The nonmaximal anti-de Sitter Penrose Inequality and original-data rigidity

Proof. Each V_a satisfies \Hess_bV_a=V_ab and \Delta_bV_a=3V_a. The scalar curvature linearization at b is

Differentiating Equation 2.8 gives the exact identity

In a sufficiently distant end write R_g+6=(DR)_b(e)+Q(e), where

The right side is O(e^{-2\tau\eta}). Since |V_a|\le C e^\eta and dV_b=O(e^{2\eta})d\eta dA_\sigma, the product V_aQ is integrable precisely with the sufficient condition 2\tau>3. Moreover,

so V_a(R_g+6) is integrable by Equation 2.4 and the uniform equivalence of g and b in the end. Equation 2.11 and the divergence theorem now make the fluxes Cauchy. The difference between their values on two escaping homologous cut boundaries is bounded by the integral of this absolutely integrable divergence over the intervening tail. This also proves the asserted exhaustion independence for boundaries enclosing the same compact region.

In the hyperboloid model the four functions in Equation 2.7 are its ambient coordinate functions. A hyperbolic isometry therefore acts linearly on their span by a Lorentz matrix. The flux expression is linear in V and natural under a background isometry. Its transformed coordinate spheres are an escaping exhaustion, to which the preceding argument applies. This proves the transformation law directly. The decay and weighted integrability remain valid because a fixed hyperbolic isometry changes \eta by a bounded amount. The more general admissible spatial chart covariance is the metric mass invariance Theorem of Chruściel–Herzlich [9, Proposition 2.2 and Theorem 2.3]: its quadratic decay threshold and weighted scalar-integrability hypotheses have just been checked. Only the background-isometry case is needed to balance the mass in the numerical proof. Finally, elementary Lorentz linear algebra sends every future-timelike covector to its rest covector.□

Proof. In the radial b-orthonormal frame, g_M-b has just one entry,

in the radial direction. Thus \mathbb U_b(V_0,g_M-b)(\nu_b)=2V_0\coth\eta\,q. The normalized integral is M(1+r^2)/F_M, which tends to M. The three other integrals vanish by the zero spherical averages of x_i. On the horizon the degenerate induced metric is the pullback of r_h^2\sigma from the space of null generators. A full spacelike cross-section projects diffeomorphically onto that sphere and therefore has its area. The relation between M and r_h follows from F_M(r_h)=0.□

Proof. Choose

Take R beyond the support of K and a fixed cutoff \chi_R=\chi(r/R), equal to one for r\leq R and zero for r\geq2R. Set q_R=b+\chi_R(g-b) on the end and q_R=g elsewhere. For large R this is positive definite. The curvature cutoff error

is supported in R\leq r\leq2R and satisfies

Indeed, the scalar-curvature formula uses two derivatives of the metric, physical derivatives of \chi(r/R) are uniformly bounded on this annulus, and every term contains g-b or one of its first two physical derivatives. The quadratic terms decay faster. The metrics q_R have uniform bounded-geometry coordinate estimates of the orders used here.

Choose a nonnegative smooth compactly supported function D\geq|K|_g, with support in the region where q_R=g, and a smooth positive w_0 equal to r^{-3-\delta} far out. We solve

Here the sign of the normal in the homogeneous Neumann condition is immaterial.

We give the existence and uniform estimates. On a finite truncation put v=0 on its outer sphere, and multiply the entire right side of Equation (3.7) by \lambda\in[0,1]. Write its gradient term as W\cdot dv+F_D, where |W|\leq D and 0\leq F_D\leq D\eta_R; for example this follows from

The maximum principle for -\Delta-W\cdot d+3, with the indicated mixed boundary conditions, bounds |v| by C\eta_R. Outside a fixed sphere containing \supp D, the radial function r^{-\beta} satisfies

uniformly in R. Comparison using Equation (3.6) therefore sharpens the bound to

The boundary constants in these comparisons are independent of the outer truncation.

For completeness, the required derivative bounds follow from scalar elliptic estimates without any coupled-system assumption. Divide the equation by \eta_R and write u=v/\eta_R. Its gradient dependence is D\sqrt{1+|du|^2}, with uniformly bounded first derivative in du. Local W^{2,p} estimates on physical balls, the corresponding Neumann or Dirichlet half-ball estimates, and first-derivative interpolation give a bound for u in W^{2,p} on smaller patches in terms of its already bounded height and the source E_R/\eta_R+w_0. One can see the absorption directly by applying the estimate to nested cutoffs and using

Choose p>3 sufficiently large. Sobolev embedding makes du Hölder, and the scalar Schauder estimate then gives C^{2,\alpha'} control. On the end the same argument is applied after multiplication by the local weight, which is comparable on every physical unit patch. Thus

Lemma A.1 supplies these scalar estimates and the finite-domain inverse. Its local reflection argument permits bounded drift, and its Schauder step is applied to the original one-sided equation after the gradient becomes Hölder. The ellipticity, boundary smoothness, source regularity and gradient-growth hypotheses have just been specified. The linearization of the finite-domain equation is

Its kernel vanishes by the same maximum principle. Its index is zero by continuation of the bounded drift from zero. It is therefore an isomorphism with these boundary conditions. The a priori bounds close the continuation from \lambda=0 to 1. Exhaustion, the uniform estimates, and a diagonal subsequence give a smooth solution of Equation (3.7).

On its exact hyperbolic tail, this solution satisfies (-\Delta_b+3)v=\eta_Rr^{-3-\delta}. Rotations commute with the operator and annihilate this radial source. Applying difference quotients, comparison by r^{-\beta'} for any fixed \beta'<3, and local elliptic estimates gives the same decay for all fixed angular derivative orders. The hyperbolic Laplacian is

Consequently, for \beta' sufficiently close to 3,

Variation of constants for the two Euler solutions r and r^{-3}, with the growing solution excluded by decay, yields

Angular difference quotients give a smooth v_3, and differentiating the radial equation gives the stated symbol estimates. In particular d_0=\eta_R/[\delta(4+\delta)]>0.

