DP(x)-DP(y), with h=|x-y|, the two near regions have size bounded by C\sum_{j\ge0}(2^{-j}h)^{\alpha_1}\le Ch^{\alpha_1}, by Equation 7.11. On the annuli of radius r\ge2h, the derivative of the gradient kernel is bounded by Cr^{-3}, giving Ch\sum_{r=2^jh}r^{\alpha_1-1}\le Ch^{\alpha_1}. The remainder Y_0-P is harmonic. Its local norms are bounded by the already established W^{2,2} bound and the potential bound. Consequently DY_0 is uniformly Hölder on an interior ball.

Choose 0<\alpha_2<\alpha_1, then a sufficiently small fixed \lambda\in(0,1/8) so that the Taylor error of Y_0 is at most \frac12\lambda^{1+\alpha_2}. Finally choose b_0 and \delta so that the uniform correction is smaller than the other half. We obtain an affine L' with

Iteration and the boundary interface. In the improvement statement b is an allowed upper bound for the error, and need not equal the actual error. Fix b_0,\delta,\lambda as above, decreasing b_0 so that Cb_0/(1-\lambda^{\alpha_2})<1/4, and then choose k_*,\varepsilon_* in Equation 7.7. Initially shrink to a radius R_0 and normalize the gradient by a fixed bound \Lambda_0 so that all errors are at most \min(\varepsilon_*,b_0\delta). After j drops the physical radius and gradient normalization are R_j=R_0r_*^j and \Lambda_j=\Lambda_0k_*^j. In particular the normalized forcing is R_jG/\Lambda_j, and is multiplied by r_*/k_*<1 at each drop. Metric and connection errors decrease at least by r_* and curvature errors by r_*^2. At the first affine entry the physical radius is R_e=r_*R_j, with the same \Lambda_j. Rescale that ball and apply Equation 7.12 with the prescribed upper bound b=b_0, even if the actual affine error is smaller. On iteration i use b_i=b_0\lambda^{i\alpha_2}. The forcing divided by b_i is at most \delta\lambda^{i(1-\alpha_2)}; the same comparison controls the connection errors. The slope increments are summable by the choice of b_0, so the axis remains separated from zero. Only constants are subtracted in the rescaled original equation, so its axis is always the actual gradient axis. These two regimes give uniform affine approximation of order r^{1+\alpha}, including the transition scale, for any sufficiently small

The affine approximation criterion for Hölder gradients can also be seen directly: affine slopes at consecutive dyadic scales differ by Cr^\alpha, hence converge; comparing overlapping balls gives the same modulus between two centers. Thus the first estimate in Equation 7.2 follows. Subtract the first jet in Equation 7.3. Its error is O(r^{1+\alpha}); connection terms are bounded. Squaring the resulting L^2 estimate proves the second estimate.

For a face use Gaussian coordinates (x',x_3), subtract its constant value, reflect z oddly and g evenly. The reflected function is C^1\cap W^{2,s}, the metric is Lipschitz, and each smooth side has bounded curvature. The coordinate Hessian estimate applies unchanged. In the trace-reversed identity the only additional term is the jump of the normal flux at x_3=0. Since the tangential gradient vanishes there,

The constant boundary value gives |\operatorname{tr}_{T}\nabla^2z|\le C|\mathrm{II}|\,|dz|. Thus the jump of the density-weighted flux is a bounded plane density J(x'), of size C|\mathrm{II}|\,|dz|^2. It can be cancelled exactly by adding the coordinate vector -J(x')\mathbf1_{\{x_3>0\}}\partial_3 to the flux; its divergence is -J\delta_{\{x_3=0\}}, and no tangential derivative of J is used. Under a spatial scale R_j and gradient normalization \Lambda_j, the normal flux jump becomes R_jJ/\Lambda_j^2. Since the current gradient bound is |dz|\le\Lambda_j, its size is at most CR_j|\mathrm{II}|, also after any number of gradient drops. The smooth-side Ricci error is at most CR_j^2|\operatorname{Ric}|. Both therefore tend to zero independently of the shrinking gradient normalization. Hence Equation 7.6, the compactness alternative, and the affine improvement all remain valid across the face. This proves the boundary assertion.□

Proof. For the bounded flux part, scale to the unit ball. Testing by (v-k)_+ gives \int_{\{v>k\}}|Dv|^2\le C\|B\|_\infty^2|\{v>k\}|. Sobolev’s inequality and the level-set iteration give \|v\|_\infty\le C\|B\|_\infty on the unit ball; the same argument for -v gives the other sign. Rescaling gives CrC_0. For the measure part the scalar Dirichlet Green function has the bound 0\le G(x,y)\le C|x-y|^{-1} for bounded measurable symmetric uniformly elliptic coefficients. This follows from the whole-space bound in [14, Theorem 7.1 and Equation (7.9)] by domain comparison. The coefficients may be extended elliptically outside the ball; symmetry is preserved. Integrating on dyadic balls centered at x gives

For completeness the measure belongs to H^{-1}: its Newton energy is bounded by the displayed potential bound times \mu(B_r), and the Newton energy identity bounds its action on H^1_0. Mollifications preserve the mass-growth bound uniformly. Indeed, at scales s<\varepsilon their density is bounded by C\varepsilon^{\gamma-2}, so their mass is at most Cs^3\varepsilon^{\gamma-2}\le Cs^{1+\gamma}; at larger scales use the original bound on a ball enlarged by \varepsilon. The bounded divergence-data estimate can first be applied to smooth matrices and then passed to the weak solution by its energy bound. The measure approximations have uniformly bounded H^{-1} norm and uniformly bounded potentials, so weak H^1_0 compactness and weak-star L^\infty compactness give the same estimate for their limit. This proves the lemma.□

Proof. The bounds in Proposition 6.6 give uniform ellipticity of A_\chi and boundedness of every smooth function of x,Z,df that occurs in the equations. The trace equation, divided by lD^{-1/2}, is

Its right side is bounded. Lemma 7.1 therefore bounds df in C^\alpha and gives \int_{B_r}|\nabla^2f|^2\le Cr^{1+2\alpha}. The exact differential identity in Equation 5.18 gives

Write the second equation as

Here M is symmetric uniformly elliptic and J is bounded. The quadratic formula for \mathcal T and Equation 7.14 imply

All volume densities and connection terms can be incorporated into M,J,F_1 without changing these assertions. Set d\mu=(1+|\nabla^2f|^2)\,dx. For a fixed 0<\gamma<\min(1,2\alpha) this measure satisfies \mu(B_r)\le Cr^{1+\gamma}.

Choose a sufficiently large fixed L and put Q_\pm=e^{\pm LZ}. For either sign the chain rule gives

The range of Z is bounded. Ellipticity and Young’s inequality show that the last term absorbs the quadratic term in Equation 7.15 if L is large. Thus

By Lemma 7.2, subtracting the zero-Dirichlet potential of the right side on B_r changes Q_\pm by at most C(r+r^\gamma) and makes it a subsolution. The corrected supremum deficit is nonnegative and is a supersolution. Apply weak Harnack to that deficit. On at least half of B_{r/2}, either Z is below the midpoint of its range on B_r, or it is above that midpoint. In the first case the Q_+ deficit, and in the second the Q_- deficit, is at least a fixed multiple of \operatorname{osc}_{B_r}Z, up to the potential error. Weak Harnack then decreases one end of the range on B_{r/4} by a fixed fraction. More precisely,

Iteration proves a uniform Hölder modulus for Z. At an outer face both unknowns have zero trace. In flattened coordinates let P=\operatorname{diag}(1,1,-1). After incorporating the volume density, reflect by Z^-=-Z^+, M^-=PM^+P, J^-=-PJ^+, and F_1^-=-F_1^+, where opposite-side values are evaluated at reflected points. The reflected total flux is -P(M^+dZ^++J^+), so its normal component has matching traces and creates no plane source. For the principal part this also follows from the vanishing tangential gradient on the constant-data face. In the actual application K_a=0 near the outer face, hence J=0 there. The equation and its growth estimate hold across the face, and the same oscillation argument applies there.

The coefficients and right side of Equation 7.13 are now Hölder. Interior and Dirichlet Schauder estimates give C^{2,\alpha'} bounds for f, with some \alpha'>0. Expanding the second equation, all derivatives of its coefficients are derivatives of smooth functions of x,Z,df. Consequently it has the form

with a uniformly Hölder principal matrix. No bound for dZ has been used to reach this point.

Here is the absorption which supplies that bound. For s>3 the local W^{2,s} estimate for a Hölder principal matrix has a uniformly bounded leading constant on sufficiently small patches. On balls or constant-Dirichlet half-balls the scaled derivative interpolation inequality is

where Q\Subset Q' are concentric patches of comparable radius r. It follows by applying the L^\infty–W^{2,s} interpolation inequality to a cutoff times Z-c; take c between its maximum and minimum. For half-balls subtract the zero boundary value and reflect, or use a boundary cutoff in flattened coordinates. The Hölder modulus makes the oscillation coefficient as small as needed by fixing a sufficiently small patch radius. To justify absorption locally without any bound depending on the number of patches, let X be the supremum of their L^s Hessian norms. Each enlarged patch is covered by a fixed number of the smaller patches. The local estimate and Equation 7.18 give X\le\frac12X+C. This supremum is individually finite on every smooth compact truncation. Thus X\le2C, independently of its radius. Sobolev embedding gives a Hölder gradient for Z. Schauder estimates applied to both equations now give second derivative bounds; differentiating gives bounds of every fixed order. The background geometry, end symbols, and outer sphere charts have uniform bounds at the chosen patch scale. The constants are therefore independent of the sufficiently large truncation radius.□

Proof. All constants here may depend on a bound for the trial’s C^{1,\alpha} norm and on this fixed domain. The global implicit relation in Equation 5.4 makes t, l, D, and A_\chi smooth functions of (x,Z,df), even at df=0. The normalized equation is

Continue it from \Delta_gf=f by using, for \sigma\in[0,1],

The coefficient of f is strictly negative in this convention. At a positive maximum df=0, so t=Z, D=1, and A_\chi=I. The maximum principle gives

for the entire continuation. If K=0 it gives f=0.

We prove a gradient bound before using uniform ellipticity. Write s=|df|. When s\to\infty with Z in a bounded interval, the implicit equation forces t\to-\infty. There p=N and l=e^{Nt}, and the defining equation gives

These estimates are uniform for bounded trial values; on a bounded s interval they follow from smoothness and positivity. The lower bound for \chi uses N>2. Notice that it does not use the floor. The axial eigenvalue \chi_\sigma=1-\sigma+\sigma\chi obeys the same lower bound, while both transverse eigenvalues equal one. Since |f|\le H, Equation 7.19 has normalized source of absolute value at most C(1+s).

Take a short fixed collar of the boundary and a smooth extension of the background metric across it. Choose a concave increasing modulus

First choose K_0 sufficiently large depending on the source and geometry bounds. Then choose a sufficiently short d_0 and a sufficiently large M so that \psi'\ge1 on [0,d_0] and \psi(d_0)>2H. Such choices are compatible: one can first make K_0d_0<1/2 and then let M\to\infty. For the distance d to the boundary, the Hessian in its gradient axis is \psi''; the transverse trace is bounded by C\psi'. Equation 7.20 gives

For large K_0 this is strictly more negative than the possible source. The negative barrier has the opposite strict inequality. At a first contact the gradients agree, and hence all the coefficients depending on the gradient agree. The strict sign and positivity of q_\sigma therefore exclude contact. The boundary values and the choice \psi(d_0)>H give

For the interior consider f(x)-f(y)-\psi(\operatorname{dist}(x,y)) for distances at most d_0, using the short geodesic distance in the smooth extended metric. The value is nonpositive on the diagonal, at distance d_0, and whenever one endpoint lies on the boundary, the last assertion by Equation 7.21 and monotonicity of \psi. At a hypothetical positive interior maximum the endpoint gradients are parallel along the short geodesic and have the same magnitude \psi'. Write e_x,e_y for that common transported direction. Varying just the first endpoint in its axial direction and then just the second endpoint gives, respectively,

Paired parallel transverse endpoint variations, summed over the two transverse directions, give

This last bound is the second variation of the short geodesic length; the endpoint Jacobi fields have equal parallel initial and final values, and their index forms are bounded by curvature times its length. Thus the difference of the principal operators is at most

The right sides of the two equations have difference bounded below by -2C(1+\psi'). Increasing K_0 gives a contradiction. Crucially, Equation 7.22 uses the two axial coefficients separately; no modulus of \chi(x)-\chi(y) is assumed. We conclude |f(x)-f(y)|\le\psi(\operatorname{dist}(x,y)) for short distances, and hence |df|\le M.

Uniform ellipticity now follows from this gradient bound. The matrix A_\sigma still has precisely the structure of Lemma 7.1, and its right side is bounded. That lemma gives a Hölder gradient. Schauder estimates then give C^{2,\beta} bounds for some \beta>0. At this stage df is Lipschitz, and the prescribed Z is C^{1,\alpha}, so the coefficients and source of Equation 7.19 are C^\alpha. Schauder estimates give C^{2,\alpha} bounds. Differentiating the equation and applying the same estimates, with the boundary estimates for tangential derivatives followed by the equation for the normal second derivative, gives C^{3,\alpha} bounds.

The linearization in f is uniformly elliptic with Hölder coefficients, bounded first-order terms, and zero-order term -q_\sigma<0. Derivatives of the gradient-dependent coefficients produce only first-order terms, because those coefficients do not depend on the value of f. The Dirichlet maximum principle and linear Schauder solvability make this linearization an isomorphism C^{2,\alpha}_0\to C^{0,\alpha}. The implicit function theorem gives openness of the set of continuation parameters. The uniform C^{3,\alpha} bounds give compactness in C^{2,\alpha} and hence closedness. At \sigma=0 the unique solution is zero, so the set is all of [0,1]. If two solutions of the final equation differed, a positive maximum of their difference would have matching gradients and an ordered Hessian, while their common q is positive. This contradicts the equations. Thus the solution is unique. Finally the same implicit function theorem, now with parameters (a,Z), gives the asserted continuous dependence. Its local branches agree globally by uniqueness.□

Proof. Fix R\ge R_{\min}(N) as in Lemma 6.1. Parametrize the homotopy by \lambda\in[0,2]:

All quantities in the trace and scalar identities use K_a=aK; the divergence equation is \operatorname{div}V=b\,u\Xi. Let X=C^{1,\alpha}_0(\overline{\Omega_{N,R}}) for a fixed \alpha\in(0,1). For a trial Z\in X solve the trace equation by Proposition 7.4, and evaluate t,w,u,A_\chi,\mathcal T and \Xi at this pair. Define T_\lambda Z=Y by the linear zero-Dirichlet equation

Only the gradient of the new unknown Y is left unfrozen. On a bounded set of trials, Proposition 7.4 bounds f in C^{2,\alpha}, and all implicit coefficients have bounded C^{1,\alpha} norms. The scalar right side has bounded C^{0,\alpha} norm, since dt is computed from Equation 5.18. Ellipticity is uniform on that bounded set by the trial gradient estimate, even when t<-\epsilon. Linear Dirichlet Schauder theory gives a uniform C^{2,\alpha} bound for Y. Thus (\lambda,Z)\mapsto T_\lambda Z is continuous and compact into X. At a fixed point the linear solve first gives Z\in C^{2,\alpha}, whereas the trace solution is already f\in C^{3,\alpha}. Thus \Xi\in C^{1,\alpha} and the principal coefficient and drift flux are C^{2,\alpha}. Schauder estimates give Z\in C^{3,\alpha}, and the trace equation gives f\in C^{4,\alpha}. Repeating gives smoothness. In particular the coupled a priori estimates apply to individually smooth fixed points; their eventual bounds for all fixed derivative orders control the chosen X norm, regardless of its initially prescribed exponent \alpha.

Choose a radius M_X larger than the X norm of every above-floor fixed point, which is possible by Proposition 7.3 and the fixed truncation. Consider the relatively open bounded set

Its openness follows from the continuous dependence in Proposition 7.4. A fixed point cannot occur at its norm boundary by the choice of M_X. At its other boundary it would have minimum t=-\epsilon and would be above the closed floor. Lemma 6.1 excludes such a minimum on the outer boundary; Lemma 6.2 excludes it in the interior. There is no inner boundary in the filled manifold.