Define

The stress transforms as \widetilde K-(\tr_{\widetilde g}\widetilde K)\widetilde g =e^{2v}(K-(\tr_{q_R}K)q_R), so its sign is retained. The conformal boundary formula is

which preserves the original expansion sign. The scalar and divergence formulas are

The momentum norm in the second formula gains a further e^{-2v}. Substituting Equation (3.7), using D\sqrt{\eta_R^2+|dv|^2}\geq |K||dv|, and using the original DEC on the support of K and its scalar version outside that support, gives

The last two terms are nonnegative. Equation (3.9) and 2\beta>3+\delta give |dv|^2\leq C\eta_R^2(1+r)^{-3-\delta}. For large R the required strict margin follows.

Uniform bilinear convergence of \widetilde g to g gives convergence of every full-cut infimum: if (1-\epsilon)g\leq\widetilde g\leq(1+\epsilon)g, the same factors bound all two-dimensional areas, hence their infima. To verify mass convergence, observe outside \supp D that

The weighted integrals of the right side on distant tails are uniformly small. The scalar-linearization divergence formula in Lemma 2.4 has quadratic remainder bounded by C V_0r^{-2\beta}, whose integral is finite because \beta>3/2. Compare the fluxes first on a fixed sphere, where local convergence applies, and then bound both tails by these integrals. Since |V_a|\leq V_0, this proves convergence for all four fluxes. Finally, for K=0, D=0 and the correction equation is linear on each fixed compact subset for all sufficiently large R. Differentiated local estimates prove the asserted smooth convergence. Relabeling a sequence R\to\infty proves the lemma.□

Proof. Use disjoint Gaussian collars [0,L]\times S_j with metric dx^2+q_j(x), where L>0 is fixed. Extend q_j smoothly to [-\varepsilon,0]. Since q_j(x)=q_j(0)+O(\varepsilon) there, a cutoff to the constant q_j(-\varepsilon) near -\varepsilon changes only the metric values by O(\varepsilon); derivatives may depend on \varepsilon. Choose a smooth map F_\varepsilon:[-\varepsilon,L]\to[0,L] equal to the identity near L, mapping endpoints to endpoints, and satisfying F_\varepsilon-x=O(\varepsilon) and F_\varepsilon'=1+O(\varepsilon/L). Its product with the identity on S_j, extended by the identity elsewhere, is a diffeomorphism \Phi_\varepsilon:\Omega_\varepsilon\to\Omega with

This diffeomorphism maps a connected exterior submanifold and its entire intrinsic boundary to another of the same type. Applying the area comparison to every such cut proves Equation (3.12). The new product face has mean curvature zero.□

Proof. Below h/2 the normalization gives t_h(s)=c_hs with c_h\asymp h. On the transition interval s\asymp h, t_h\asymp h^2, and b_h\in[1,2]. Above it the inequalities follow from t/s^2>0 and b>0 on a bounded interval. The radial and tangential eigenvalues of the covector derivative are t_h' and t_h/s. Metric differentiation costs at most a constant times |p|_g in differentiating |p|_g, and s t_h'=b_h t_h\leq2t_h. These facts prove (3.20). For example, outside a slightly larger bounded-gradient interval one may interpolate the logarithmic derivative to 2 and then continue by a quadratic flux law. Positive lower and upper bounds for that logarithmic derivative give the required global extension. This extension is used only after the gradient bound; it is not an existence argument for the saturating law.

We specify the hypotheses of the regularity theorem. For -\partial_i A^i(x,u,Du)=B(x,u,Du) on a C^{1,\alpha} domain, Appendix Theorem A.1 of [28] permits

an analogous bound for |D_pA|, and the spatial and scalar continuity condition

For bounded weak solutions it yields a Hölder continuous gradient up to a C^{1,\alpha} Dirichlet boundary; the same local argument gives the interior estimate. We apply it with p_0=3, \mu=h, no scalar dependence, and B=0. The metric bounds and (3.20) verify every displayed condition; constants on a fixed height and gradient range are uniform in h. At a finite Dirichlet face the datum is constant; subtracting that constant reduces a boundary patch to the homogeneous theorem. The finitely many smooth boundary components are disjoint, so these patches can be chosen separately. At conformal infinity the datum is already zero. The interior gradient estimates originate in [29], and the frozen boundary estimate used in the proof is the boundary regularity estimate of [21]. We use the stated Appendix theorem as the precise boundary input.

For completeness, normalization removes any additive constant in the interior gradient bound. Extend the unregularized density with D_pA\asymp|p| and |\partial_xA|\leq C|p|^2. If m=\operatorname{osc}_{B_1}f>0, subtract the infimum and put u=f/m. The equation for u has density A_m(x,p)=m^{-2}A(x,mp), whose structural constants do not depend on m. Since 0\leq u\leq1, the local gradient estimate for u is uniform. Multiplication by m proves (3.21). The case m=0 is immediate.□

Proof. At a noncritical point put s=|df|_g, e=\nabla f/s, and

The maximum principle gives the height bound. Let d be the inward distance from the finite boundary. Its level sets have mean curvature H(d)=\Delta_g d\leq C d in a fixed collar, since H(0)\leq0. Consider

The radial operator on this function is

Uniformly in h, b_h(s)(1+s^2)\geq c_0>0, and hence

Choose the collar length \ell and c>0 small first; then choose d_*>0 sufficiently small. The last bound makes (3.23) nonnegative on [0,\ell], while \gamma(\ell)=c\operatorname{arsinh}(\ell/d_*)\geq D. Comparison with \underline f and the constant upper barrier gives a uniform finite-boundary gradient bound c/d_*.

On the end, for F(r)=r^{-\xi}, the leading radial operator is \xi(b_h\xi-2)r^{-\xi}, with relative error tending to zero uniformly for 0<b_h\leq2. Thus F is a strict supersolution beyond a fixed sphere, uniformly in any positive multiple of F. Comparison with K_D(r^{-\xi}-R^{-\xi}) gives both the last estimate in (3.22) and the outer-boundary gradient bound. Choose K_D at the fixed inner sphere before sending R to infinity; for R at least twice its radius the required K_D is uniform.

It remains to exclude a large interior gradient. The following calculation is for s\geq s_0, where b_h=b. Writing x=t^{3/2} gives

Thus, for fixed sufficiently large s_0, b<1 and b+s b'<0. At a point with first frame vector e, differentiating Lf=0 and commuting covariant derivatives yields the exact identity

To see the coefficients, differentiate A in the e direction: its contraction with \Hess f is b'\Hess f(e,e)^2+ 2(b-1)s^{-1}|\Hess f(e,\cdot)|_{e^\perp}^2. The two derivatives of \log s then leave exactly the three quadratic terms displayed. In the curvature commutator the additional (b-1)e\otimes e term vanishes by antisymmetry, leaving \Ric(e,e). The Ricci curvature is bounded below on the whole exterior, so L\log s\geq-C. Moreover L(f^2)=2b s^2 and \inf_{s\geq s_0}b s^2>0. Choose c_1 so large that

An interior maximum in this range is impossible. The two boundary gradient bounds and 0\leq f\leq D therefore bound the maximum of \log s+c_1f^2, and then s, everywhere. At a critical point \log s=-\infty, so such points cause no difficulty in this maximum argument.□

Existence, uniqueness, and finite-boundary regularity in Theorem 3.4. Fix h>0 and a finite smooth truncation. Use the homotopy of boundary values \lambda D, 0\leq\lambda\leq1, for the original regularized saturating equation. At \lambda=0 the zero solution exists. For fixed h and on the gradient range of Lemma 3.6, the operator is uniformly elliptic and smooth in the covector. Its linearization has no zeroth-order term, is invertible with homogeneous Dirichlet data by the maximum principle and linear elliptic solvability, and gives openness. For closedness, first apply Lemma 3.5 using the already established gradient bound. Its extension agrees with the original flux on every solution. This gives a common C^{1,\theta} bound; the ordinary uniformly elliptic Schauder estimates for this fixed h then give C^{2,\theta'} bounds and the required compactness. The continuity method therefore reaches \lambda=1; the linear Dirichlet solvability and Schauder estimates used here follow from Lemma A.1, including its general-principal all-Dirichlet conclusion. No uniform ellipticity as h\downarrow0 is asserted at this step.

Now use the estimates that are uniform in h and in the outer truncation. On every fixed interior or finite-boundary patch, the C^{1,\theta} estimates permit a diagonal C^1 limit as h\downarrow0 and R\to\infty. The fluxes converge locally uniformly, so the weak equation and finite Dirichlet trace pass to the limit. The common bound C_Dr^{-\xi} gives the zero limiting value at infinity. Interior elliptic bootstrap applies on every set where df\ne0.

The unregularized flux is strictly monotone in the covector: its radial potential has derivative t(s), with t'>0 for s>0. For two solutions f_1,f_2, test the difference of the weak equations by (f_1-f_2-\varepsilon)_+. This test has compact support, since both functions tend uniformly to zero at infinity, and has zero trace on the finite boundary. Strict monotonicity forces its gradient to vanish. It is therefore zero; letting \varepsilon\downarrow0 proves f_1\leq f_2, and interchanging the solutions proves uniqueness. The same comparison proves monotonicity in D. Positivity can be propagated by a direct local comparison. On a compact geodesic annulus r_0<\varrho<r_1 lying below the injectivity radius, take w=A(e^{-k\varrho^2}-e^{-kr_1^2}). Its radial operator is b w''+w'\Delta_g\varrho. First choose k so large that b_0(2k\varrho-1/\varrho)\geq\Delta_g\varrho on the annulus, where b_0=\min_{0\leq s\leq1}b(s)>0. Then take A>0 small enough that |dw|\leq1 and the inner boundary value of w is below a given positive minimum of f. Since w''/(-w')=2k\varrho-1/\varrho, this is a positive subsolution, zero at the outer sphere. Weak comparison propagates positivity from the inner ball to the larger ball. The finite-boundary barrier supplies an initial positive collar; a finite chain of interior balls along any path proves f>0 throughout the connected interior. Apply the same argument to D-f, which solves the same odd-flux equation and is positive near infinity, to obtain f<D. This completes the asserted existence and finite-boundary statements.□

Proof. There is a useful radial identity that also applies to the families considered below. For g=z^{-2}\bar g, put u=|dz|_{\bar g}^2 and \bar e=\nabla_{\bar g}z/\sqrt u. If F'>0, the operator on F(z), divided by z^2u, is

Indeed \Hess_g z=\Hess_{\bar g}z+ 2z^{-1}dz\otimes dz-z^{-1}u\bar g and \Delta_g z=z^2\Delta_{\bar g}z-zu. For (3.15),

uniformly for 0<b\leq2. More generally, if \bar g-h_0=O_2(z^\beta) with h_0=(1+z^2)^{-1}dz^2+\sigma and \beta>1, then E=O(z)+O(z^{\beta-1}), again uniformly in b.

Choose F_+(z)=A(z-c_2z^2) with a fixed c_2>0. The term (b-2)F_+'/z is nonpositive. Make z_0 small and take A sufficiently large for F_+(z_0)\geq f(z_0,\cdot). The order of these choices can be made consistent with small physical slope: the coarse bound allows A=2C_0z_0^{\xi-1}, so z\sqrt u F_+'\leq C C_0z_0^\xi. Thus b\geq1 on the whole collar after reducing z_0. The negative term -2Ac_2b then dominates E F_+'. Hence F_+ is a supersolution, and comparison gives f\leq F_+. For the lower barrier take F_-(z)=a_0(z+c_2z^2), with a_0>0 small enough to lie below the positive minimum of f on z=z_0. Here b=2-O(a_0^3z^3), so the unfavorable term (b-2)F_-'/z is O(a_0^4z^2), while bF_-'' has a positive constant multiple of a_0. Reducing the collar if needed makes this a subsolution. For the upper comparison test by (f-F_+-\varepsilon)_+; for the lower comparison use (F_--f-\varepsilon)_+. The limiting zero boundary values make both tests compactly supported for every \varepsilon>0, and their traces vanish on the inner sphere. Monotonicity and then \varepsilon\downarrow0 justify both comparisons at infinity. They prove the height bounds in (3.25).