For completeness the variation of the domain in Equation 7.25 presents no degree obstruction. The images of the bounded parameter-trial cylinder under T have compact closure. Therefore any convergent sequence of parameters and associated boundary fixed points would converge to a boundary fixed point, already excluded. Around any fixed parameter the compact set of its fixed points has an open neighborhood whose closure lies inside its slice of \mathcal U. By compactness the same neighborhood contains every fixed point in the relevant slices for all sufficiently nearby parameters; otherwise a contrary sequence converges to an excluded fixed point. Excision and the usual fixed-domain homotopy invariance of Leray–Schauder degree show that the degrees on the slices are locally constant. A finite interval covering makes them constant along the full path. At \lambda=2 we have K_a=0, f=0, and b=0. The linear solve in Equation 7.24 is therefore identically zero for every trial. Its sole fixed point is Z=0, with t=0>-\epsilon, and the degree of I-T_2 on the terminal slice is one. Hence the initial slice has degree one and contains a fixed point. This proves existence on every sufficiently large truncation, with the estimates already established.

We next verify the end before taking the exhaustion limit. All bounds in the remainder of the proof are uniform in the truncation radius at this fixed N. Outside a fixed sphere K=0, so Equation 7.13 is the homogeneous proper equation

The eigenvalues of A_\chi stay in a fixed positive interval. For a sufficiently small fixed c>0, the functions C e^{-c(r-r_0)} have (A_\chi:\nabla^2-q)C e^{-c(r-r_0)}<0 for r_0 sufficiently large. Indeed their Hessian is bounded by C(c^2+c/r)e^{-c(r-r_0)}, whereas the zero-order term is at least (\eta/e)C e^{-c(r-r_0)}. Choose c first, then r_0. The negatives give the opposite inequality. Choose C to dominate the uniformly bounded values on r=r_0; the outer Dirichlet data are zero. Comparison gives |f|\le Ce^{-cr} on every truncated end. The local estimates in Proposition 7.3 give uniform smooth coefficients for Equation 7.26; its interior and boundary estimates then give exponential decay of every fixed derivative of f. Alternatively, interpolation between this exponential height bound and the uniform higher derivative bounds gives the same conclusion, with a possibly smaller positive exponent for each order. The implicit relation implies that t-Z and its fixed derivatives are exponentially small.

On the bounded fixed-N range of Z, define the smooth increasing function \Phi by

Its derivative is bounded above and away from zero on that range. Putting f=K=0 in Lemma 5.5 gives exactly

The exponential bounds just proved and the uniform local derivative bounds show that the actual end equation differs from this one by O_j(e^{-c_jr}) for every fixed order. Thus

This scalar Poisson bound is independent of the radius R.

Here are explicit barriers fixing the end constant. Put \gamma=\beta/2 and, after increasing r_0, let W(r)=r^{-1}-r^{-1-\gamma}>0. The prepared metric has g-\delta=O_j(r^{-1}). Since \Delta_\delta r^{-1}=0 and \Delta_\delta r^{-1-\gamma} =\gamma(1+\gamma)r^{-3-\gamma},

for large r_0. Choose a constant multiple of W to dominate the inner boundary values and the right side of Equation 7.28; this is possible because \gamma<\beta. The zero outer boundary values lie between the positive and negative barriers. The maximum principle therefore gives |\Phi(Z)|\le Cr^{-1}, and the bounds for \Phi' give

on every truncated end. In particular no undetermined additive constant is introduced in the exhaustion.

We record the derivative decay rather than infer it from an unscaled estimate. Rescale an annulus of radius r to unit size. The bounded right side of Equation 7.28 and Equation 7.29 give, by local Poisson W^{2,s} estimates for s>3, |d\Phi(Z)|\le Cr^{-2} and a scaled Hölder bound for its gradient. The exponentially small error has all unscaled derivatives exponentially small, so it remains negligible after this rescaling. The other right-side term is a smooth function of Z times the prescribed weight \rho_0; its scaled Hölder norms are bounded by Cr^{-3-\beta}. Schauder estimates give D^2Z=O(r^{-3}). Differentiating the transformed equation and repeating the scaled estimates inductively gives Z=O_j(r^{-1}) for every fixed j. Since t-Z is exponential, the same holds for t. Finally the untransformed equation gives

because \beta<1 and |dZ|^2=O(r^{-4}).

Take any sequence R\to\infty. The compact-patch estimates and a diagonal subsequence give a smooth solution on \Omega_N, with all bounds above and the stated end estimates. Initially the limit has t\ge-\epsilon. If equality held at a point, it would be an interior minimum of a smooth solution and the same strict contradiction in Lemma 6.2 would apply. Hence t>-\epsilon. No limit in N has been taken. Since Z\to0 and N\epsilon>2, the cutoff \vartheta(N(t+\epsilon)) vanishes sufficiently far out. The exact zero-trace expression for \mathcal T and the exponential trace error give \mathcal T=O(r^{-4})+O(e^{-cr}) there. Also \eta a_w|df| is exponential, \rho=O(r^{-3-\beta}), and u=e^Zl(Z)+O(e^{-cr})\to1. These functions are integrable over the three-dimensional end, and smoothness handles the compact part. Thus u\Xi\in L^1(\Omega_N) and the final assertions follow.□

Proof. The end conclusion of Theorem 7.5 gives, at fixed N, exponential decay of f and each of its fixed-order derivatives, and

Here the constants and the positive c_j may depend on N. Since K=0 sufficiently far out, \mathcal T=O(r^{-4}) up to exponentially decaying terms, u=1+O(r^{-1}), and \eta a_w|df| decays exponentially. For N\epsilon>2, the penalty cutoff m_0=\vartheta(N(t+\epsilon)) vanishes sufficiently far out. Consequently u\Xi is absolutely integrable. The divergence equation and the absence of any boundary in \Omega_N prove the flux identity in Equation (8.4).

The definition of V, A_\chi=\Id+O_j(e^{-c_jr}), and the same end estimates show that

Thus its limiting flux is twice the limiting flux of dt. The graph term l^2df^2 makes no contribution to ADM energy. Direct substitution of e^{2t}g into the ADM integral gives

The terms containing (g-\delta)dt, t\,\partial g, or t\,dt have integrals O(R^{-1}); changing between Euclidean and g normals and measures has the same vanishing cost. This proves the coefficient and sign in Equation (8.4) without requiring a separately chosen monopole coefficient for t.

It remains to estimate the possibly negative part of the integral. Write \sigma=|df|_{g_N}. On the support of m_0 the floor gives -\epsilon<t<-\epsilon+N^{-1}, and therefore \ell\le l\le e\ell for large N. Since u=e^t l\sqrt D and D=1+l^2\sigma^2, direct calculation gives

The negative term in u\Xi is consequently bounded in absolute value by

Young’s inequality, using a fixed fraction of the positive \delta_0\eta u a_w\sigma, bounds the second summand by

Here \rho_0=r^{-3-\beta} and \rho_1=r^{-2b_1}. Their required integrals have polynomial bounds: the compact contribution is O(1+N), while on the end \rho_0, \rho_0^2, and \rho_1^2 are integrable, the last because 4b_1>3. In particular, no integral of \rho_1 is being assumed finite. Positivity of \mathcal T, proved in Proposition 5.3, now gives

Equation (8.5) follows.□

Proof. We give the construction when \sigma_i=1. Use the signed collar coordinate s increasing toward the original end. Choose a small fixed number d_0>0. On the cap put the comparison height equal to -d_0. There K=-L_0g_N, so the trace of K in A_\chi is -L_0(2+\chi) and this constant height is a supersolution once L_0 is large.

On the adjoining product cylinder choose a nonnegative slope a(s) which starts at zero, becomes a truncated increasing exponential, and ends at a small fixed positive value a_0. It can be arranged that

Indeed, grow exponentially from a value of order \eta to a_0, smoothly cutting off near the first value and making the last value constant. The required length is O(1+|\log\eta|)=O(1+N), and the integral is bounded by a fixed multiple of a_0. Let H_0(s) be the resulting height, starting at -d_0.

For a height test H_0, the Hessian tensor in the trace equation is

On a product cylinder only its axial entry can be nonzero. The bound on a' gives

Multiplication by \chi\le1 can only improve this estimate. A fixed choice of L_0 therefore makes H_0 a supersolution on the entire product, including its small-slope starting portion.

Continue with a small fixed positive slope through the fixed transition and up to S, keeping the total increase below d_0/4. On these leaves the prepared extension satisfies H_s+\operatorname{tr}_{S_s}K<0. Increase the slope within the fixed strict collar on the exterior side until the test height exceeds the uniform bound C in Equation (8.3) at its outer face. All slopes on these fixed transitions are bounded below by a fixed positive number, and all their derivatives are fixed. At a putative comparison contact the gradients of the solution and the test agree, so the solution’s floor gives \eta/l\le\eta/\ell=\ell^{1/2}. At such contacts the transverse Hessian trace converges uniformly to H_s, while \chi\to0 uniformly and the axial Hessian contribution vanishes. Thus the full test trace converges uniformly to H_s+\operatorname{tr}_{S_s}K. First choose d_0 smaller than the absolute strict expansion bound on these fixed collars, and then choose N large. The trace is then below -d_0, hence below the test height, at every possible contact in the remaining ramp.

For completeness, the comparison uses the actual solution Z as a fixed function. At a positive maximum of h-H_0, their gradients agree, so the defining equation for t gives the same t and the same positive matrix A_\chi for both functions. Their Hessians are ordered. Consequently F[h]\le F[H_0]\le H_0<h, contradicting F[h]=h. There is no inner boundary, and the test dominates on the outer collar face. It follows that h\le H_0, which is less than -d_0/2 on the cap, neck and original boundary. A slightly smaller bound holds on a neighborhood by continuity. For \sigma_i=-1 the same construction applies to -h and -K. There are finitely many components, so one may take the minimum of their positive height gaps as b_2.□

Proof. Since F=h=\eta f, differentiation gives w(F)=\eta a_w|df|. Substitute the equations into the scalar identity of Proposition 5.2. On |h|\le b_3, using Equation (8.8) and dropping the nonnegative penalty contribution, one obtains

We estimate the cost of \psi directly through the trace equation. The graph inverse is A_d, and its volume density is \sqrt D. Differentiation gives the identities

For example, \sqrt D\,A_d\nabla h=(\eta/l)w; the divergence formula for \bar g, followed by the three-dimensional conformal Laplacian formula, proves Equation (8.15). Differentiating w=lD^{-1/2}\nabla f proves Equation (8.16).

In the axial decomposition of Section 5,

Since |d-\chi|\le d and 0<d\le1, the coercivity estimate controls (d-\chi)j, d\,w(t), and |dt|_{A_d} by \mathcal P_N\sqrt{\mathcal T}. On the compact original region where \zeta' or \zeta'' can be nonzero, Equations (8.15)–(8.16) therefore give

This estimate uses no quantitative bound on second derivatives of the solution beyond the structural coercivity inequality.

In dimension three the conformal curvature identity reads

The absolute value of its two error terms is at most

Here

Young’s inequality absorbs the \sqrt{\mathcal T} term into half of (1-\delta_0)\mathcal T, with remainder tending to zero even after the polynomial factors. The remaining errors are absorbed by the positive minimum of \rho on this fixed transition region. Off that region \psi is constant, so there is no error. This proves Equation (8.11).

Compact support of \psi follows from the height bound. It leaves the end energy unchanged. The estimates in Theorem 7.5, the strong prepared end, and the exponential graph error imply e^{2t}\bar g-\delta=O_2(r^{-1}). Its scalar curvature is integrable: on the end the conformal formula involves the integrable prepared R_g, \Delta_g t=O(r^{-3-\beta}), |dt|^2=O(r^{-4}), and exponentially decaying graph errors. The compact part is smooth. Finally, constant scaling by e^{2\epsilon} followed by y'=e^\epsilon y multiplies ADM mass by e^\epsilon. Lemma 8.1 proves Equation (8.12).□

Proof. Choose a constant matrix \mathsf R\in SO(2) which sends the charge vector (Q_E,Q_B) to (Q,0). For Q>0 one can take

for Q=0 take the identity. Put (X_e,X_o)^T=\mathsf R(\EE,\BB)^T and \alpha=\iota_{X_e}dV_g. A determinant-one rotation preserves both the sum of squared norms and the cross product, so

To identify the first two terms, choose an oriented g-orthonormal frame in which w=a_we_1. Then

and the components of \alpha=\iota_{X_e}dV_g give

The identity remains valid at w=0 by continuity.

Combine Equations (8.11), (8.20), and (8.21), retaining the different scaling powers of scalar curvature and two-form norm:

The third inequality uses \lambda\ge1 from Equation (8.10).

Define

Both forms are closed. Orthogonality of \mathsf R and the isometry between vector and contracted volume-form norms in dimension three identify the sum of squared vector norms with |\alpha|_{\widetilde g}^2. This proves Equation (8.19). Closedness is precisely divergence freedom in \widetilde g. The total flux vector is \mathsf R^{-1}(4\pi Q,0)=(4\pi Q_E,4\pi Q_B).

The field associated with \alpha itself is explicitly

At fixed N, D-1 and its derivatives decay exponentially, t=O_j(r^{-1}), and \psi=0 near infinity. Hence the original O_1(r^{-2}) field decay gives the asserted decay of both new fields; the constant coordinate dilation has no effect on its order. Flux integrals of two-forms are invariant under this coordinate change.

This construction preserves the two total charges. It imposes no claim about the individual charges of the boundary components. All forms have been used only on the original exterior; their extension through a cap is unnecessary.□

Proof. Truncate beyond the confining radius and equip the compact region with the auxiliary second fundamental form zero. The inner boundary has expansion zero with respect to the normal pointing into the region; the outer sphere has positive expansion with respect to the normal pointing out. These are the weak inner and strict outer barriers in [2, Theorem 5.1]. In particular neither strict stability of the inner boundary nor an energy condition is required. Definitions 7.1–7.2 of that paper prescribe the entire inner boundary in the bounding class of trapped sets. Theorem 7.3 and Remark 7.4 give the smooth outer boundary of their union and its outermostness. The weak-barrier version permits coincidence with inner components. Since the auxiliary second form vanishes, the resulting marginal surface is minimal. Keeping the full frontier of the component incident on the end, or equivalently filling bounded complementary pockets, gives the asserted full cut. The outer mean-convex spheres exclude a further compact minimal enclosure outside the truncation. This proves outermostness in the complete exterior as well as in the compact barrier problem.□

Proof. Choose a fixed original coordinate sphere beyond the radius containing \{|h|\ge b_3/4\} and beyond a fixed further neighborhood. It encloses O_N for all large N and lies where \psi=0. Its area in e^{2(t+\epsilon)}(g+l^2df^2) is at most e^{\mathcal P_N} by Equation (8.3). This proves the proposed upper bound for the infimum with a radius independent of N.

We justify compactness separately at fixed N. Strong asymptotic flatness gives, sufficiently far out, a foliation by spheres with positive definite outward second fundamental form. Follow their unit-normal trajectories, parametrized by the radius. Tangential metrics increase along these trajectories. Projection from an outer leaf to an inner leaf consequently contracts two-dimensional area. For almost every sufficiently large radius R, clipping a bounded competitor at that leaf does not increase its perimeter: along each normal trajectory where clipping creates boundary, the bounded competitor must exit farther out; projection of that portion of its old boundary covers the new trace, with area counted with multiplicity. BV slicing and the area formula give the same inequality for finite-perimeter competitors. Thus the infimum over all bounded competitors equals the infimum in one fixed sufficiently large compact truncation. BV compactness and lower semicontinuity [1, Theorem 3.23 and Proposition 3.6], and the closed containment condition B\supset O_N almost everywhere give a minimizer. Since it minimizes among all bounded competitors, the outer truncation imposes no local constraint. Large mean-convex spheres also exclude a free minimal contact with that outer truncation.

We prove that the perimeter support has no contact with O_N. Write

On the 2s_N-neighborhood of O_N, the plateau is constant, and

Only pointwise metric comparison is asserted here.

Suppose q\in O_N\cap\operatorname{supp}|D\mathbf1_{B_N}|. Because B_N contains O_N almost everywhere, such a point is in \partial O_N. Take a shortest background geodesic of length s_N from q to H_N. The ball of radius s_N/4 about its midpoint lies in O_N\cap B_{s_N}(q). Uniform local background geometry gives

We obtain a lower bound for the complementary volume without using regularity of either frontier. Put v(r)=|B_r(q)\setminus B_N|_{g_N} for 0<r<s_N. Full minimality against B_N\cup B_r(q), cancellation outside the ball, and removal of the constant factor e^{2A_N} imply, for almost every r,

The last derivative is the background area of the complementary trace on the sphere, by coarea. In the uniform normal charts, background metric balls correspond to Euclidean balls and volume and perimeter are uniformly comparable. Thus the Euclidean absolute and relative isoperimetric inequalities [1, Theorem 3.46 and Remark 3.50] hold here with uniform constants. Local absolute isoperimetry for B_r(q)\setminus B_N, including its boundary on the sphere, gives

Every ball at a point of perimeter support has positive complementary volume. Integrating the resulting inequality for v^{1/3} from \delta to r and then letting \delta\downarrow0 yields

Relative isoperimetry in B_{s_N}(q), together with Equations (8.26) and (8.28), therefore forces

Restoring the constant area multiplier in Equation (8.25) gives

For fixed \epsilon>0, this contradicts Equation (8.24) when N is large.

The minimizing frontier is now disjoint from the obstacle. It is locally perimeter minimizing without a constraint and is therefore smooth and embedded by three-dimensional perimeter regularity [12, Regularity Theorem 1.3(iii)]. Filling every bounded complementary pocket cannot increase perimeter and preserves the obstacle condition. The frontier of the remaining exterior is the whole smooth minimal boundary incident on the end. It is nonempty because a bounded set containing the nonempty open obstacle cannot have zero perimeter in the connected filled manifold. This proves all assertions.□

Proof. Apply Lemma 8.5 to the exterior of \Gamma_N^0 from Lemma 8.6. The comparison end is strongly asymptotically flat, so its large coordinate spheres have positive outward mean curvature and confine compact minimal surfaces by the maximum principle at a largest-radius point. We obtain an outermost full minimal enclosure \Gamma_N and take its exterior \widetilde\Omega_N.

The obstacle contains every cap and a neighborhood of the entire original boundary. Both \Gamma_N^0 and its outer enclosure are therefore full cuts in the original exterior, with every component counted. Their surviving exterior is disjoint from H_N=\{|h|\ge b_3\}. Lemmas 8.3 and 8.4 consequently apply everywhere on \widetilde\Omega_N, up to its boundary. No part of a filling is retained. The construction supplies smoothness, completeness with boundary, one end and the stated asymptotics. Finally, metric dominance on each tangent two-plane and the full-cut definition give

and Equation (8.12) gives the mass bound.□

Proof. Put r_*=(a_g(S)/(4\pi))^{1/2}. We prove the family of bounds

The numerical input is the neutral enclosing-area theorem [17, Definitions 1.1–1.2 and Theorem 1.3]. In dimension three it states that a smooth connected oriented complete one-ended exterior with nonempty compact boundary, integrable neutral constraint densities satisfying \mu\ge|J|, finite future-timelike ADM vector, decay g-\delta=O_6(r^{-q_0}), K=O_5(r^{-1-q_0}) for some 1/2<q_0<1, and weakly future-trapped boundary on every component obeys m\ge\tfrac12(a/(4\pi))^{1/2} when its full-cut infimum a is positive. We verify this contract for the auxiliary data constructed below.

Retained curvature and a flux form.. Work on the same filled manifold at fixed \epsilon,N, and put

Then z\ge0, g_b\ge g_N, and |dh|_{\bar g}\le\eta/l\le\ell^{1/2}. Rotate the two original fields by the constant matrix \mathsf R in the proof of Lemma 8.4, and retain \alpha=\iota_{X_e}dV_g. It is closed on the original exterior and has flux 4\pi Q. For Q=0 the identity rotation suffices. Extend \alpha smoothly into the filling, cutting it off on fixed collars inside the original boundary; denote the extension by \alpha_{\rm ext}. It need not be closed there. Define

The second field is uniformly bounded on the fixed filling collars, is O_1(r^{-2}) on the end, and has L^2 and L^6 norms bounded polynomially in 1+N. On the original exterior E_b is divergence free. The square completion in Equations (8.20)–(8.21) gives

We need a sharper consequence of the existing quadratic identity:

Here is a direct check. In Equation (5.22), minimize over M,F,j, keeping the transverse and axial components y,x of dt fixed. The transverse coefficient is 2s_p-v/4\ge3/4. The remaining axial coefficient, divided by d, is

This is increasing in p\ge0, and C(0,v)=(8+v)/(8+4v)\ge3/4 for 0\le v<1. This proves Equation (8.34), including zero slope by the invariant definition of \mathcal T.

Let \mathcal U_N be the part of the original exterior where |h|\le b_3. Equation (8.13) and \delta_0<1/24 show that throughout \mathcal U_N, R_{g_b}/2\ge|E_b|_{g_b}^2. On the fixed original band b_3/4\le|h|\le b_3, the positive function \rho has a fixed positive lower bound c_*; hence, since (1-\delta_0)3/4>2/3, we have the stronger bound

The end estimates already proved give all-order decay g_b-\delta=O_j(r^{-1}) after the constant dilation, integrable scalar curvature, and

A smooth bounded flux profile.. Fix s\in(0,1) and write y_0=\log(1-s^2)<0. For every t_0>0 choose smooth functions L>0 and H_0 on the real line with

To construct them, set c=1-s^2 and first use L_{\rm opt}(y)^2=(1-ce^{-y})/s^2 for y>y_0. The right side in the bound for H_0' is then 2e^y/s, whose integral from y_0 to zero is 2s. For a small \sigma>0, replace L_{\rm opt}' below y_0+\sigma by \chi_\sigma L_{\rm opt}', where \chi_\sigma is nondecreasing, zero below y_0+\sigma/3, and one above y_0+2\sigma/3; integrate backwards from the unchanged value at y_0+\sigma. This gives a positive smooth nondecreasing L on the whole line, constant at its lower end. Take H_0'=\zeta_\sigma\,2e^y/s, where 0\le\zeta_\sigma\le1 is smooth and supported in (y_0+\sigma,-\sigma) and tends to one on (y_0,0). Put H_0(y)=\int_{-\infty}^yH_0'(v)\,dv and decrease \sigma until the asserted value at zero holds.

Define B(0)=0, B'(y)=e^yL(y). This is strictly increasing, its range on [y_0,\infty) is [B(y_0),\infty), and B(y)\ge e^y-1 for y\ge0. Set Y_0=B(y_0)<0 and extend

This is smooth and bounded, vanishes near and below Y_0, and is constant for all sufficiently large Y.

The second scalar solve.. There is a smooth solution on the entire filled manifold of

We supply the estimates needed for this assertion; no estimate uniform in s or in the profile tolerance is required. Set Y=B(y). In the background measure its equation is exactly

Indeed the volume ratio is e^{3z}\sqrt D, and its product with E_b is E_{\rm ext}. The axial and transverse eigenvalues of A are e^z/\sqrt D and e^z\sqrt D; thus their upper bounds and positive inverse lower bounds are at most \exp(\mathcal P_N).

The filled background has the Sobolev inequality

for compactly supported v, with the same bound on every outer truncation with zero trace. To verify the dependence on N, cover the compact core and its product necks by O(1+N) uniformly controlled patches with bounded overlap. Chains of at most O(1+N) overlapping patches join them to a fixed end annulus. Radial integration from infinity controls the annulus’s L^2 norm by the end gradient energy, because \int_R^\infty r^{-2}\,dr<\infty. Local Poincaré inequalities control the successive differences of patch averages. Summing along the chains bounds the core L^2 norm by a polynomial multiple of the global gradient energy. The local Sobolev inequalities and the Euclidean end inequality then give the stated bound.

Solve Equation (8.37) on expanding smooth outer truncations with zero Dirichlet data. Multiplying its right side by any homotopy parameter in [0,1] leaves the following estimates unchanged. The vector F_Y=\mathfrak h(Y)E_{\rm ext} has bounded L^2 and L^6 norms, independent of Y and of the outer radius. Testing by Y gives \|dY\|_2+\|Y\|_6\le\exp(\mathcal P_N). For a>0, test by (Y-a)_+. Hölder and Sobolev give

For b>a this implies

Starting at level one, whose superlevel measure is bounded by the energy estimate, and taking levels 1+d_*(1-2^{-j}) proves \sup Y\le\exp(\mathcal P_N) for a sufficiently large d_*\le\exp(\mathcal P_N). The identical argument for negative levels bounds \inf Y.

On a fixed truncation, freezing F_Y and solving the linear uniformly elliptic Dirichlet equation defines a continuous compact map on the continuous functions with zero boundary values: linear W^{1,p} estimates with p>3 give compactness. The preceding bounds hold for every fixed point of its homotopy with zero. Leray–Schauder degree therefore gives a fixed point. Divergence-form local Hölder estimates, then divergence Schauder estimates and differentiation, make it smooth. The same estimates are uniform on each fixed compact subset as the truncation radius tends to infinity, so a diagonal limit solves the equation on the filled manifold. These are the standard uniformly elliptic Dirichlet and interior estimates [11, Chapters 6–8]. Testing by (Y_0-Y)_+ gives zero gradient, since \mathfrak h=0 on its support and its outer trace vanishes. Consequently Y\ge Y_0 both on each truncation and in the limit. Inverting B proves y\ge y_0 and, from its exponential growth, y\le\mathcal P_N.

On the original end \operatorname{div}_{g_b}E_b=0, so 2\Delta_{g_b}Y=\mathfrak h'(Y)E_b(Y), with bounded drift O(r^{-2}). For fixed 0<\beta'<1, positive multiples of r^{-1}(1-r^{-\beta'}) are supersolutions of the absolute drift comparison equation outside a fixed radius. Their negative Laplacian has order r^{-3-\beta'}, which dominates the metric error and drift terms of order r^{-4}. Comparison with both signs and the zero data at each outer boundary proves Y=O(r^{-1}) uniformly during exhaustion. Scaled estimates give the first two derivative orders. Thus y\to0 and eventually H_0(y)=H_0(0); on that end Y is harmonic. The all-order metric bounds then give Y,y=O_j(r^{-1}) for every fixed j. Integration on the whole filled truncation, which has no inner boundary, gives the exact limiting flux

Indeed the flux identity first holds with Y in place of y; B'(0)=1 and the O_j(r^{-1}) decay make their difference tend to zero. This also explains why source terms in the artificial filling impose no omitted inner flux condition.

A neutral exterior.. Set \gamma=e^{2y}g_b. On \mathcal U_N, expansion of the scalar equation and the conformal curvature identity give

The last inequality is precisely the profile discriminant bound. On the band, the terms in Equation (8.35) beyond the field norm remain available. Moreover, when y\ge0 one has H_0'=0, so the full additional |dy|^2 term is retained. Therefore

Everywhere on the filled manifold \gamma\ge e^{2y_0}g_N. The conformal ADM calculation, with normalizing factor 1/(16\pi), and Equation (8.38) give

The metric has all-order O(r^{-1}) decay. Its scalar curvature is integrable: far out B(y) is harmonic, so \Delta_{g_b}y=-(1+L'/L)|dy|^2=O(r^{-4}), and R_{g_b} is integrable.

Choose a fixed smooth function b with b=0 on (-\infty,0], 0\le b'\le1, and b'=1 for all sufficiently large arguments. Then C(y)=e^{y-b(y)} is bounded above and below by positive constants on [y_0,\infty). We shall use a compactly supported pure-trace tensor

where \mathcal H vanishes on [0,b_3/4] and is a constant M on [b_3/2,b_3]. For such a tensor the neutral densities are \mu'=R_\gamma/2+3a^2 and J'=2\,da. Multiplication by e^{2(z+y)} bounds its momentum contribution by

Young’s inequality absorbs the first term into the two gradient terms of Equation (8.39) at a cost at most

Where y<0, b'=0, so no unavailable y-gradient term is used. The remaining energy coefficient is 3-5/2=1/2>0. Consequently the neutral DEC follows if

for a fixed sufficiently small c'>0, depending only on c_* and the two bounds for C. This condition can be enforced for every finite plateau M by a single sufficiently large N: choose a fixed smooth cutoff \chi_h from zero to one between b_3/4 and b_3/2, and set

Then |\mathcal H'|/(1+\mathcal H^2)\le (\pi/2)\|\chi_h'\|_\infty independently of M, whereas \eta/l\le\ell^{1/2}\to0. Its composition with |h| is smooth because it vanishes on a neighborhood of zero.

After these scalar solves, choose a regular value c_h\in(b_3/2,b_3) of |h|. Let \mathcal D_N be the closure of the component of \{|h|<c_h\} containing the end. The height gap keeps it entirely in the original exterior. Its full boundary is compact, smooth, embedded, two-sided and nonempty. Since this boundary is fixed and compact, take the plateau M sufficiently large that, with normal toward the end,

on every component. The choice of N need not be changed when M is chosen. Restrict \gamma,K' to \mathcal D_N. They are smooth, satisfy the neutral DEC, and K' is compactly supported. This is a connected oriented one-ended exterior with compact complement of its coordinate end. It is complete with boundary: its domain is closed in the original complete exterior, and its metric is bounded below by e^{2y_0}g. All-order metric decay and compact support of K' imply the neutral theorem’s O_6/O_5 assumptions for any 1/2<q_0<1. The densities are integrable and its momentum is zero.

Every full cut in \mathcal D_N is a full cut of the original exterior. Counting all components and contact portions, the metric comparison therefore gives its full-cut infimum a_N' the bound

If e_N\le0, use the energy-raising conformal family constructed in the proof of Lemma 4.4, with zero fields and with its radial function constant on a core containing K' and the boundary. The neutral DEC and trapping persist, areas do not decrease, momentum remains zero, and the energy becomes e_N+2\lambda. The all-order end decay persists for this radial construction. For arbitrarily small e>0, choose \lambda=(e-e_N)/2. The neutral theorem would imply e\ge\tfrac12e^{y_0}r_*>0, a contradiction. Thus e_N>0, and applying that theorem directly gives

At fixed prepared data, s, profile, and \epsilon, let N\to\infty. Then let \epsilon\downarrow0 and finally t_0\downarrow0. This proves Equation (8.33). No convergence of comparison horizons or uniform estimate at s=0,1 is required.

Letting s\uparrow1 first yields E_*\ge Q. If r_*>Q>0, choose s=Q/r_* to get E_*\ge(r_*+Q^2/r_*)/2; for Q=0 the same conclusion follows by s\downarrow0. Since r_*>Q, this is equivalent to r_*\le E_*+\sqrt{E_*^2-Q^2}. If r_*\le Q, that upper-radius bound follows directly from E_*\ge Q. This proves both assertions of Equation (8.32). The original componentwise signs enter only in the height-gap construction. Every component of the new auxiliary boundary has the common future sign required by the neutral theorem.□

Proof of Theorem 2.3. First suppose the original ADM vector is future timelike, and write m=\sqrt{E^2-|P|^2}. Proposition 4.7 supplies strict prepared rest exteriors indexed by j with

The area convergence is for the full original cut class. More explicitly, the lower metric comparison g_j\ge(1-\delta_j)g, with \delta_j\to0, gives a_{g_j}(S)\ge(1-\delta_j)a_g(S); local smooth convergence on any fixed full cut gives the reverse upper limit after taking its area arbitrarily close to the infimum. Thus no original minimizing cut is needed. The preparation keeps the fixed sign on each boundary component.

For each j, Proposition 8.8 has already taken the limits in the order: outer Dirichlet exhaustion at fixed N, then N\to\infty, then \epsilon\downarrow0. Apply its conclusion and only now let j\to\infty. It follows that

Since E>0, the first inequality is equivalent to E\ge\sqrt{|P|^2+Q^2}.