Physical balls of a fixed small radius in this collar have uniform metric bounds; z is comparable to its central value throughout each ball. The height estimate gives oscillation O(z) there. Apply (3.21) using the bounded-gradient extension furnished by the global bound (or the asserted uniform local bound for a family). It gives |df|_g=O(z). All the choices above use only the quantities listed in the statement, proving the uniform version.□

Proof. Put q=|df|_{\bar g}. In dimension three, change of the volume density and inverse metric gives the exact equation

For reference, the conformal factor in (3.15) can also be cancelled exactly: with Q=|df|_{h_0} the equation is \divg_{h_0}[a(ze^{-2v}Q)Q\nabla_{h_0}f]=0. The bounds (3.25) give q\leq Q_0, so f is Lipschitz up to z=0 in the compact metric and has zero trace there. In a \bar g-orthonormal frame, the covector derivative of the compact flux has eigenvalues

They are uniformly comparable to q, and the first spatial derivatives of the coordinate-density flux are bounded by Cq^2 on this bounded covector range. A continuation beyond that range gives the global three-Laplace structure used in Lemma 3.5.

The interior equation has the Dirichlet weak formulation up to z=0. Indeed, cut a smooth zero-trace test off in z<\varepsilon. The compact flux is bounded, the test is O(z), and the cutoff layer has volume O(\varepsilon); the additional integral is therefore O(\varepsilon). No assertion of zero normal flux is made. The boundary gradient estimate now yields f\in C^{1,\theta} up to z=0. The lower bound f\geq cz implies f_z(0,\omega)\geq c. Thus a smaller full collar has gradient bounded away from zero and the compact equation is uniformly elliptic there.

The compact metric is generally C^{3,\delta}, rather than smooth in its normal variable. Every one of its angular derivatives has the same finite normal regularity. Choose 0<\gamma<\min(\theta,\delta). The first boundary upgrade can be obtained without assuming a second derivative in advance. Take angular difference quotients of the divergence equation. Their principal coefficient is the average of A_p between two gradients; it has a common Hölder modulus and is uniformly elliptic. Their divergence source is the difference quotient of the explicit spatial dependence of the flux, with the same Hölder control. The boundary trace remains zero. The linear divergence boundary estimate [1, Theorem 9.1, p. 675, with l=1] therefore bounds these difference quotients in C^{1,\gamma} on smaller half-balls, using their already bounded C^0 norm. Passing to the limit gives tangential second derivatives and mixed derivatives. The equation then gives the normal derivative of the normal flux from its tangential derivatives. Its derivative with respect to f_z is bounded away from zero, so the implicit function theorem recovers the remaining normal second derivative with its Hölder bound. Thus f\in C^{2,\gamma}, and subsequent ordinary Schauder estimates use Lemma A.1. To record explicitly why arbitrary angular derivatives are allowed, write (3.27) in density form \partial_j A^j(x,Df)=0 and put B^{jk}=A^j_{p_k}(x,Df). For an angular multi-index \nu\ne0, w_\nu=\partial_\omega^\nu f satisfies

After the top derivative B Dw_\nu is removed, the chain-rule remainder F_\nu contains only explicit angular derivatives of the flux and D\partial_\omega^\mu f with |\mu|<|\nu|. Inductively B,F_\nu\in C^{1,\gamma}, and the preceding stage gives w_\nu\in C^{1,\gamma}. Expansion of (3.28) gives a uniformly elliptic nondivergence equation whose right side is C^{0,\gamma}. Linear boundary Schauder estimates give w_\nu\in C^{2,\gamma}. Finite covers and nested boundary patches complete this induction for each fixed \nu, with constants allowed to depend on \nu.

Set C_f=f_z|_{z=0}. It is positive and smooth angularly. Taylor’s formula applied to each w_\nu gives \partial_\omega^\nu(f-zC_f)=O(z^2). The identities

prove the two symbol derivative estimates. Finally q=C_f+O(z), and the outward normal to the physical end points toward decreasing z. Substituting in the physical flux, with dA_g=z^{-2}e^{4v}dA_\sigma, gives (3.18). This proves the remaining assertions of Theorem 3.4.□

Proof. Increase s_0 in (3.24) if necessary. Since b\to0 and -(b+s b')/b\to1, that identity implies

Choose a smooth positional cutoff \psi that is negative near the ball boundary and positive on the smaller ball, with controlled first two derivatives. Choose \varepsilon>0 depending only on H so that \phi=\psi+\varepsilon f is still negative near the outer boundary and uniformly positive on the smaller ball. Because Lf=0, L\phi=L\psi is uniformly bounded. Moreover, when s is sufficiently large,

using bs^2\to4/3. On the relatively compact region \phi>0, put q(\phi)=e^{k\phi}-1 and consider W=\log s+4\log q(\phi). At an interior maximum with large gradient, d\log s=-4(q'/q)d\phi. Equations (3.29)–(3.30) then give

Here q'/q\geq k and q''/q=kq'/q. A fixed sufficiently large k makes the right side positive, a contradiction. Since W\to-\infty at \phi=0, and q is bounded below on the smaller ball, a bound at the remaining, bounded-gradient maxima gives the desired interior gradient bound. Every calculation takes place where the gradient is large and the solution is smooth, so critical points introduce no additional regularity requirement.□

Proof. First obtain an upper height bound without using D_i. Write \eta=\operatorname{arsinh}r and choose a large fixed \eta_0. On x=\eta-\eta_0>0 take

Uniformly in i, |d\eta|_{g_i}^2=1+o(1) and

for 0<b\leq2. The radial operator is therefore b w''|d\eta|^2+w'(2+o(1)). A direct differentiation gives

Far from the pole, b\leq2 and \xi<1 give a strict supersolution after fixing \eta_0 sufficiently large. Near the pole, |w'|\asymp A/x^2, w''\asymp A/x^3, and b(|w'||d\eta|)=O(x^4/A^2): the positive term is O(x/A), whereas the negative term has order -A/x^2. On the remaining compact interval, increasing A makes b uniformly small. Thus one fixed A makes w a supersolution on the entire end, for every i.