Finally, Proposition 3.2 gives a_g(S)>0 for every original exterior in the theorem. We recall explicitly how Lemma 4.4 eliminates the remaining ADM endpoint using the timelike conclusion just proved. Its conformal energy-raising family preserves P,Q_E,Q_B, the matter condition and the componentwise boundary signs, increases every full-cut area, and changes the energy to E_s=E+2s. If E\le|P|, take s>(|P|-E)/2. The family then has a future-timelike ADM vector, so the preceding result applies and in particular gives

Here we used x+\sqrt{x^2-Q^2}\le2x for x\ge Q. Letting s\downarrow(|P|-E)/2 makes the left side tend to zero, a contradiction. Hence E>|P| in the original data. The already proved timelike case completes the theorem, with no boost of speed one and no additional area or timelikeness hypothesis.□

Proof. This is the envelope formula of Proposition 3.8; the following two inequalities specify how it is used here. On the common compact truncation, the area density has the expansion

where the remainder is uniform in the point and the two-plane T. Every fixed M\in\mathcal M_B is a competitor at positive t, giving the upper derivative bound. Conversely choose minimizing sets M_t at positive t. Metric comparison and the same expansion show P_g(M_t)\to A. A subsequence converges in measure to a member M_0\in\mathcal M_B, and its perimeter convergence is strict. Continuity of Equation (9.3) gives

The uniform remainder is bounded by a constant times the uniformly bounded perimeter. Taking a subsequence realizing the lower limit proves Equation (9.4).□

Proof. One-point compactify the active-ray space at infinity. If it is already compact, adjoin an isolated point instead. Assign the limiting values just calculated to the direction functions. The boundary and minimizing family are compact. Finite linear combinations V of all the directions give continuous functions on their disjoint union through

We claim its linear image misses the open cone of functions that are strictly positive everywhere.

Indeed, a direction in that cone has positive V_*-coefficient, by its value at infinity. Add its compact and prototype changes linearly and apply the indicated positive conformal change. The forms stay closed and retain their end fluxes. On a compact region, strict positivity on the closed active set makes the first constraint derivative uniformly positive there. Continuity gives this conclusion in a neighborhood of that set; outside the neighborhood the unperturbed constraint has a positive minimum. A uniform Taylor remainder therefore gives the DEC for all sufficiently small positive parameters. The same argument gives the strict chosen boundary expansion.

Here the far end requires a uniform estimate, rather than compact continuity. The conformal identities retain a positive multiple of the original nonnegative ball constraint. Their additional term is a positive constant times t\rho_*. The nonconformal linear errors are tO(r^{-3-q}+r^{-5}). The quadratic metric and tensor prototype errors are O(t^2r^{-4}); the remaining conformal and mixed errors are O(t^2\rho_*), after the background nonnegative term has been retained. Since \beta<q and \beta<1, first choose the end radius large enough for the linear errors to be absorbed, and then choose t>0 small enough for the quadratic errors to be absorbed. This proves feasibility simultaneously on every end ray. The estimates also show integrability of the changed constraints and their Taylor remainders. The conformal gradient flux and the explicit prototype fluxes give finite, differentiable ADM limits. The weak decay, completeness, and cut hypotheses are preserved. Thus these paths belong to the numerical class; no preservation of equality-subclass outermostness is needed.

Equation (9.4) and positivity of the third component in Equation (9.10) imply c\,a'_+(0)>16\pi\mathcal L'. This contradicts the numerical bound and Equation (9.2). The claim follows.

Separation of a linear subspace from a disjoint open convex cone gives a nonzero continuous functional that annihilates the subspace and is nonnegative on positive functions. The representation of positive functionals on continuous functions gives positive measures on the three compact test spaces. If the measure on \mathcal M_B had zero mass, evaluation on V_*, using Equation (9.8) also at the adjoined point, would be strictly positive. This contradicts annihilation. Normalize that mass to one. On the finite active rays divide the resulting measure by \rho_*, obtaining \Lambda_B. The possible atom at infinity contributes zero to compact tests and the four prototypes. Rearranging the annihilation identity proves Equation (9.9) and the weighted finiteness assertion.□

Proof. On a neighborhood of the support prescribe a Gaussian metric \mathbf G=-dt^2+g_t with g_0=g, \partial_tg_t|_0=2K, and future normal n=\partial_t. Prescribe a two-form \mathbf F whose initial electric and magnetic fields are the given ones. The spatial pullbacks are \mathbf F|_{t=0}=\alpha_2 and (*_{\mathbf G}\mathbf F)|_{t=0}=-\alpha_1. The spatial closure equations are precisely d\alpha_1=d\alpha_2=0. The components of d\mathbf F=0 and d(*_{\mathbf G}\mathbf F)=0 with one normal slot determine all six normal derivatives of the two-form from its spatial derivatives and the already prescribed first metric jets. Thus both Maxwell equations hold at the slice. For the graph variation with velocity sn, Cartan’s formula gives explicitly

These are compact closed variations.

Let

The constraint equations give T_\delta(n,n)=\mu_m and T_\delta(n,\cdot)|_{T\Omega}=J_m. Choose the second normal metric jets so that

This is a free prescription: the dependence of the spatial Einstein tensor on H=\partial_t^2g_t|_0 is \tfrac12(H-\operatorname{tr}_g(H)g). Its inverse sends a desired coefficient A to 2A-(\operatorname{tr}_gA)g in dimension three. Smooth time collars realizing these finite jets can be obtained by a quadratic Taylor polynomial and a cutoff. The contracted Bianchi identity and the Maxwell equations at the slice give \nabla^a(T_\delta)_{ab}=0 there. In particular they determine the normal derivatives of its normal contractions from spatial derivatives; no further evolution equation is imposed.

We justify a limit used in this calculation. If smooth \mu\ge|J| on a compact set, then

At a zero of \mu, nonnegativity and domination give d\mu=dJ=0. In a fixed smooth frame,

These bounds are uniform in \delta. On a sufficiently small neighborhood of the compact zero set the right sides, including the first derivatives, are uniformly small. On its complement the denominator has a positive lower bound and convergence is smooth. This proves the claim, including differentiability with derivative zero on the zero set.

Apply it to Equation (9.14). Where \mu_m>0, the limiting tensor is T=\gamma\otimes\gamma/\mu_m, where \gamma(n)=\mu_m and \gamma|_{T\Omega}=J_m. The covector \gamma is causal. At an active ray with positive \mu_m, one has

Parallel transport \ell along any spatial curve. Its pairing with the causal covector field \gamma is nonnegative and vanishes at the point under consideration. Its derivative there is therefore zero. It follows that (\nabla_eT)(Z,\ell)=0 for spatial e and arbitrary Z. The divergence identity, in an orthonormal frame, now gives

At \mu_m=0, both the limiting tensor and its spatial covariant derivatives vanish, so the same conclusion holds for every active |v|\le1.

For completeness the varying arguments in this contraction also matter. Under the graph variation and metric-square-root transport,

The tangential term sK^\sharp v in the derivative of a graph tangent vector is canceled by its square-root transport. Differentiating the constraint therefore gives

At a positive-density active ray, the last term is J_m(\nabla s)+\mu_m v(s)=0, and the middle term is \delta J_m(\nabla s)/(\mu_m+\delta), which tends uniformly to zero. At a zero-density ray both terms vanish. Together with the proved C^1 convergence and Equation (9.15), this proves uniform convergence on the compact active set. The variations have no boundary or ADM contribution. Since \Lambda_B is locally finite, Equation (9.9) and dominated convergence give Equation (9.11).□

Proof. There is no cancellation between different tangent planes in Equation (9.11). Indeed, perimeter submodularity and the common lower bound A show that unions and intersections of two minimizers are minimizers. Their regularity excludes a transverse crossing. Lemma 3.9 further shows that the plane incidence set is closed: strict convergence, uniform local density, and graphical regularity prevent a sheet from disappearing or its tangent plane from changing at a limiting point. Thus the average of the tangent-plane area measures has the form d\eta(x)\,\delta_{T(x)}, with a single plane over each relevant spatial point. Equation (9.11) reads

for every compact smooth s. Hence its integrand vanishes \eta-almost everywhere. Fubini’s Theorem gives the claim on \pi_B-almost every frontier. On each free smooth sheet it then holds everywhere by continuity.□

Proof. Every minimizing set for an enlarged obstacle is also a minimizing set for the original obstacle, because both minima equal A. Choose one of the frontiers supplied by Lemma 9.4. Away from S this is a free original minimizer, so it is smooth and minimal, including at contact with an interior component of B. On the coincidence set with that component, its second graph derivatives agree almost everywhere with those of B. Thus H_B=0 there. Its chosen marginality gives \operatorname{tr}_{T_M}K=0, independently of the chosen sign. Off the coincidence set the same assertion follows from Lemma 9.4. Continuity consequently gives H=\operatorname{tr}_{T_M}K=0 throughout the frontier in \operatorname{int}\Omega.

At contact with S, the obstacle regularity gives a C^{1,\alpha}\cap W^{2,2} graph. On its coincidence set with S the second derivatives agree almost everywhere, and the original condition H_S+\operatorname{tr}_S K=0 applies. Off that set the same equation has just been proved. The frontier therefore satisfies the uniformly elliptic expansion graph equation almost everywhere through the contact patch. Writing its principal part in divergence form, local elliptic regularity first gives C^{2,\alpha} and then smoothness, since the coefficients and the original data are smooth. The strong comparison principle for two touching ordered solutions of this graph equation gives local coincidence with S. The contact set is consequently both open and closed in connected S. If it is nonempty it is all of S, whose area already equals A; no additional frontier component remains.

A frontier disjoint from S would instead be a smooth full enclosing surface in the interior with \theta_+=0, contrary to the original outermostness hypothesis. Thus the chosen frontier must be exactly S. Its regular bounded filled side is the original obstacle, so it cannot contain the additional open set of an enlarged obstacle. This proves the exclusion. With B=S, the same reasoning applies to almost every member selected by \pi_S, showing that their area functionals all equal a'_S.□

Proof. For compactly supported interior tensor variations k, the part of \mathfrak C' depending on k integrates to

There is no interior area term by Proposition 9.5. Integration by parts in distributions and invertibility of spatial trace reversal give Equation (9.17).

For a compact interior metric variation h, the second-order principal term is the scalar curvature adjoint \nabla^2u-(\Delta u)g. Connection variation in the momentum constraint has at most one derivative of h; after integration by parts it has at most one derivative of u,X. The variations of the fixed flux forms’ norms and of the transported ray are algebraic. All other terms in the metric equation are therefore smooth linear expressions in u,X,\nabla u,\nabla X. Taking the trace solves for \Delta u, since in three dimensions \operatorname{tr}(\nabla^2u-(\Delta u)g)=-2\Delta u, and hence solves for the full Hessian of u.

Taking a divergence of Equation (9.17) and using its trace \operatorname{div}X=-u\tau gives a vector Laplace equation for X with only first derivatives of the moments on the right. Together with the traced metric equation this is a linear elliptic system with diagonal Laplace principal symbol. Local distributional elliptic regularity makes the moment measures smooth. To see that no derivatives remain unaccounted for, differentiate the symmetrized equation and combine its three index permutations. For C_{ij}=-2uK_{ij} this expresses 2\nabla_i\nabla_jX_k as \nabla_iC_{jk}+\nabla_jC_{ik}-\nabla_kC_{ij} plus curvature contractions with X. Thus every second derivative of X, as well as every second derivative of u, is a smooth linear function of the first jets. This is the asserted closed system.

It also proves extension through the boundary. In a smooth collar, restrict the system for the first jets to each inward normal segment from a fixed interior collar slice. It is a linear ordinary differential equation with coefficients smooth up to S, so its solution extends smoothly to S. Smooth dependence on the tangential initial point gives a smooth extension of all the moments. Uniqueness of the normal equations preserves the identities between these extended jets and derivatives of the extended fields.

It remains possible a priori that \Lambda has a boundary moment not represented by these densities. Take a metric test with h=\nabla h=0 on S and arbitrary normal second derivatives of its tangential block. In collar coordinates the principal term R'_g=\nabla^i\nabla^jh_{ij}-\Delta\operatorname{tr}h-\langle \operatorname{Ric},h\rangle then has arbitrary boundary value, through -\partial_\nu^2\operatorname{tr}_S h. The momentum, electromagnetic, transported-ray, area, and expansion variations vanish on S. After integrating the smooth interior moments by parts, their boundary terms also vanish because h has zero first jet. Equation (9.9) consequently annihilates every boundary test against the scalar moment mass. That mass is zero. Its vector moment is dominated by the scalar mass and is zero as well.□

Proof. Write Y=(u,X) in the end coordinates. Inspection of the preceding adjoint and its prolongation gives

The derivative coefficients contain \partial g,K; the value coefficients contain \partial^2g,\partial K, their quadratic products, and squared electromagnetic fields. These are controlled by the original O_2/O_1 hypotheses; no extra derivative bound on the original data is being used.

Here is the growth argument. On each radial segment, first bound |Y(r)| by its value at a fixed sphere plus \int_R^r|\partial Y(t)|dt. Substitution in the integrated form of Equation (9.19) and Gronwall’s inequality with the integrable kernel t^{-1-q_0} give uniformly bounded first derivatives and Y=O(r). The same equation then gives \partial^2Y=O(r^{-1-q_0}), so \partial Y has a limit on each ray with error O(r^{-q_0}). Two ray values on a sphere can be joined by an arc of length at most \pi r; their derivative difference is O(r^{-q_0}). Thus the limiting derivative matrix L is independent of the ray, and Y=Lx+O(r^{1-q_0}).

The inequality u\ge|X|_g holds in every direction at infinity. Applying it on opposite rays first forces the linear part of u to vanish, and then forces every linear part of X to vanish. Consequently \partial Y=O(r^{-q_0}) and Y=O(r^{1-q_0}). Reinsert these bounds in Equation (9.19) to obtain \partial^2Y=O(r^{-1-2q_0}). Its radial integral from infinity gives \partial Y=O(r^{-2q_0}). Since 2q_0>1, Y is bounded. One more insertion gives \partial^2Y=O(r^{-2-q_0}), \partial Y=O(r^{-1-q_0}), and a common constant limit Y_\infty with error O(r^{-q_0}).

Integrate the smooth adjoint identity against the four prototypes, truncated at radius R, and let R\to\infty. The limiting scalar boundary term is 16\pi u_\infty E', and the vector boundary term is 16\pi X_\infty^iP'_i. The errors involve a decaying coefficient times a prototype or its derivative, or a derivative of Y times a prototype, and their sphere integrals are O(R^{-q_0}). The volume integrals converge by the weighted finiteness in Lemma 9.2. The prototypes vanish near S, so Equation (9.9) identifies these limits with 16\pi(b^0E'+b^iP'_i). Their four independent flux values give Y_\infty=(b^0,b^i), proving Equation (9.18).

If u vanished at an interior point, nonnegativity would give du=0 there and domination would give X=0 and \nabla X=0 there. The entire first jet would be zero. Uniqueness along curves for the closed first-order system in Lemma 9.6 would make u,X identically zero on the connected exterior. This contradicts u_\infty=E/m>0.□

Proof. Define \mathbf F uniquely by stationarity and its electric and magnetic contractions with the future slice normal n=u^{-1}(\partial_z-X). With the given orientation its spatial parts are \alpha_2 and -\alpha_1 for \mathbf F and *_{\mathbf G}\mathbf F. Contraction with \xi=un+X gives Equation (9.23), including both cross-product signs.

In Equation (9.9), vary \alpha_1 by any smooth compactly supported closed two-form \eta in the interior. The only contribution is -4\int\beta_E\wedge\eta, hence it is zero. Varying \alpha_2 gives the same assertion for \beta_B. We explain why this test class implies exactness on arbitrary topology. Testing first with exact \eta=d\gamma proves d\beta=0. For any embedded oriented loop choose a trivialized oriented normal disk bundle and a compact disk two-form of integral one, extended constantly along the loop and by zero through the tube sides. It is a compactly supported closed two-form. Since \beta is closed, its pairing with this form equals the period of \beta around the central loop: all parallel loops in the tube have the same period. Thus every such period is zero. Smooth one-cycles can be represented by finite sums of embedded loops in a three-manifold, so every period is zero. Path integration defines a global potential. This argument uses all compact closed two-form variations, not only the exact ones.

The forms in Equation (9.23) are smooth up to S. In a collar, integration in the normal direction extends their potentials smoothly there; local potentials on overlapping charts differ by the same constants as in the interior, so the extensions agree. Finally, stationarity and Cartan’s identity give i_\xi d\mathbf F= -d(i_\xi\mathbf F)=0 and the corresponding identity for *\mathbf F. Their spatial exterior derivatives vanish because d\alpha_1=d\alpha_2=0. These two conclusions give d\mathbf F=d(*\mathbf F)=0 on the product.□

Proof of Proposition 9.8. The slice metric is g, and stationarity of Equation (9.20) gives its second fundamental form -u^{-1}\operatorname{sym}\nabla X=K, by Equation (9.17). The definition of \mathbf F recovers the two original fields. Lemma 9.9 proves Maxwell closure. Causality and the end normalization follow from Equations (9.16) and (9.18); \xi is nonzero because u>0.

We derive the stress assertion to fix its normalization. Write \mathsf B(L,M)=L:M-\operatorname{tr}L\operatorname{tr}M. In a compact metric variation keep the lapse and shift fixed, and set b=-u^{-1}\operatorname{sym}\nabla X, so b=K at the base point. Integrating the momentum term by parts gives the gravitational functional

Its first variation at b=K is that of \int u\{R+\mathsf B(b,b)\}\,dV, since their difference is -\int u\mathsf B(K-b,K-b)dV, whose first derivative is zero. The Gauss scalar-curvature identity identifies the latter functional, up to a divergence, with the spacetime scalar action per unit Killing time of Equation (9.20). Integrating the scalar-curvature variation twice by parts gives its coefficient in a covariant spatial metric variation as -u\mathbf{Ein}_{ij}. This calculation uses the coframe dx^i+X^i dz: its dual spatial vectors are tangent to the slices, so the indicated Einstein slots are exactly the spatial ones.

With a fixed flux form, the vector is its contravariant density divided by \sqrt{\det g}. Consequently

The coefficient for twice the lapse times this expression is -u\mathsf S_{ij}. Also \langle\mathcal E\times\mathcal B,X\rangle dV is independent of the metric when the two vector densities and X are fixed. The transported-ray derivative in Equation (9.5) adds -J_{m(i}X_{j)}h^{ij}. Including a volume variation costs nothing because u\mu_m+J_m(X)=0. The zero interior metric variation in Equation (9.9) therefore yields Equation (9.21).

The normal-normal and normal-spatial components of \mathbf{Ein}-\mathsf S are \mu_m,J_m by Gauss–Codazzi and the original constraints. If \mu_m=0, the DEC gives J_m=0, and Equation (9.21) makes all components zero. If \mu_m>0, complementarity and the two inequalities |J_m|\le\mu_m, |X|\le u imply

Since \xi^\flat(n)=-u and its spatial restriction is X^\flat, these normal components and Equation (9.21) give Equation (9.22). This also proves its stated support property. In these formulas \mu_m,J_m are geometric densities; there is no additional factor of 8\pi.□

Proof. The integration normal of the exterior at its boundary is -\nu. An arbitrary compact K-variation k therefore leaves the boundary expression

The mixed normal-tangential components first give X_{\mathrm{tan}}=0, and the tangential trace gives d\zeta=2X_\nu dA.

Next use a compact scalar test s and the variation (h,k)=(2sg,sK), with the flux forms fixed. The differential part of \mathfrak C' is 4(-\Delta s+K(\nabla s,v)), by Lemma 4.1. The expansion derivative at a marginal boundary is 2\partial_\nu s. After the already proved interior adjoint cancels the interior terms, Equation (9.9) becomes

The independent boundary value and normal derivative of s give X_\nu=u and \partial_\nu u+K(X,\nu)=c/2. This proves Equation (9.24).

Let L_+ be the expansion linearization for outward normal motions of S; its principal part is -\Delta_S. Apply the multiplier identity to Lie derivatives of all four original data under a compact vector field that equals s\nu on S. These are legitimate jets up to the boundary: extend the data smoothly into a collar solely to take the derivative. The flux-form Lie derivatives are exact. At an interior active ray, the transported DEC quantity is a nonnegative function with value zero, so its Lie derivative vanishes. There is no boundary constraint atom by Lemma 9.6. The area and expansion derivatives are \int_SH_Ss\,dA and L_+s, respectively, while the ADM derivative is zero. Since d\zeta=2u\,dA, we obtain

Outer area minimization gives H_S\ge0, by nonnegative outward normal area variations. Thus u\ge0 is a supersolution of a smooth uniformly elliptic operator on connected compact S. The strong minimum principle, applied locally with the bounded zeroth-order coefficient, implies either strict positivity everywhere or identical vanishing. No positive stability eigenvalue is assumed.

If u>0 on S, then W=0 there. Tangential derivatives of W vanish, and the normal component of Equation (9.17) gives

This is Equation (9.25) and gives the positive deleted collar. If u=0 throughout S, then X=0 there and all its tangential covariant derivatives vanish. The mixed and normal components of Equation (9.17) force its normal derivatives to vanish as well. Equation (9.24) gives \partial_\nu u=\kappa, proving Equation (9.26) and again W>0 for sufficiently small positive s.

Finally Equation (9.18) gives W\to1 at infinity. Equation (9.22) makes \mu_m=0 wherever W>0, so h_* vanishes on both the boundary collar and a far end neighborhood. It is smooth in the remaining compact interior region because u>0 there. Extension by zero proves the last claim.□

Proof. Write \mathfrak A=I^-(\mathfrak E); the future case has the same proof with time orientation reversed. The end has timelike Killing orbits, so \mathfrak E\subset\mathfrak A. The set \mathfrak A is open and invariant under every Killing translation. It is connected: a point of \mathfrak A can be joined to \mathfrak E by a timelike curve entirely in \mathfrak A, and \mathfrak E is connected. Hence \mathfrak A=\mathbb R\times A for a connected open subset A of \operatorname{int}\Omega containing its distant end.

Suppose that its relative boundary \mathfrak H=\partial_{\mathfrak R} \mathfrak A is nonempty. A boundary of a chronological past is achronal and locally separates its past side from an excluded future side. To recall the local regularity used here, choose a small coordinate cylinder whose vertical lines are timelike. Achronicity permits at most one boundary point on each such line. The openness of the past set and local timelike cone bounds show that the boundary is a graph over a spatial ball, with a uniform Lipschitz bound. Shrinking the cylinder removes any edge of this graph. This description also shows that both of its local sides are nonempty.

The boundary is invariant. At every differentiability point its tangent hyperplane therefore contains \xi. An achronal tangent hyperplane cannot contain a timelike vector. Since \xi is nonzero and causal, it is null there, and the hyperplane is precisely \xi^\perp. Continuity gives W=0 throughout \mathfrak H. This proves smoothness as well. Indeed, in the preceding graph chart write the graph as x^0=f(x^1,x^2,x^3). The smooth plane field \xi^\perp is transverse to the vertical direction. Its graph slopes are smooth functions a_i(x^0,x); the weak derivatives of f satisfy \partial_i f=a_i(f(x),x) almost everywhere. The right sides are continuous. Mollification, or integration on coordinate lines, shows that these are the classical derivatives of a C^1 function. This identity then gives successively C^2,C^3,\ldots regularity. In particular no regular-level-set assumption on W is involved.

Intersecting with z=0 gives a smooth embedded two-sided surface C=\partial A in \operatorname{int}\Omega. The intersection is transverse because the z-slice is spacelike while the null normal \xi is nonzero. Invariance identifies \mathfrak H with \mathbb R\times C. Since W>0 in a deleted collar of S and on the end, C lies in a compact subset of \operatorname{int}\Omega. It is closed there and thus compact. A compact smooth embedded surface has finitely many components: finitely many connected local submanifold charts cover it, and each component contains one of these charts. In particular there is no accumulating family of extra frontier components.

Choose the normal \nu_C into the end-side exterior. On each connected component of C,

where n is the future normal to z=0 and the sign is constant on that component. The null second form of \mathfrak H along its Killing normal is zero: for tangent vectors Y,Z to a spacelike section it is symmetric by hypersurface orthogonality and has symmetric part \tfrac12(\mathcal L_\xi\mathbf G)(Y,Z)=0. Its trace therefore vanishes. Thus C is marginal for one of the two expansion signs on each component.

The connected collar of the connected surface S contains no frontier point. It is consequently wholly contained in A or wholly excluded from A. The closure of A in \Omega is a connected closed smooth exterior with full boundary C\cup S in the first case and C in the second. It contains the distant end. The local excluded side at every point of C shows that its complementary obstacle contains an additional nonempty open subset outside the original obstacle. Its components have the marginal signs just determined, together with the original sign on S if that component is present. This contradicts Proposition 9.5. Hence \mathfrak H is empty. Connectedness of \mathfrak R proves the assertion.□

Proof. Choose a smooth spatial orthonormal frame e_i on an original collar patch and extend it, and n, by stationarity. For spatial directions the spacetime connection is nonsingular:

In particular spatial covariant differentiation preserves smooth one-sided coefficients. The spatial derivatives of \xi=u n+X are

Skew symmetry of \boldsymbol\nabla\xi determines its remaining components without division by u. The boundary identities u=0, X=0, DX=0, and du=\kappa\nu^\flat give Equation (10.3). Also [\xi,e_i]=[\xi,n]=0, so the connection coefficients along \xi are the coefficients of \boldsymbol\nabla_{e_i}\xi and \boldsymbol\nabla_n\xi and are bounded and smooth in the same sense.

The order-zero Maxwell coefficients are the original electric and magnetic fields. Gauss–Codazzi determines \mathbf R_{ijkl} and \mathbf R_{nijk} from g,K and their spatial derivatives. In the deleted collar the Ricci tensor is

because the Maxwell stress is trace free. The contraction identity

then supplies the remaining curvature block (with the same curvature convention in Gauss–Codazzi and this identity). All order-zero coefficients therefore have smooth limits.

Here is an induction which controls every normal derivative. Write F_j=\boldsymbol\nabla^j\mathbf F and R_j=\boldsymbol\nabla^j\mathbf{Rm}, with derivative slots listed from outermost to innermost. Suppose all orders below j have smooth limits. Any component of order j having a spatial derivative slot has a smooth limit: commute that derivative to the outermost slot and use the displayed spatial connection formulas. The commutators are sums of contractions of lower derivatives of curvature with lower derivatives of the tensor being differentiated. Their total derivative order is j-2, so the induction hypothesis controls every term.

It remains to treat derivative slots n,\ldots,n. Apply j-1 covariant derivatives to closure and co-closure of \mathbf F, and evaluate their free derivative slots on n. Metric compatibility allows these tensor identities to be contracted directly; derivatives of a chosen extension of the frame do not enter. For spatial i,l they read

Here n^j means j derivative slots equal to n. Every right-hand side has a spatial derivative slot and was already controlled. These identities give all of F_j first.

The differentiated differential Bianchi identity similarly gives

By pair symmetry this covers every curvature component except the block with two normal tensor slots, R_j(n^j;n,i,n,l). Differentiating Equation (10.5) supplies that component. The differentiated Ricci tensor is a sum of products of the already controlled F_0,\ldots,F_j. This closes the joint induction. Each right-hand side has smooth coefficients on the original closed collar, so the conclusion is smooth one-sided dependence, including all spatial derivatives, rather than merely bounded limits. No identity in the induction solves a time evolution equation by dividing by u.

For T=\boldsymbol\nabla^j\mathbf F or \boldsymbol\nabla^j\mathbf{Rm}, Killing invariance of the connection gives \mathcal L_\xi T=0. Written covariantly this is

The first term tends to zero: it is u\boldsymbol\nabla_nT+X^i\boldsymbol\nabla_{e_i}T, and the next-order jets have just been controlled. Thus B\cdot T=0 at the edge. In the null basis n+\nu,n-\nu the two eigenvalues of B are \kappa,-\kappa. A component of an invariant covariant tensor can be nonzero only when its total boost weight is zero. Counting its derivative slots as well as its tensor slots, it therefore contains equal numbers of the two normal null directions. Their simultaneous sign reversal has even total degree. This proves the I_b invariance.□

Proof. In the first case Taylor’s formula to arbitrarily high order gives

where b_L is smooth. Put p=r^2. Repeatedly applying \partial_p=(2r)^{-1}\partial_r to the remainder, up to order h\leq L, bounds the result by C p^{L+1-h}; the same estimate holds after any prescribed tangential derivatives. Hence a(\sqrt p,y) has the derivatives at p=0 prescribed by the polynomial, continuously to every order. In the second case, Hadamard’s formula writes a=r b with b smooth; the Taylor coefficients of b have the parity of the first case. This proof also treats a smooth flat remainder and makes no analyticity assumption.□

Proof. First suppose u|_S>0. The equation dW=2\kappa X^\flat on S makes W a global positive defining function on a boundary collar. Use coordinates (W,y) there. Division of a smooth tensor vanishing on the boundary by its defining function gives a smooth one-form \beta with

Set

Substitution into Equation (10.1) cancels the singular terms and gives

where \beta,g are pulled back from (p,y); in such pullbacks dW=VdU+UdV. This is smooth at both null faces and at their intersection. At the intersection its normal block is \kappa^{-2}dU\,dV and its tangential block is g|_{TS}.

To extend the Maxwell tensor, write in the original regular chart

At W=0 the vector \xi is the null normal of a smooth Killing horizon. Its null second form vanishes by the Killing equation. The null Raychaudhuri equation, either with an affine generator or with its nonaffine term proportional to the zero expansion, gives \mathbf{Ric}(\xi,\xi)=0. Electrovacuum holds up to this horizon by continuity from the deleted collar. Thus

The one-form \iota_\xi\mathbf F annihilates \xi. On the orthogonal space of a null vector the metric is positive semidefinite with precisely its null line as kernel. The last equality therefore implies that e is proportional to the metric dual of \xi, hence to dW, at the horizon. Smooth division gives e=b\,dW+W\gamma with b,\gamma smooth on the collar. The potentially singular terms become

All remaining terms are smooth pullbacks from (p,y).

Now suppose u|_S=0. In original normal distance s the boundary identities give u=\kappa s+O(s^2) and X=O(s^2) with smooth Taylor factors. Thus the tensors

are smooth up to S, and h_s is positive definite and equals g there. The metric is \mathbf G=-W(dz-A_s)^2+h_s. Let (r,y) be the Gaussian normal coordinates of h_s at S, and put

In these coordinates A_s-dH has no radial component. It follows that G_{rr}=1 and G_{ra}=0 for a\ne r. Therefore the curves r\mapsto(z',r,y) are unit spacelike geodesics, and Y_A=\partial_{y^A} and \xi=\partial_{z'} along them are Jacobi fields. Notice that r is a smooth original defining function with r=s+O(s^2).

We prove the precise parity of their coefficients. The connection along these geodesics is smooth one-sided: their velocities are smooth linear combinations of original spatial directions and \xi, whose connection coefficients were controlled in Lemma 10.2. A parallel orthonormal frame with a prescribed limiting basis at r=0 is therefore obtained by a linear ODE with smooth coefficients on a closed half interval. The radial velocity has limiting value \nu. The values Y_A(0) are tangent to S, while \xi(0)=0 and D_r\xi(0)=\kappa n.

For completeness, D_rY_A(0) is normal to S. The Killing identity \boldsymbol\nabla(\boldsymbol\nabla\xi)=\mathbf{Rm}*\xi shows that the tangential derivative of B=\boldsymbol\nabla\xi vanishes at the edge. Hence the projector P=B^2/\kappa^2 onto \operatorname{span}(n,\nu) is parallel there in every tangent direction. The vector (1-P)\partial_r vanishes at the edge, with its tangential derivatives. Differentiating this equality tangentially yields (1-P)\boldsymbol\nabla_{Y_A}\partial_r=0. Since coordinate fields commute, this is the assertion about D_rY_A(0).

Let I_b act in the parallel frame as in Lemma 10.2. The jth radial derivative of curvature at zero is its jth covariant derivative contracted with j copies of \nu. Its transformation under I_b consequently acquires the factor (-1)^j. Differentiate the Jacobi equation

The initial values are fixed by I_b and their initial derivatives change sign. Induction in this equation gives I_b D_r^jY(0)=(-1)^jD_r^jY(0) for both Y=Y_A and Y=\xi. The contractions \mathbf G(Y_A,Y_B), \mathbf G(\xi,Y_A) and \mathbf G(\xi,\xi) therefore have only even Taylor coefficients. The initial values and Equation (10.3) give

For \mathbf F, the same differentiated contraction rule uses all its covariant jets. Contractions on two Jacobi fields have even parity, and contractions with one radial velocity and one Jacobi field have odd parity. Also \mathbf F(\xi,Y_A)=O(r^2) because \xi(0)=0. Lemma 10.3 now gives the following actual smooth coefficient formulas, with p=r^2:

All coefficients in these displays are smooth in (p,y) up to p=0. Smooth dependence of the geodesic and parallel-frame ODEs on y makes these conclusions uniform with every tangential derivative. The statements are tensorial on overlapping charts of S, so they define global tensors on its collar.