For each finite D_i compare first on a truncated domain starting sufficiently near the pole that w>D_i. At its outer boundary the Dirichlet solution is zero. Passage to the exhaustion limit gives f_i\leq w. In particular f_i\leq C_0z^\xi beyond a fixed sphere, independently of i,D_i. Applying Lemma 3.9 on uniform physical balls there gives a common local physical gradient bound. This is the bound needed before extending the flux to a three-Laplace structure law on these patches.

For a uniform positive lower value on a fixed sphere, let u_i be the height-one solution for g_i. Comparison gives f_i\geq u_i. The fixed-height estimates have uniform constants in this family: local smooth convergence controls the finite-boundary collars and compact geometry, while (3.32) controls the end. Thus every subsequence has a further subsequence converging locally in C^1 to the height-one solution for g, with zero limiting boundary value at infinity. This limit is positive in the interior by the local annular comparison used above. Uniqueness gives the same limit for every subsequence, so the minima of u_i on a fixed interior sphere have a positive common lower bound. Reducing it to handle finitely many initial terms if necessary supplies the desired lower bound. Lemma 3.7 now gives (3.33), using its general radial formula (3.26).

We spell out the regularity conversion for the metrics. If E_i=z^2(g_i-b), tensor-to-coordinate conversion in (3.32) gives

Indeed, a tensor of rank 2+j with b-norm O(z^\beta) has coordinate components O(z^{\beta-2-j}), and \Gamma_b=O(z^{-1}), \partial\Gamma_b=O(z^{-2}). The compact metrics are consequently uniformly C^{1,\alpha} for any 0<\alpha<\min(\beta-1,1). To verify the only nontrivial case 1<\beta<2, let two points be separated by coordinate distance d. If d is smaller than half the larger z value, integrate the second derivative bound along a path on which z is comparable to that value:

For the remaining pairs use the first derivative amplitude Cz^{\beta-1} at the two endpoints. Pairs on the boundary follow by continuity. Smooth interior convergence and this uniform estimate also give compact C^{1,\alpha'} convergence for \alpha'<\alpha. In particular, a cutoff annulus moving toward z=0 does not require any extra ordinary-derivative bound.

The compact flux in (3.27) has uniform three-Laplace structural constants, and (3.33) gives a common compact gradient bound. The weak zero-trace formulation and Lemma 3.5 give uniform C^{1,\theta} estimates up to z=0. The lower height bound implies f_{i,z}(0,\omega)\geq c. The Hölder gradient estimate therefore makes a common collar noncritical. On it ordinary quasilinear boundary Schauder estimates yield a uniform C^{2,\gamma} bound for 0<\gamma<\min(\alpha,\theta). Compactness gives the stated convergence, and smooth interior convergence of the metrics permits elliptic bootstrap on each compact subset of the collar away from z=0.

Finally, in a smooth limiting compactification \widehat g=\rho^2g, comparability of \rho and z gives c'\rho\leq f\leq C'\rho and |df|_{\widehat g}\leq Q'. Apply the compactified boundary argument afresh with \rho. It gives a noncritical collar; all subsequent boundary Schauder bootstraps are now allowed because the limiting compact metric is smooth. Hence the limit is smooth up to that conformal boundary.□

Proof. Put X_D=t(|df_D|)\nabla f_D/|df_D|, with value zero at a critical point. This vector field is continuous, divergence free in distributions, and has norm at most one. The end expansion gives |df_D|=zC_D+O(z^2) and t(|df_D|)=z^2C_D^2+O(z^3). Its flux through a distant sphere, with outward normal, tends to -\mathcal F_D. The divergence theorem therefore gives the inward flux \mathcal F_D on every full cut separating the inner boundary from infinity. Consequently \mathcal F_D is bounded by the area of every admissible cut. This is also valid for continuous divergence-free fields: apply the distributional equation to smooth approximations of the indicator of the region between a cut and a distant sphere, or first mollify the density on that compact region.

Integration of \operatorname{div}(f_DX_D)=t(|df_D|)|df_D| over a truncation gives the first identity in Equation (3.36). The outer boundary term tends to zero because f_D=O(z) while the flux has a finite limit. The integrand is integrable at infinity: it is O(z^3) and the volume element is O(z^{-3})\,dz\,d\omega.

Fix 0<\lambda<1. The coarse radial supersolution in Theorem 3.4 gives f_D\leq cD r^{-\xi} for a fixed 0<\xi<1, with c independent of D. Hence

uniformly in D. The function \sigma(1-t(\sigma)) is bounded on [0,\infty): it tends to zero both at zero and at infinity, since 1-t(\sigma)=O(\sigma^{-2}) at infinity. Thus for a constant C_0,

We justify the coarea lower bound with the exact full-boundary convention. For almost every level a\in(0,D), the set \{f_D>a\} has finite perimeter, contains an inner collar of T, and is bounded. Its complement has a unique component containing the end. Fill the other complementary components. This operation can only decrease the perimeter, and the remaining entire boundary separates the end from every component of T. Approximate the filled set by smooth sets in a fixed compact region while preserving a smaller inner collar and the distant exterior; their perimeter converges to the finite-perimeter value. Such approximation follows by convolving the characteristic function in finitely many interior charts and using regular levels; the inner and outer collars are held fixed, and bounded complementary components are again filled. Their full boundaries are smooth admissible cuts. It follows that the original level perimeter is at least A_g(T). Coarea, valid for the locally Lipschitz f_D, now gives

No smooth Sard assertion at critical points of a C^{1,\theta} solution is needed. We obtain

Let D\to\infty, then \lambda\downarrow0. Finally, Equation (3.37) is Hölder’s Inequality on the sphere with its probability measure d\omega/(4\pi).□

Proof. Let

Differentiating \sigma=\sqrt t/k and k^2=1-t^{3/2} gives

Consequently W is convex and \mathcal P=Wg-\sigma W'e\otimes e. The weak equation says that f is a local minimizer of \int W(|df|)\,dV_g: convexity gives the minimizing inequality against every compactly supported additive variation. Composing with a compactly supported smooth diffeomorphism is another admissible local variation. Differentiating this inner variation yields \int\mathcal P:\nabla X\,dV_g=0 for every compactly supported smooth vector field X. The differentiation is valid for the locally Lipschitz minimizer by change of variables and dominated convergence. This proves distributional conservation.