Set

Then

Equations (10.12)–(10.13) are thus smooth in (U,V,y). Their metric at U=V=0 has normal block dU\,dV and tangential block g|_{TS}.

In each case extend the finitely many smooth coefficient tensors in p\geq0 to |p|<p_0, keeping the displayed invariant forms. A smooth extension exists by collar extension in each coordinate patch and a partition of unity on the compact S. Lorentz signature and time orientation hold near B_0 by the nondegenerate leading blocks. They hold on the entire sufficiently small band: the boost U\mapsto e^{-\kappa t}U, V\mapsto e^{\kappa t}V takes any point with p\ne0 to one with |U|=|V|=\sqrt{|p|}, and takes a point on a punctured null face arbitrarily close to B_0. The extended formulas are invariant under these boosts.

Glue this open band to the deleted collar of \mathfrak R by Equation (10.9) or Equation (10.14). This is a collar gluing of smooth manifolds. To check its separation property, a sequence in the overlap which tends to an added null face has p\to0 or unbounded original time and cannot have a limit in \mathfrak R; a sequence tending to the outer collar boundary has no limit in the open added band. No pair of distinct retained points is therefore identified by a limiting overlap sequence. Local collar charts and the old charts give a Hausdorff, second-countable manifold. The translations and boosts agree on the overlap, and both are defined for every real parameter. Their resulting Killing flow is complete.

In the positive-lapse case the original regular chart gives U=0,V=e^{\kappa z} on S; in the zero-lapse case r=0 gives B_0 and the original z=0 slice has a smooth finite z' offset. These observations also identify the induced metric on B_0 and the indicated original boundary. The extension was constructed for this geometric purpose; only its restriction to \mathfrak R is used in the field equations below.□

Proof. Proposition 9.8 gives a constant limiting metric for Equation (10.1), with error O_2(\rho^{-q_0}), and a unit timelike limiting \xi, since (b^0)^2-|b|^2=1. A constant linear change of coordinates puts this metric in rest form \eta while keeping the new spatial coordinates constant along Killing orbits. Explicitly, in the limiting product coordinates one can take t_*=z-b_i x^i and a spatial linear map whose Euclidean quadratic form is \delta_{ij}+b_i b_j. The old and new spatial radii are comparable. Maxwell components remain O_1(\rho^{-2}) under this constant change.

We describe the coordinate improvement to account for the weak initial derivative bounds. For a stationary scalar the wave operator has the form

It is uniformly elliptic on a sufficiently distant end because W>0 there. Solve L\chi^a=-\boldsymbol\Box x_*^a, for the three linear spatial rest coordinates and for x_*^0=t_*. The sources are O_{0,\gamma}(\rho^{-1-q_1}); this Hölder estimate follows on scaled annuli from the available second metric derivatives, after decreasing q_1 and, if necessary, \gamma.

Here is a construction of the right inverse at this weight. Extend the scalar operator to the Euclidean Laplacian inside a large sphere with a smooth cutoff. This extension is only for the coordinate equation. For a source f=O_{0,\gamma}(\rho^{-1-q_1}) the Newton kernel with its constant term subtracted on the far tail defines

where \chi_\infty=1 outside a fixed ball and vanishes near zero. The tail is absolutely convergent because the difference of kernels is O(|x|\,|y|^{-2}). Splitting the integral into |y|<|x|/2, |y|\asymp |x|, and |y|>2|x| gives |Pf(x)|\leq C(1+|x|)^{1-q_1}. For example, the last region is bounded by C|x|\int_{2|x|}^{\infty}s^{-1-q_1}\,ds; the other regions give the same power. Differentiation of the kernel for first derivatives and scaled interior Schauder estimates for second derivatives give the weighted C^{2,\gamma} bound. Thus P is a bounded right inverse from weight -1-q_1 to weight 1-q_1.

After rescaling the cutoff radius to one, the coefficient norm of L-\Delta between these spaces tends to zero with that radius. The convergent Neumann series for 1+P(L-\Delta) gives the required corrections \chi^a=O_{2,\gamma}(\rho^{1-q_1}). They are smooth by local elliptic regularity. Their first derivatives tend to zero, so the new spatial coordinates are a diffeomorphism on a further tail. Since the corrections are stationary,

In particular \xi remains exactly \partial_t.

In these harmonic coordinates the reduced Ricci equation has scalar principal part

where \mathcal Q is quadratic in first derivatives with smooth bounded coefficients near \eta. The coordinate construction starts with scaled C^{1,\gamma} bounds for the metric error and scaled C^{0,\gamma} bounds for the Maxwell field. The Ricci source is quadratic in \mathbf F, since the end is electrovacuum. Schauder estimates in Equation (10.16) therefore give scaled C^{2,\gamma} bounds for the metric. The Maxwell equations then give one additional field derivative. One way to see the latter step precisely is to square closure and co-closure:

By stationarity this is elliptic. Its coefficients have the metric regularity just obtained and its zeroth-order source has scaled C^{0,\gamma} bounds. Interior W^{2,p} estimates first improve the bounded first field derivatives to C^{0,\gamma'} bounds for any fixed \gamma'<1 by taking p large; Schauder estimates then apply. Equivalently the stationary Hodge system is elliptic because the symbol of d+\delta squares to \mathbf G^{ij}\zeta_i\zeta_j>0 for a nonzero spatial covector. Repeating these two estimates gives, on every scaled annulus,

for every fixed j. The constants may depend on j. This bootstrap uses only the original two metric derivatives and one field derivative to start, and only ellipticity on the distant timelike region.

It follows from Equation (10.16), replacing its principal coefficients by \delta^{ij} and using the just-proved second derivative estimate, that

for every j. The right side is integrable in three dimensions because 2q_1>1. Multiply a component of the decaying perturbation by a cutoff on the end and call its Euclidean Laplacian f. The unsubtracted Newton integral of f is now O(\rho^{-1}): the inner region is bounded by C\rho^{-1}\|f\|_{L^1} and the comparable and outer regions are O(\rho^{-2q_1}). Its difference from the cutoff perturbation is an entire harmonic function tending to zero, hence vanishes by the maximum principle. Scaled estimates for the Poisson equation and its derivatives prove the first estimate in Equation (10.15) to every order. The field estimate is already known. One further derivative implies each desired scaled Hölder estimate. Finally h_{s,ij}=\mathbf G_{ij}-\mathbf G_{0i}\mathbf G_{0j}/\mathbf G_{00} and W=-\mathbf G_{00} give the asserted orbit estimates.□

Proof. The covector in Equation (10.18) is timelike on a sufficiently small positive collar. This can be checked at U=V=\sqrt p in the smooth metrics of Proposition 10.4 and then everywhere on the positive collar by boost invariance. In the positive-lapse case it is dz-(2\kappa W)^{-1}dW. Choose a smooth radial cutoff \chi(W) equal to one near zero and zero outside this collar and set

The integration constant is chosen so that the displayed local formula holds exactly. Its differential is a convex combination of dz and dz-(2\kappa W)^{-1}dW. Both covectors have value one on the future causal \xi, so they lie in the same component of the timelike covector cone; that cone is convex. This proves timelikeness and extends s_0 to z plus a constant off the collar.

In the zero-lapse case Equation (10.18) is z'=z-H(r,y). Cut off H in a sufficiently thin original collar. The additional spatial differential stays bounded because H=O(r) and a cutoff derivative of order r^{-1} is multiplied by H. For any bounded spatial one-form \lambda,

Here u=\kappa r+O(r^2) and X=O(r^2). Thus the negative term dominates uniformly on a sufficiently thin collar, including the cutoff annulus. This again continues the desired time as z plus a constant.

On the far end replace this time by the harmonic rest time t plus a constant. Both time covectors have value one on \xi; their limiting Minkowski covectors, and the segment joining them, are uniformly timelike. Their difference of functions is a spatial linear function plus O(\rho^{1-q_1}) and a constant. If the interpolation cutoff varies by one over a sufficiently long interval of \log\rho, the extra differential is at most C/L, where L is the length of that interval. Choose L large, and then its starting radius large. This error is smaller than the uniform timelike margin of the convex segment. The resulting smooth function has all the asserted local formulas and is temporal globally. Each modification was a globally defined stationary function, so \xi(s_0)=1 is preserved.

We check the Cauchy assertion quantitatively. Write r=\sqrt{UV} in the positive collar and use (s_0,r,y) there. At s_0=0 the smooth extension has a positive radial coefficient, a positive tangential metric, and time coefficient a negative constant times r^2. The remaining terms in the formulas of Proposition 10.4 can be absorbed into these leading quadratic forms for small r. Uniformly for a causal tangent vector one obtains

Here |dy| is measured in a fixed metric on the compact S. Boost invariance extends the bound to every s_0. In particular |d\log(UV)/ds_0|\leq C, so a causal curve cannot reach the deleted inner end in a finite amount of s_0 time. On the outer end, Equation (10.15) gives |dx/ds_0|\leq C, excluding escape to spatial infinity in finite time. On the compact part between these two regions, temporality and stationary smooth coefficients give the same bounded-speed estimate in any fixed auxiliary spatial metric. This argument also covers compact loci where W=0, since u and ds_0 remain nondegenerate there.

If an inextendible future causal curve had a finite upper endpoint for its s_0 range, these estimates would confine it to a compact spatial set away from both ends and make its spatial coordinates Cauchy as s_0 approaches that endpoint. It would have a limit in \mathfrak R and could be extended by a short timelike curve, a contradiction. The past argument is identical. Thus its s_0 range is all of \mathbb R, and strict monotonicity gives exactly one intersection with every level.□

Proof. We retain only a sufficiently small open band |UV|<p_0 on the added side. The invariant smooth metric formulas imply

where a,b,c are smooth and bounded for |p|<p_0. Choose a fixed large C and define

A direct computation gives

The analogous formula interchanges U,a with V,b. Taking C large and then p_0 small makes the braces strictly negative. The two gradients are nonzero even on their respective null faces. With the right-wedge future orientation, \mathcal U is nonincreasing and \mathcal V nondecreasing along future causal curves in the band. Their difference \mathcal T=\mathcal V-\mathcal U has timelike differential: the two causal covectors have opposite orientations and are linearly independent. In particular \mathcal T is temporal through the intersection of the faces.

These monotonicities also control curves which leave a coordinate neighborhood. First the right wedge is causally convex in the full attachment. A future curve leaving it across a null face must cross U=0 into U\leq0,V>0. It cannot return across that face because \mathcal U cannot increase to positive values. It cannot leave the added band elsewhere except through the right wedge, the only old region continuing beyond the band. The corresponding past statement uses V=0. The same sign argument excludes a departure and return through the right wedge for endpoints in any one of the other open quadrants. Thus the temporal functions s_0 and \mathcal T establish strong causality at points off the faces.

There is one additional global issue at a face: a curve could otherwise travel far into the right wedge and return near that face. Fix a small p_1>0 at which the exact logarithmic formula (10.18) still holds. Suppose a future causal curve with endpoints a,b sufficiently close to a face leaves p<p_1 through e, and subsequently returns through f. Both crossing points have p=p_1 in the right wedge. The segment between them has nondecreasing s_0, by causal convexity and Lemma 10.7. Hence

Monotonicity on the initial and final band segments then gives

If either factor has the wrong sign, such a curve is already impossible. Otherwise the left side tends to zero as both endpoints approach any fixed point on a null face, contradicting the fixed positive lower bound. This excludes the excursion.

For a curve that stays in the band, both corrected coordinates stay between their endpoint values. The map (U,V)\mapsto(\mathcal U,\mathcal V) is a local diffeomorphism throughout the smaller band, since its Jacobian is e^{-2Cp}(1-2Cp)>0. Thus endpoints near a fixed face point confine U,V to a fixed small coordinate box. In that box and for all y\in S, compactness and the timelike covector d\mathcal T give a uniform causal speed bound |dy|\leq C\,d\mathcal T. A sufficiently small difference of endpoint \mathcal T values prevents angular escape from any prescribed neighborhood of the point. Combined with Equation (10.24), this is the defining local no-departure-and-return property of strong causality at every face point, including B_0.

The open part of \Sigma is acausal by its temporal level-set description and causal convexity of the right wedge. No causal curve can join it to B_0. A future curve from B_0 starts with \mathcal U\leq0,\mathcal V\geq0 and cannot enter the right wedge; a past curve has the opposite obstruction from \mathcal V\leq0. A hypothetical connection in the other direction is the same statement with endpoints reversed. A causal curve between two points of B_0 would have both corrected coordinates constant zero. It would then be tangent to B_0, whose metric is positive definite, and must be constant. Hence the closed hypersurface \Sigma is acausal.

Every inextendible causal curve in the full spacetime through a point of the right wedge has a maximal portion in that wedge which is inextendible as a curve of the wedge. By Lemma 10.7 this portion intersects s_0=0. Thus \mathfrak R\subset D(\Sigma), and openness gives the inclusion into its interior.

For the reverse inclusion, in the left wedge U,V<0 the complete Killing orbits are timelike and avoid \Sigma. In the future quadrant U<0,V>0, a timelike curve can be continued to the past into the left wedge while staying in the band. To verify this directly, boost a point to (U,V)=(-d,d), where d=\sqrt{|UV|}. At fixed y follow the past-directed segment

Its leading normal metric norm is a strictly negative constant times d^2, and the perturbation is smaller uniformly when the band is small. It is therefore timelike. Along it |UV|\leq d^2<p_0, and it ends in the left wedge. Continue there on a complete timelike Killing orbit in the past direction; in the future direction remain in the future quadrant until an inextendible end of the curve. Both portions avoid \Sigma. The past quadrant has the time-reversed construction. Corners can be rounded inside the open timelike cones. Thus every point of the three other open quadrants lies on an inextendible timelike curve avoiding \Sigma. None belongs to D(\Sigma) even for the timelike-curve convention. Finally each null face has empty interior and borders an excluded quadrant. It cannot contribute to \operatorname{int}D(\Sigma). This proves Equation (10.21) with either convention.□

Proof. We apply the class-(b) existence theorem of Chruściel and Wald [9, Definitions 2.1, 2.3 and 2.4, Equation (4.1), Theorem 4.2]. We verify its geometric conditions on the particular spacetime just constructed.

The spacetime is smooth, time oriented and strongly causal by Proposition 10.4 and Lemma 10.8. The hypersurface \Sigma in Equation (10.20) is connected, closed, acausal, spacelike and smooth up to the compact boundary B_0. It is the union of a compact core and one Euclidean end. The smooth Killing field has a complete flow on the full spacetime and fixes B_0 pointwise. The end lies in a stationary chart with uniformly negative G_{00}, uniformly positive spatial metric, and \xi=\partial_t. Equation (10.15) supplies all derivatives and the weighted Hölder bounds of their asymptotic definition. In particular we may use differentiability order two and the fixed exponent q_2\in(1/2,1).

Let \mathfrak M_{\rm ext} denote the Killing development of this end, as in their definition of black and white hole regions. It is exactly a stationary end of the original right wedge. Equation (10.21) identifies the open domain of dependence as \mathfrak R, and Proposition 10.1 gives

Thus neither black nor white hole points meet that open domain; this is their Equation (4.1). All hypotheses of [9, Theorem 4.2] are now satisfied. We record the analytic details that give regularity at the fixed boundary and the derivative estimates in this application. Let \tau_0 be the smooth time function furnished by [9, Proposition 4.2], and let \Sigma_* = \{\tau_0=0\}\cap D(\Sigma) be its zero hypersurface in the closed domain, with boundary B_0. Its interior is Cauchy in \mathfrak R. On the end \tau_0 is Killing time; choose the additive constant of the harmonic rest coordinate so that \tau_0=t there. On this strong end we first make the reference graph maximal outside a compact set. Here is a direct construction. Rescale a sufficiently distant end by x=R y, t=R\theta, and extend its stationary metric coefficients to Minkowski coefficients inside |y|=1. The extended coefficients satisfy

with the same scaled Hölder bounds. This auxiliary extension is used only to solve the end graph equation. For a stationary graph \theta=a(y) its normalized maximal equation is \Delta a+\mathcal Q(a)=0. It depends on Da,D^2a, not on a. Use the norm controlling

and the corresponding scaled Hölder seminorm, with a(0)=0. The Euclidean Newton operator with its constant tail subtracted, followed by subtraction of its value at zero, is a bounded right inverse from C^{0,\gamma}_{-2} to this space. Indeed, splitting the integral into |z|<|y|/2, |z|\asymp|y|, and |z|>2|y| gives the logarithmic bound for its value and O(\langle y\rangle^{-1}) for its first derivative; local scaled Schauder estimates give the second derivative and its seminorm. The subtracted kernel is O(|y||z|^{-2}) in the last region, so that integral converges absolutely.

On a ball of radius C\varepsilon in this norm, \mathcal Q(0)=O_{0,\gamma}(\varepsilon\langle y\rangle^{-2}) and the Lipschitz norm of \mathcal Q into C^{0,\gamma}_{-2} is O(\varepsilon). These estimates follow by writing the principal coefficients as their Euclidean values plus O(\mathbf G-\eta)+O(|Da|^2); all remaining terms contain a metric derivative. The contraction map a\mapsto-P\mathcal Q(a) therefore gives the required graph. Differentiating its uniformly elliptic equation on scaled annuli gives all higher derivatives. Returning to x and cutting off the graph on a fixed annulus gives a smooth spacelike reference hypersurface \Sigma_e equal to \Sigma_* on the compact core, and maximal on the distant end, where its height a_e satisfies

The cutoff is spacelike because its gradient is small on that annulus. The end Killing time \tau_e=t-a_e still has \mathbf G-\eta=O_j(\rho^{-1}) in its coordinates.

Subtract the same cut-off end correction from \tau_0 and call the resulting time function \tau. Its differential remains timelike: it is unchanged on the compact core, and the correction has uniformly small gradient in the transition annulus and the end. Thus \tau is smooth through B_0, has \Sigma_e as its zero hypersurface in D(\Sigma), and equals \tau_e on the distant end. Its zero level is Cauchy in \mathfrak R: the compact and inner ends are unchanged, and the end correction is sublinear with vanishing derivative, so the end causal speed bound and no-escape argument, or [9, Lemma 4.2], apply. Exhaust \Sigma_e by its compact truncations, whose boundaries are B_0\cup C_i with C_i an end coordinate sphere. Compactness of their domains of dependence is supplied by [9, Theorem 3.2], using the communication condition already verified. It is this compactness, not merely compactness of the truncations, that permits the smooth Dirichlet construction of [3, Theorem 4.2]. As in the cited proof, its domain and edge conditions hold for these acausal truncations. We obtain smooth maximal spans M_i with the exact boundary B_0\cup C_i.

For clarity, the uniform height argument permits this fixed inner boundary. Choose a fixed sufficiently distant cylinder \rho=R_0, and translate M_i by a Killing parameter b_i so that its least \tau value on the cylinder is 1. Spacelikeness bounds the oscillation on that cylinder uniformly. The compactness statement [9, Corollary 3.6] then confines the translated compact portions to one compact set. The geometric constants in Bartnik’s tilt estimates are uniform on these surfaces: on the compact portion this follows from the smooth regular time \tau, and on the end from its stationary identification with \tau_e and the strong coefficient bounds. In particular the curvature, the lapse and its first logarithmic derivative, and the derivative of the reference unit normal are bounded. If b_i\leq0, the translated outer boundary has time b_i, while B_0 is fixed and has time zero. Comparison on the end with the maximal reference translates also bounds the translated height above: its interface height is bounded, and its outer height is nonpositive. Together with the compact-core bound this gives a uniform upper bound for the translated height. Thus the cylinder is separated by at least one unit of time from both boundary values, and its distance below the supremum of the translated height is uniformly bounded. The interior tilt estimate [3, Theorem 3.1(iv), Equation (3.14)] gives a uniform tilt bound there. The exterior test-function argument in [3, proof of Theorem 5.3, Equations (5.13)–(5.18)] now bounds -b_i. That argument integrates only over the end, after subtracting an exterior barrier and the outer boundary time. Its cylinder trace has bounded oscillation and bounded tilt; these are precisely the two inner-boundary estimates used in its boundary term. The fixed surface B_0 is outside this integration region and contributes no additional term. The test function gives \log(2+|b_i|)\leq C, with C independent of i. For large positive b_i, an additional bounded translation puts the greatest cylinder time at -1; applying the same argument with time reversed bounds b_i. This is the translation argument in [9, proof of Theorem 4.2, Equations (4.22)–(4.26)]. Consequently |b_i| is uniformly bounded. Compactness on the interior and comparison with the maximal reference translates on the end give |\tau|\leq C on the original M_i.

The ensuing tilt estimate includes the actual boundary. Both components of \partial M_i lie in \{\tau=0\}; their ambient mean-curvature vectors have a uniform bound, since B_0 is fixed and smooth and C_i are end spheres. The boundary maximum argument in [3, Theorem 3.1(iii), Equations (3.10)–(3.12)] therefore gives a uniform tilt bound on all of M_i, including B_0. Its hypotheses bound the geometry of the prescribed edge; they do not assume the unknown boundary gradient. In smooth coordinates across B_0 this makes the graph equation uniformly elliptic with fixed smooth boundary data. Boundary gradient Hölder estimates and Schauder estimates, followed by bootstrapping, give uniform bounds of every order on each fixed closed collar. The interior estimates give the same conclusion on every compact subset. A diagonal subsequence consequently converges smoothly, up to the same embedded B_0, to a spacelike maximal hypersurface.

One can also read off its end estimates directly. Relative to \Sigma_e, the graphs M_i have uniformly bounded heights. For any fixed 0<\beta<1, the graphs a_e\pm C\rho^{-\beta} are upper and lower barriers outside a fixed large sphere. The principal term has the strict sign of \Delta\rho^{-\beta}=\beta(\beta-1)\rho^{-\beta-2}; the coefficient and drift errors are lower order, since Da_e=O(\rho^{-1}), D^2a_e=O(\rho^{-2}) and the stationary metric has the strong decay above. Choose C to dominate the bounded inner trace. The outer trace relative to \Sigma_e is zero, so comparison gives the same bound uniformly in i. Scaled elliptic estimates give all derivative and Hölder bounds for the limiting difference. Thus its full end graph has f=O(\log\rho) and \partial^j f=O(\rho^{-j}) for j\geq1, which imply (10.25) for every q_2<1. In particular its induced metric has two controlled asymptotic derivatives and its second form has one, as required below. The Cauchy assertion and covering translates follow from [9, Lemma 4.2 and Theorem 4.2].

Finally, the complete nonzero causal orbit of \xi through any point of \mathfrak R is an inextendible causal curve there, so it meets the Cauchy hypersurface exactly once. It is transverse to that hypersurface: a nonzero causal vector cannot be tangent to a spacelike hyperplane. The intersection map is smooth by the implicit function theorem, giving the stated global graph and covering property directly. Its future lapse is strictly positive in the interior. This application uses the existence part of the cited theorem; the additional alternatives stated there for a foliation are not needed.□

Proof. Only the boundary assertion requires more than the global graph already proved. In a smooth collar of \widehat\Sigma choose a defining function q\geq0. Its boundary is the actual embedded surface B_0, and its inward spacelike tangent has a positive spacelike component in the (U,V) normal plane. Because its interior is in U,V>0, the two components of this normal-plane vector are both strictly positive. Subtracting its component tangent to B_0 does not change that conclusion, since the normal and tangential blocks are orthogonal at B_0. Hadamard’s formula consequently gives

It follows that r=q\sqrt{ab} is a smooth defining function, and the orbit projection (q,y)\mapsto(r,y) has an invertible boundary derivative. Its boundary map on S is the identity in these natural labels. Also the graph time s_0=(2\kappa)^{-1}\log(b/a) is smooth to the boundary. This proves the collar diffeomorphism. In the positive-lapse case r=\sqrt W; it is not the original defining function W. In the zero-lapse case it is the Gaussian coordinate of Equation (10.14).

The Killing contractions \iota_\xi\mathbf F and \iota_\xi(*_{\mathbf G}\mathbf F) are invariant and annihilate \xi. Their pullbacks under any two sections of the orbit projection are therefore the same one-forms on the orbit space. Their global exactness on the original interior implies exactness on \widehat\Sigma\setminus B_0. Both contractions are smooth on the attachment and vanish on B_0, since \xi=0 there. Their restrictions to the closed hypersurface are closed by continuity from its interior. Local collar primitives consequently extend the global primitives smoothly: the difference of two primitives has zero differential on an overlapping connected interior patch and is a constant. A finite collar cover, or integration along its radial curves, matches these constants. The resulting boundary differential is zero, and B_0 is connected, so each potential has one constant value on the whole boundary. This reasoning uses no simple-connectivity assumption on \Omega.□

Proof. Each Killing orbit in the right wedge is a complete, nonzero causal curve. It is inextendible there: in the original product its spatial label is fixed and its z coordinate traverses all of \mathbb R. The Cauchy property of \operatorname{int}\Sigma' therefore gives exactly one intersection with each orbit. A nonzero causal vector cannot be tangent to a spacelike hypersurface. Consequently the restriction of orbit projection is a local diffeomorphism, and its bijectivity makes it a global diffeomorphism. Flowing the section thus gives a diffeomorphism

For a compact cut its two sections bound the compact chain swept out by the intervening finite orbit segments, which proves the homology assertion without a filling behind either boundary.

For clarity, the identification of boundary labels does not assert that the two hypersurfaces have the same transverse defining function. In the attachment coordinates of Proposition 10.4, B_0=\{U=V=0\} and the right wedge has U,V>0. A spacelike hypersurface through B_0 is transverse to both null faces. If s\geq0 is a smooth defining function on that hypersurface, then

Thus \sqrt{UV}=s\sqrt{ab} is a smooth defining function, and the restriction of y to B_0 is its original boundary label. This also proves that the graph approaches precisely that boundary and no other limiting orbit. The end assertion follows from the rest-chart graph estimates in Proposition 10.9.□

Proof. The exact potentials of Lemma 9.9 remain exact after rotation and pullback to the global graph. In particular,

The primitive extends smoothly to B_0. To see this without an assumption on its periods at the boundary, take a smooth collar primitive locally, subtract its constant difference from the interior primitive, and glue; smoothness of the one-form supplies the local primitives. Since \xi=0 on B_0, its differential vanishes there. Connectedness of B_0, inherited from S, makes \psi one constant \psi_0 on the whole boundary.

At infinity b'=O(r^{-2}). Radial integration gives a finite limit along each ray, and joining two rays along a sphere gives oscillation at most Cr^{-1}. Hence there is a single constant \psi_\infty with \psi-\psi_\infty=O(r^{-1}). Since \operatorname{div}_{g'}\mathcal B'=0 and its total flux is zero, the divergence theorem on the portion of \Sigma' inside a large coordinate sphere gives

The outer error is O(R^{-1}) and the inner total flux is zero. This is precisely where a single connected boundary is needed: zero total magnetic charge would not cancel independently chosen constants on several components. Equation (11.3) therefore yields

We check the normalization and sign of the other integral. Proposition 9.8 gives

Here h_* is supported away from the inner and outer collars. For the geometric constraint densities used throughout this paper, \mathsf S(n',e_i)=2(\mathcal E'\times\mathcal B')_i. Also \mathbf G(\xi,n')=-u' and \mathbf G(\xi,e_i)=X'_i. The mixed Einstein constraint on the maximal slice is consequently

There is no 8\pi in this identity. The 8\pi factor belongs to the ADM momentum flux and to conversion from physical matter densities, not to these geometric densities.

Multiply Equation (11.5) by X'^i and integrate. Integration by parts and the Killing equation give

The inner boundary term vanishes because X'=0 there. At infinity X'=O(r^{-q_2}) and K'=O(r^{-1-q_2}) for some q_2>1/2, so its absolute value is O(R^{1-2q_2})\to0. All volume integrals converge: the gravitational integrand is O(r^{-2-2q_2}), the Maxwell squared norms are O(r^{-4}), and h_* has compact support. Substituting Equation (11.4) gives the exact nonnegative identity

It follows that K'=\mathcal B'=0 in the interior and hence at its boundary by smoothness.

Now X' is a spatial Killing field and vanishes on the whole boundary. Its covariant derivatives in boundary tangent directions are zero. Skew symmetry of \nabla X' then forces all remaining components of \nabla X' to vanish there as well. The Killing identities form the first-order linear system \nabla X'=A, \nabla A=\operatorname{Rm}_{g'}*X' along any curve. Its zero initial data at a boundary point give X'=0 along every such curve, and connectedness gives X'=0 everywhere. Thus W=(u')^2>0 in the interior. Every orbit meets this slice, and W is invariant under the flow, so the same conclusion holds throughout the right wedge. The support condition now forces h_*=0. Flowing the orthogonal slice gives Equation (11.2); invariance and the vanishing of the tangential restriction of \widehat{\mathbf F} give its stated form. Restriction back to z=0 proves \mu_m=J_m=0 in the original open exterior. Smoothness of the original data extends these equalities to S.□

Proof. Smoothness through B_0, compactness of the core and completeness of the end follow from Proposition 10.9 and Lemma 11.1. The symbol bounds for the stationary end in Proposition 10.6, together with Equation (11.7), give the stated strong asymptotic flatness and field bounds. Fluxes were checked in the proof of Proposition 11.2.

We spell out why the mass is that of the original data. In the unimproved stationary coordinates the spacetime perturbation of its limiting constant Minkowski metric has differentiated decay O_2(r^{-q_0}), q_0>1/2. It is electrovacuum near infinity. The skew-slot flux in Lemma 4.2 therefore has spacetime divergence

Stokes’ Theorem between cuts in the original asymptotic plane and cuts in a plane orthogonal to the limiting Killing direction has error O(R^{1-2q_0})+O(R^{-1})\to0: the connecting chains have volume O(R^3) and remain at radii comparable to R. The same estimate permits replacing comparable large cuts by coordinate spheres. With the positive-K convention, Lemma 4.2 pairs a translation (b^0,b^i) with b^0E+\sum_i b^iP_i. Here the limiting unit Killing translation furnished by Proposition 9.8 is

Its flux on the original section is therefore

On the orthogonal rest plane that same flux is its ADM energy. Its spatial metric and h_s differ by the quadratic shift term, which is O_1(r^{-2q_0}) in this chart and contributes O(R^{1-2q_0}) to the ADM flux.

Finally the stationary harmonic correction has spatial coordinate displacement O_2(r^{1-q_1}), q_1>1/2, as in Proposition 10.6. Its leading metric change is a symmetric flat gradient 2\partial_{(i}a_{j)}. The ADM divergence of this term is \Delta a_i-\partial_i\operatorname{div}a, whose divergence vanishes identically. Extend a smoothly across the inside of a coordinate sphere; the flux of this leading term is exactly zero by the Euclidean divergence theorem. Products of coordinate and metric errors have flux O(R^{1-2q_1})\to0, with a smaller decay exponent chosen if necessary. The energy is consequently still m in the strong asymptotic coordinates. This argument uses the original invariant ADM flux, not the mass of a deformation.□

Proof. Put N=u'. We first verify the electrostatic system, including its potential and asymptotic normalization. By Lemma 9.9 and the constant duality rotation, the one-form \iota_\xi\widehat{\mathbf F}=N(\mathcal E')^\flat has a global primitive. Choose its negative, denoted by \Psi, and add a constant so that \Psi\to0 at infinity. This normalization is possible because d\Psi=O(r^{-2}): radial integration gives a limit along each ray, and the oscillation on a sphere is O(r^{-1}). Thus

The primitive extends smoothly to B_0, as in the proof of Proposition 11.2. Its differential vanishes there because \xi=0, and connectedness makes its boundary value a single constant \Psi_0. This is a scalar Killing potential; no global vector potential or simple connectivity is needed.

The spatial Ricci tensor of -N^2dT^2+g' is \operatorname{Ric}_{g'}-N^{-1}\nabla^2N. The spatial component of the purely electric Maxwell Ricci tensor is |\mathcal E'|^2g'-2(\mathcal E')^\flat\otimes(\mathcal E')^\flat. Together with Equation (11.8) and \operatorname{div}_{g'}\mathcal E'=0, this gives

These signs agree with the convention \widehat{\mathbf F}=d\Psi\wedge dT.