Trace reversal in Equation (3.39) gives

Both eigenvalues are negative and their quadratic constraint combination is 6. As \sigma\to0, k=1+O(\sigma^3), so the definitions extend continuously across critical points, independently of e.

Positivity of K-(\tr K)g implies \tr K\leq0 and \tr_{e^\perp}K\leq0. Thus

The flat Hamiltonian of K+L is

and its momentum is \mathcal J_g(K) by conservation. Negative definiteness of L gives the strict boundary expansion decrease.

We spell out the smoothing assertion because pointwise tensor approximation alone would not control momentum. In a coordinate chart let Q^{ij}=\sqrt{\det g}\,\mathcal P^{ij}. Conservation means

Convolution gives smooth symmetric densities converging uniformly to Q, whose divergence differs from the required connection term by a commutator tending uniformly to zero. In a boundary chart x^3\geq0 use instead

Its kernel lies on the original side even at the face, and the same commutator estimate holds. After a partition of unity, additional divergence errors are sums of d\chi\,(Q_\epsilon-Q), which also tend uniformly to zero. Divide by the original volume density and trace reverse. The algebraic terms and the momentum therefore converge uniformly on the compact set. Strict DEC and strict negative expansion survive sufficiently fine smoothing. A separate smoothing scale may be chosen for each member of a sequence.□

Proof. Choose a spherical orthonormal frame \bar E_a, independent of s, and set E_a=s^{-1}\bar E_a. At a point where the spherical connection vanishes, the connection of q is

Away from this point the second identity also contains the spherical connection divided by s. Thus the leaf shape operator is \sqrt F I/s. For lapse N=F^{-1/2} the scalar Gauss identity is

Substituting H=2\sqrt F/s gives Equation (3.43). Moreover,

which proves Equation (3.44) by differentiating F. For P=B-(\tr B)q, its blocks are P_{nn}=2k, P_{nT}=G and P_T=(h+k)I+U. In the normal divergence, the two acceleration terms give -2G\cdot\nabla_\gamma\log\sqrt F/s, while the shape terms give 2\sqrt F(k-h)/s. In the tangential divergence the shape terms give 3\sqrt F G/s and the acceleration terms give ((k-h)I-U)\nabla_\gamma\log\sqrt F/s. Adding the ordinary block derivatives proves the two momentum identities.□

Proof. We may assume 0<\varepsilon<1. All modifications take place beyond a radius chosen at the end of the proof.

Area coordinate and initial coefficients. Set s=e^{2v}r. Its radial derivative is positive far out. In coordinates (s,\omega) the metric is

where the angular differential on the right is initially taken at fixed r. Expansion of Equation (3.47) gives

The mixed term has relative orthonormal size |d_\omega v|/\sqrt{F_g}=O_2(s^{-4}). Define

Then F=1+s^2k^2-2m/s agrees with Equation (3.53) up to an O_2(s^{-2}) term. Thus F^{-1}ds^2+s^2\gamma differs from g by O_2(s^{-4}) in hyperbolic tensor norm. Equation (3.48) gives the blocks B_T=-kI+O_1(s^{-4}), B_{nn}=-h+O_1(s^{-4}), and B_{nT}=O_1(s^{-4}). In particular the approximate spherical pair differs from the original pair by errors of exactly the orders needed by the constraints.

For completeness, the time mass of e^{4v}b in the stipulated hyperbolic metric flux is \fint 8v_3. Indeed, for e=\lambda b its flux integrand is -2V_0d\lambda+2\lambda dV_0, and \lambda=e^{4v}-1=4v+O(r^{-6}). The leading integrand on r=R is 32v_3R^{-2}+o(R^{-2}), whose integral divided by 16\pi is 8\fint v_3. This uses precisely the metric time flux; no charge of B is included.

Exactification and finite angular approximation. Over a fixed interval in \log s, interpolate the metric and tensor to the pair in Equation (3.54). For example interpolate the metric covariantly, and interpolate B+q in the same fixed coordinates. The interpolation is positive definite for a sufficiently distant annulus. Its constraint error is O(s^{-4}), since physical derivatives of the logarithmic cutoffs are bounded on this hyperbolic part. In the spherical pair the mass term supplies

All remaining terms in its constraints are O(s^{-4}): in particular \nabla_\gamma k=O(s^{-3}), h-k=O(s^{-3}), and \nabla_\gamma F=O(s^{-1}). The margin in Equation (3.55) therefore absorbs the interpolation error. Next cut k to 1 on another fixed logarithmic annulus, retaining h=k+s\partial_sk, G=U=0, and the same m. The new terms are again O(s^{-4}) and are absorbed by the same margin.

Fix \eta>0. Choose a finite spherical harmonic sum \mu_N, with the same mean as \mu, such that \|\mu_N-\mu\|_{C^4}<\eta. Such sums approximate a smooth function in C^4: repeated integration by parts with \Delta_\gamma makes its harmonic coefficients decay faster than every power of the degree, and hence the differentiated series converges uniformly. On one further logarithmic annulus replace \mu by \mu+\chi(\mu_N-\mu), for a nondecreasing cutoff \chi. Add to m a radial nondecreasing function \rho with \dot\rho=C\eta\dot\chi. The signed contribution 4\dot\chi(\mu_N-\mu) to s^3\mathcal H is bounded by 4\eta\dot\chi, so a fixed C>1 absorbs it. Here k=1 and t=sk=s; the remaining scaled angular errors are O(s^{-2}). This step costs at most C\eta in mean mass. From this point onward the freely prescribed angular coefficient fields belong to one fixed finite collection of harmonic spaces. Nonlinear expressions such as F^{-1} need not be finite harmonic sums; their angular derivatives are estimated directly in the formulas below.

Reduction from large radial boost. For radial t=sk, put

The constraint identities become the exact formulas

For example these follow from \sqrt F\Delta_\gamma F^{-1/2} =(\Delta_\gamma m)/(sF)+3|\nabla_\gamma m|^2/(s^2F^2). Thus no expansion involving an unbounded \dot t is needed.