We next establish the lapse expansion required by electrostatic uniqueness. Proposition 10.6 gives N=1+O_j(r^{-1}), g'-\delta=O_j(r^{-1}), and \mathcal E'=O_j(r^{-2}) for every fixed j, with scaled Hölder bounds. Hence the lapse equation implies \Delta_\delta(N-1)=O_j(r^{-4}). Multiply N-1 by an end cutoff and extend it by zero to \mathbb R^3. Its Euclidean Laplacian b is integrable and is O_j(r^{-4}) off a compact set. The Newton potential -(4\pi)^{-1}\int b(y)|x-y|^{-1}\,dy equals this cutoff function, since their difference is an entire harmonic function tending to zero. Subtracting the monopole term and splitting the integral into |y|<r/2, |y|\asymp r, and |y|>2r gives an O(r^{-2}\log r) remainder. Scaled Poisson estimates give the corresponding first two derivative bounds. Consequently, for a constant \mu,

For completeness, this coefficient is the original invariant mass. Let \mathcal G_{ij}=\operatorname{Ric}_{g',ij} -\tfrac12R_{g'}g'_{ij}. Expanding about the Euclidean metric, the linearized contracted Bianchi identity gives

Extend the metric perturbation smoothly through a fixed ball. The flux of this divergence-free vector field is zero. The nonlinear terms have size O(r^{-4}) and give an O(r^{-1}) error in the resulting flux identity. Thus, with n_\delta and dA_\delta Euclidean,

The electrostatic equations give \mathcal G=N^{-1}\nabla^2N -2(\mathcal E')^\flat\otimes(\mathcal E')^\flat =\partial^2N+O(r^{-4}) in these coordinates. The last flux is therefore -8\pi\mu+o(1), proving \mu=E_{\mathrm{ADM}}(g')=m by Lemma 11.3. Also, with \nu pointing from B_0 toward the end, integration of the lapse equation gives

Here \partial_\nu N=\kappa>0 follows from the bifurcation attachment and the normalization of \xi.

We can now apply the connected-horizon electrostatic uniqueness theorem of Borghini, Cederbaum, and Cogo [5, Theorem 3.1 and Proposition 6.1]. The spatial manifold is connected, oriented, one-ended and complete up to its compact connected boundary by Lemma 11.3. The metric, lapse and potential are smooth up to that actual boundary, N>0 inside, N=0 and |dN|=\kappa>0 on B_0, and \Psi is constant there. The displayed electrostatic system and asymptotics are exactly the required ones. The additional charge expansion imposed in that theorem when \mu=0 is irrelevant because \mu=m>0. It follows that the entire system is the subextremal Reissner–Nordström system with mass m and some signed charge q, including its normalized lapse and potential N=\sqrt{1-2m/r+q^2/r^2} and \Psi=q/r. Its isometry extends smoothly to the nondegenerate boundary.

The sign is fixed by the actual flux, not by a choice of potential convention: \mathcal E'=-N^{-1}\nabla\Psi for our convention, so the charge of this model is q. Lemma 11.3 therefore gives q=Q. This also covers Q=0. More directly, in that case integration of \operatorname{div}(\Psi N^{-1}\nabla\Psi) =N^{-1}|d\Psi|^2 gives

The boundary expression is regular because N^{-1}d\Psi=-(\mathcal E')^\flat; the term at infinity tends to zero. Thus \Psi=0, and the zero-charge case of the same uniqueness theorem is Schwarzschild.

The identified horizon radius is m+\sqrt{m^2-Q^2}=r_h. The boundary isometry is smooth in proper-distance collars, where N vanishes simply. If the isometry initially reverses orientation, compose it with a reflection of the round sphere. That reflection fixes the radial lapse, potential and electric field, and makes the resulting identification orientation preserving. This proves all three identifications in Equation (11.9) on the closed exterior.□

Proof. The global product in Equation (11.2) and Proposition 11.4 identify the right wedge with the static Reissner–Nordström exterior, with the form in Equation (11.10). The original section is a global graph there by Lemma 11.1. All original tensors are already induced by \mathbf G,\mathbf F on that section; in particular its second fundamental form is the original K, not the zero second fundamental form of \Sigma'. Its unit normal need not be the static normal. The contraction identities defining the original fields consequently recover both full fields even when they are nonradial or nonparallel on that section.

It remains to verify smoothness and injectivity at the original boundary. On the open original orbit base write

The last equality is global: T is the Killing-flow coordinate based on the orthogonal global section \Sigma', so T-z is a single-valued function of the orbit label. On the original section it says dT=-A_s. Let r_* be the Reissner–Nordström tortoise coordinate, dr_*/dr=f^{-1}. Since f'(r_h)=2\kappa and W=f(r),

for a smooth function a near W=0. To verify this expansion, use the smooth inverse r=r(W) of f near r_h and subtract (2\kappa W)^{-1} from dr_*/dW; the difference is smooth. The normalization of \kappa is the one from Proposition 9.10, equivalently the surface gravity of the now normalized static Killing field.

If u|_S=0, Proposition 10.4 gives smooth A_s and a smooth defining function \sqrt W in the original collar. Hence T|_{z=0} extends smoothly there. The orbit metric is a smooth nondegenerate metric in that collar, and the base isometry extends smoothly by its normal geodesic collars; its angular boundary label is a diffeomorphism. The exterior Kruskal coordinates

are, by Equation (11.12), smooth multiples of \sqrt W on this section. Both vanish simply in its inward normal parameter, while the angular variables parametrize S. Their derivative vector is a nonzero spacelike direction in the normal two-plane, since the two nonzero factors have opposite signs. This gives a smooth immersion through the bifurcation sphere.

The exterior in these Kruskal coordinates has \mathsf U<0<\mathsf V; the attachment coordinates of Section 10 have U,V>0.

If u|_S>0, the other part of Proposition 10.4 gives W as a smooth original boundary defining function and

where \beta is smooth in (W,y). Equations (11.11) and (11.12) show that the advanced time

has smooth differential dv=-\beta+a'(W)dW on the original section, and hence has a finite smooth extension to S. In these coordinates the spacetime metric is

The sign of the logarithmic cancellation therefore selects the future horizon, where r=r_h and v is finite.

Here the angular map also is smooth in the original defining function W, a fact that requires more than smoothness in \sqrt W. Put s=\sqrt W. The formula for X_g^\flat shows that

extends as a smooth nondegenerate tensor in (s,y) invariant under s\mapsto-s. At s=0 its normal coefficient is \kappa^{-2} and its tangential metric is g|_{TS}. Reflection is consequently a local isometry fixing the boundary. In normal geodesic collars the angular labels are constant along the normal geodesics and invariant under this reflection. The angular components of the base isometry are thus smooth even functions of s. Taylor’s formula with parameters makes every such function smooth in s^2=W on the one-sided collar. The radial coordinate r=r(W) is smooth as well. We have obtained a smooth map in the regular coordinates (v,r,\text{angles}), and dr/dW=(2\kappa)^{-1} at S is nonzero. Together with the boundary angular diffeomorphism this proves immersion at the future horizon and shows that S maps onto a full horizon cross-section.

In either case the limiting pullback of the ambient metric is the original smooth positive definite g, so the immersion is spacelike up to S. The induced future normal is a smooth algebraic function of its first derivatives and the ambient metric; it therefore extends smoothly as well. The second fundamental form and the two field contractions agree in the open exterior and extend by smoothness, proving their equality with the original tensors on S.

The open graph is injective, its boundary angular map is injective, and the boundary image is disjoint from the open exterior. Finally its end escapes properly: the base radius tends to infinity and the rest-chart graph has the original strictly spacelike asymptotic plane slope, up to sublinear errors. Its image approaches spatial infinity in that end. Compactness of the remaining core and the boundary collar then make the injective immersion proper, hence an embedding. The spatial orientation was fixed in Proposition 11.4; flowing the future normal preserves the corresponding spacetime orientation. This proves the stated orientation and all the required original-data matching.□

Proof of Theorem 2.5. The equality variation and stationary construction in Propositions 9.8, 9.10, and 10.9, together with Lemma 9.9 apply to the original equality data. Proposition 11.2 proves static electrovacuum in their entire right wedge. Lemma 11.3 preserves the original invariant mass, and Proposition 11.4 identifies its orbit metric, lapse, and rotated electric field. The inverse duality formula (11.10) recovers the prescribed two charge signs. Proposition 11.5 then realizes the entire original data, with smooth future normal and fields at its horizon boundary. The vanishing of original non-electromagnetic matter was proved in Proposition 11.2.□

Proof of Theorem 2.6. Write Q_*^2=q_E^2+q_B^2 and R_+=M+\sqrt{M^2-Q_*^2}. The degenerate induced metric on the future horizon is R_+^2\sigma_2, with its null direction along the generators. Every full smooth cross-section, including the bifurcation sphere, therefore has area 4\pi R_+^2. The assumed original cut condition implies

The converse assumes that the actual invariant ADM mass is M and that the actual outward fluxes are q_E,q_B. Substitution now gives the required equality. The examples below verify those assumptions for three explicit types of slicing.□

Proof. Use the converse parameters M,q_E,q_B and set Q_*^2=q_E^2+q_B^2. Let

Start with the static exterior down to the bifurcation sphere. It is smooth there in proper distance or Kruskal coordinates; its future normal and both induced Maxwell fields are smooth, K=0, and its boundary is a connected minimal sphere with \theta_+=0. Its metric is complete with that boundary included. Consider also time graphs T=h(y) in areal Cartesian variables y=r\omega of the following two types:

1. 1.

   h=\varepsilon h_0(r), where h_0 is smooth with compact support in (R_+,\infty), and has a critical point r_0 with h_0''(r_0)\ne0;

2. 2.

   h=\chi(r/L)\,v\cdot y, where \chi=0 for r/L\leq1, \chi=1 for r/L\geq2, L is sufficiently large, and |v| is sufficiently small and nonzero.

All graphs agree with the static slice near S. Their induced metric is

For the radial graph it is (f^{-1}-f(h')^2)dr^2+r^2\sigma_2 and is positive for small \varepsilon. For the cutoff graph, bounded derivatives of \chi give f|dh|_{g_0}^2\leq C|v|^2. Choose |v| such that this is at most a fixed \eta<1. Then

These estimates prove spacelikeness and completeness, and the unchanged inner collar supplies smoothness of the graph and its future normal at the bifurcation sphere.

These are electrovacuum initial data with exactly the normalizations of Definition 2.1. One can check this directly in the static coordinates: the mixed diagonal components of the Einstein tensor are

which equal r^{-4}(-Q_*^2,-Q_*^2,Q_*^2,Q_*^2), the components of 2(F_{ac}F_b{}^c-\frac14\mathbf G_{ab}F_{cd}F^{cd}). The forms

are closed. Gauss–Codazzi with K(U,V)=\mathbf G(\nabla_U n,V) gives \mu=|\mathcal E|^2+|\mathcal B|^2 and J=2(\mathcal E\times\mathcal B)^\flat on every graph; the spatial pullbacks of the closed forms give both divergence constraints. Thus \mu_m=J_m=0 and the matter DEC holds. For every graph sphere r=R, dr vanishes on its tangent space, so directly

The orientation is toward increasing r. These are precisely the induced magnetic and electric fluxes, respectively. Closure identifies them with the fluxes on the actual end coordinate spheres, even when those coordinate spheres are not the surfaces r=R.

We next check the full cut condition in the original graph metric. On the product [R_+,\infty)\times\mathbb S^2 let

It is closed and has comass R_+^2/r^2\leq1 in the static and radial graph metrics, since their radial and angular blocks are orthogonal and the angular metric is r^2\sigma_2. For the cutoff graph, Equation (11.14) shows that its comass is at most R_+^2/((1-\eta)r^2) in the changed region. Choose L>R_+/\sqrt{1-\eta}. On the unchanged inner region the previous exact bound applies, so the comass is at most one everywhere on this graph too.

If \Gamma=\partial D is any full cut as in Definition 2.2, truncate D by a large sphere outside \Gamma. Stokes’ Theorem for this compact region, with all its inner boundary components and coincident obstacle portions counted, gives

The comass bound yields |\Gamma|_{g_h}\geq4\pi R_+^2=|S|_{g_h}. It covers disconnected cuts and cuts meeting the obstacle. Taking D=\Omega shows the infimum equals this value. No extension or filling behind the horizon has entered the calculation.

The boundary expansion remains zero on the unchanged collar. Every coordinate sphere with r>R_+ has strictly positive \theta_+ on these examples. On the static slice its value is 2\sqrt f/r. For the compact radial change, the value varies continuously with the height in C^2 on its fixed compact support, so sufficiently small \varepsilon preserves strict positivity; outside that support it has the static value. For the cutoff graph choose L large enough that 2\sqrt f\geq1 for r\geq L. On any annulus with radii comparable to R\geq L, rescale spatial coordinates and time by R. The background metric and its derivatives are uniformly bounded and converge to the Minkowski ones as R\to\infty, and the rescaled height has C^2 norm at most C|v|, with a constant independent of R and of whether the annulus meets the cutoff. The induced normal, mean curvature, and tangential K trace depend smoothly on these derivatives as long as the graph stays uniformly spacelike. Consequently

pointwise on each sphere. Taking C'|v|<1/2 proves the claim.

To deduce the required outermostness, suppose a compact smooth enclosing surface entirely in the open graph had \theta_+\leq0. At a point of largest r it is tangent to the coordinate sphere, with the same outward normal: the radially outward side is in the component containing infinity. The inner surface’s local graph lies on the inner side of the coordinate sphere. In coordinates on their common tangent plane, the second derivative test and the mean-curvature principal part -\Delta give H_{\mathrm{surface}}\geq H_{\mathrm{sphere}} at contact. Their tangential K traces are equal there because their tangent planes coincide. Hence its expansion is at least the strictly positive sphere expansion, a contradiction. The argument applies to a largest-radius point among all components, so it also excludes disconnected enclosing weakly outer-trapped surfaces.

The radial example has nonzero original second fundamental form. At its critical radius the future normal is the static normal, and the graph Hessian formula gives

Thus these examples are not time-symmetric.

For completeness we calculate the ADM fluxes, including the momentum sign of the boosted examples, directly. The static metric in areal Cartesian coordinates is

Its energy flux is M and its momentum flux is zero, as also follows from the calculation below at v=0. The compact radial graph has exactly the same end data.

Rotate the spatial axes for the cutoff graph so v=\beta e_3, where \beta may have either sign. On its uncut tail set

The induced Minkowski metric in x is Euclidean. Let A_0=\operatorname{diag}(1,1,\gamma^2), w=A_0x, and r^2=x\cdot A_0x. The leading induced metric is

Because \partial_i r=w_i/r, differentiation gives

On |x|=R put \mu=x^3/R and \rho^2=1+d\mu^2, so r=R\rho. The 16\pi flux therefore is

where the primitive is \mu/\sqrt{1+d\mu^2}.

To compute K with its sign, the leading spacetime perturbation in (T,y) is h_{00}=2M/r, h_{0a}=0, h_{ab}=2My_ay_b/r^3. Its required connection coefficients are

up to O_1(r^{-3}); the remaining mixed time coefficients at this order are zero. Put F_0=T-\beta y^3 and \sigma=\sqrt{-\mathbf G^{-1}(dF_0,dF_0)} =\gamma^{-1}+O(r^{-1}). The future normal is n=-\sigma^{-1}\nabla F_0. For the constant tangent vectors e_i to the plane this gives

Writing z=x^3, substitution yields

The trace at this order is

Contracting with the Euclidean outward normal x/R gives

Consequently the actual 8\pi momentum flux is

Indeed an antiderivative is \mu[1+(1+2d)\mu^2]/(1+d\mu^2)^{3/2}, whose endpoint difference is 4/\gamma. The two transverse momentum components vanish by reflection in the corresponding coordinate. Terms quadratic in asymptotic errors and the charge-squared terms have integrated error O(R^{-1}) and do not alter either flux. Rotating back gives P=\gamma Mv. Thus the actual invariant mass is \sqrt{E^2-|P|^2}=M, and the momentum is nonzero when v\ne0.

Equations (11.16) and (11.18) also verify g_h-\delta=O_2(R^{-1}) and K=O_1(R^{-2}) in the actual Euclidean end chart. Pulling back the smooth Maxwell forms and contracting with the uniformly timelike graph normal gives both fields O_1(R^{-2}). Their stress, and hence \mu and |J|, are O(R^{-4}) and integrable. The compact core causes no integrability issue. Together with Equation (11.15), the cut calibration, and the expansion barrier, these checks establish every initial-data and connected rigidity-subclass hypothesis for all three families. The converse just proved supplies equality.□