Choose a large fixed P. Over a bounded logarithmic interval turn t=s into a constant of comparable size, then lower that constant monotonically to P over a second bounded logarithmic interval. Explicitly, starting at s=R_1, take x=\log(s/R_1) and t=R_1\exp(\int_0^x\kappa(u)\,du), where \kappa decreases smoothly from one to zero and is constant near the endpoints. Its final value is T\asymp R_1. On the next interval take t=P+(T-P)(1-\chi(x)) with a nondecreasing cutoff flat at its endpoints. These profiles have P\leq t\leq s. In the first interval t is comparable with its starting radius and |\dot t|\leq Ct; in the second interval t is nonincreasing. Accordingly

The contribution of 1+t follows from the bounded interval lengths; for the monotone part the derivative contribution is \int_P^T(1+t^2)^{-1}dt\leq P^{-1}.

As long as |m| is bounded and the starting radius is large, F\geq(1+t^2)/2. Equations (3.56) show that a smooth nonnegative radial addition with

where the angular errors need repair makes their contribution to \mathcal H-2|\mathcal D| nonnegative. A smooth majorant of |\dot t| can be used, with the same integral bound. The addition starts before t departs from s; in this initial overlap the errors already have the smaller orders treated above. Its cost is at most C_N/P. This also verifies the boundedness of m used in the argument: choose P so that this cost is at most one and then take the radius large. Continue this repair into a short overlap on which t=P.

The bounded radial boost equations. Now t follows a fixed smooth profile from P to zero on a bounded logarithmic interval. Write S_0=\sqrt{1+t^2} and allow

where u is tracefree. Decompose m=\bar\mu+\rho-ds^{-\delta}+m_{\rm ang}, with m_{\rm ang} of zero mean. For bounded coefficients and their needed derivatives, F=S_0^2+O(s^{-1}). Substitution in Lemma 3.14 gives

For instance the two leading terms in the normal component are -\sqrt F\divg b/(S_0s^3) and \divg b/s^3, which cancel to the stated order. The term 4k(h-k-s\partial_sk) gives 2t\divg b/(S_0s^3). In the tangential component, \sqrt F(\partial_sG+3G/s)=\sqrt F(\dot b+b)/s^3, and \nabla\log\sqrt F=-\nabla m/(sF). These observations account for every leading term in Equations (3.60) and (3.61); the quadratic G,U terms are O(s^{-4}).

We cancel the leading angular terms by solving

Let Y be a spherical harmonic with \Delta_\gamma Y=-\lambda_\ell Y, \lambda_\ell=\ell(\ell+1). Commutation of a covariant derivative with the Hessian, using \Ric_\gamma=\gamma, gives

For \ell\geq2 set the corresponding component of b equal to zero. Its mass coefficient M_\ell and tracefree tensor are then

For each dipole write m_{\rm ang}=MY and b=A\nabla_\gamma Y. Since \lambda_1=2, the remaining equations are

This finite linear system has a unique smooth solution throughout the chosen bounded interval. All of its norms are finite once the profile, P, and the harmonic cutoff have been fixed. Its equations preserve the mean of m_{\rm ang}.

To join smoothly to the previous stage, use an overlap of fixed logarithmic length on which t=P. Start with the prescribed mass coefficients and A=0, and multiply the right sides of the derivative equations (3.65) and (3.66), and the formula for u_\ell, by a cutoff increasing smoothly from zero to one. At constant P, the equations imply on this overlap

The first estimate follows directly from the scalar equation for A by variation of constants; its source is PM/S_0^3=O(P^{-2}) and its zeroth-order coefficient is bounded. The mass coefficients remain bounded by the same finite-dimensional estimate. Inserting these bounds into the uncancelled parts of Equations (3.60) and (3.61) bounds them by C_N/P. Continue the radial repair with a smooth derivative of this size until the cutoff is identically one. It can decrease to zero in proportion to one minus that cutoff; its total cost is at most C_N/P. Thereafter the displayed angular equations hold exactly and no further radial mass bump is needed.

The final dipole adjustment. Take t=0 after the preceding profile, flat at the join. Equations (3.65) and (3.66) then give \dot M_\ell=-\lambda_\ell M_\ell/2, \dot A=0, and u=0. Let C_1 denote the sum of the resulting dipole potentials, so initially b=\nabla_\gamma C_1. On one further bounded logarithmic interval set

where

and \zeta rises smoothly from zero to one, flat at both endpoints. Continue \dot m_{\rm ang}=\Delta_\gamma m/2 and keep \rho constant. The leading tangential momentum in this convention is s^{-3}\nabla_\gamma\partial_{\log s}(a+A_1), which vanishes identically. The normal leading term also cancels. There is no new leading Hamiltonian term: direct substitution of Equation (3.67) gives

In particular all new errors are O(s^{-4}). The formula uses F=1-2m/s, not 1+s^2k^2-2m/s; the difference belongs to the displayed quadratic order. The join is smooth because initially a=0, and the preceding t profile is already zero with all derivatives.

At the endpoint a=A_1=C_1/2 is independent of s and \Delta_\gamma a=-2a. Thus

This is exact tracefreeness, rather than an asymptotic trace estimate.

Error bounds and the order of choices. Here is the constraint ledger; its errors are expressed after multiplication by s^3. The positive contribution retained at every stage is 4\delta d s^{-\delta}.

The constants in the last three rows also depend on the fixed smooth profiles. Choose first \eta and the finite harmonic approximation with C\eta<\varepsilon/3. Next choose P so large that the sum of the two C_N/P costs is less than 2\varepsilon/3. Then fix the bounded boost, overlap, and final dipole profiles and solve their finite linear systems. Only now choose the starting radius R large enough that F>0 and

for every remaining error constant. This is possible because \delta<1. The large boost formulas are exact, so the possibly large derivatives of that radius-dependent profile do not enter C_{N,P} in Equation (3.71). Smooth repairs may overlap the successive stages as described above; where a repair is turning on before t departs from s, the errors have the initial smaller order, and where it turns off the angular cutoff is already becoming identically one. This proves a lower bound c s^{-3-\delta} for the new end. On the unchanged compact part the original strict inequality has a positive minimum. Together these give Equation (3.50).

Decay, energy, and the metric comparison. Let s_* be the beginning of the final tail. Its coefficients have the form

Every nonconstant mode is O_\infty(s^{-1}). Since \dot m=\delta ds^{-\delta}+\Delta_\gamma m/2, the exact scalar formula is

For example the exact endpoint momentum is

It is O_\infty(s^{-4}). Equations (3.72) and (3.70) prove all the symbol estimates in Equation (3.51); conversion to Cartesian derivatives loses one power of s per derivative. In particular q-\delta_{\mathrm{Eucl}}=(F^{-1}-1)ds^2=O_\infty(s^{-1}).

For this metric the ADM integral at s equals \frac{s}{2}\fint(F^{-1}-1). Indeed writing q_{ij}=\delta_{ij}+f\omega_i\omega_j with f=F^{-1}-1, the radial contraction of \partial_jq_{ij}-\partial_iq_{jj} is 2f/s; angular derivatives cancel in this contraction. Its limit is m_\infty. All angular changes have zero mean, and the radial additions are nonnegative with total at most \varepsilon, proving Equation (3.52).

Throughout the large boost reduction t\leq s, and throughout the bounded boost t\leq P\leq s. Thus F\leq1+s^2+C/s everywhere after the initial interpolation. On the final dipole adjustment and tail the stronger bound F\leq1+C/s holds. The original inverse radial coefficient is 1+s^2+O(s^{-1}), and its mixed relative term is O(s^{-4}). Consequently comparison of the radial coefficients and the unchanged s^2\gamma blocks gives q\geq(1-o_R(1))g uniformly on the modified end. The initial interpolation has the same bound. Increase R once more to obtain Equation (3.49). This lower comparison also gives completeness. The boundary and the number of ends are unchanged. Every admissible full cut is the same embedded submanifold before and after the construction. Restricting q\geq(1-\varepsilon)g to its two-dimensional tangent planes gives \Area_q(\Gamma)\geq(1-\varepsilon)\Area_g(\Gamma). Taking the infimum over the full intrinsic boundaries, including S when D=\Omega, proves the asserted area comparison.□

Proof of Theorem 3.1. If K-(\tr K)g\leq0, replace K by -K. Indeed \tr_TK\geq0 for every two-plane, so this decreases the future expansion and preserves DEC. We may therefore assume K-(\tr K)g\geq0.

Apply Lemma 3.2 and first fix one prepared data set, suppressing its index. Add the short inner collar from Lemma 3.3. Solve the saturating Dirichlet problem there with height D, and restrict its stress to the original exterior. Lemma 3.13 gives strict flat-cosmological DEC and strict future trapping for (g,K+L). The stress is smooth on an end, and it can be smoothed on the remaining compact region without changing either the end or the strict conditions.

Proposition 3.15, with arbitrarily small error, produces a smooth asymptotically Euclidean exterior satisfying all hypotheses in Equation (3.2), with

Theorem 4.1 therefore gives, on letting \epsilon\downarrow0,

For each fixed added collar, let D\to\infty and use Lemma 3.12. Then let the collar width tend to zero and use Equation (3.12). This proves Equation (3.1) for the prepared data. Finally, remove the preparation using the mass and area convergence in Lemma 3.2.

Every limiting operation just used is a limit of scalar inequalities. No smooth limiting stress as D\to\infty, or regular limiting metric at equality, is required for this argument. The fixed-chart proof did not use a sign or timelikeness assumption on the flux covector. If it is future timelike, a background hyperbolic isometry makes its time component equal to its Lorentz norm. The geometric and decay hypotheses, the compact support and sign of the constraint stress, and the cut area are unchanged. Lemma 2.4 then proves the final assertion.□

Proof. This is the total-region theorem of [3, Definitions 7.1–7.2 and Theorem 7.3], together with its strict-barrier existence theorem and stability conclusion [3, Theorems 3.1 and 4.1]. The conventions in Section 2 of that paper allow an empty required inner boundary for the total-region theorem. Apply these results on each connected component of M: its outer part is nonempty, and strict-barrier existence is used only when its own inner part is nonempty. A component with empty inner part contributes a frontier only if its collection of trapped domains is nonempty. Take the finite union of the resulting regions. Each free frontier is oriented toward the designated outer part. Their domains include the entire required inner boundary, which is not discarded as an empty relative frontier.□

Proof. Choose a countable dense set x_i and L larger than the diameter of the compact space. Set

For every i, the bounded nonincreasing function c\mapsto d_c(x_i) has at most countably many right discontinuities. Avoid their countable union. The functions d_c are uniformly 1-Lipschitz, so convergence at the dense set implies uniform convergence by a finite net argument. Nesting gives one Hausdorff inclusion; if points in E_{c_j} stayed a positive distance from E_c, evaluating the uniformly convergent distance functions at those points would contradict d_{c_j}=0 there. If E_c is empty, uniform convergence to L is incompatible with a zero of any d_{c_j}.□

Proof. For normal graphs v over \Sigma, let \Phi(v)=\Theta_Q(\Sigma(v))-c, pulled back to \Sigma. Its linearization L is the scalar MOTS stability operator for Q-(c/2)g, with principal part -\Delta_\Sigma. Stability gives a principal eigenvalue \lambda\ge0 and a positive smooth eigenfunction \varphi. If \lambda>0, the graphs v=s\varphi have \Phi(s\varphi)=s\lambda\varphi+O(s^2)>0 and positive velocity for sufficiently small s>0.

If \lambda=0, the kernel of L is spanned by \varphi>0, and its adjoint kernel is spanned by \varphi^*>0. These scalar elliptic principal-eigenfunction properties also follow by applying the maximum principle after dividing a kernel element by \varphi, together with Fredholm index zero. The derivative of the augmented map

is

It is an isomorphism. For a prescribed first component f, pairing with \varphi^* uniquely fixes a=-\int\varphi^*f/\int\varphi^*; adding a unique multiple of \varphi then fixes the second component. The implicit-function theorem supplies v(s),a(s) with output (0,s). Differentiation gives a'(0)=0 and v'(0)=\varphi/\int_\Sigma\varphi>0. Thus the graphs are ordered and have constant expansion c+a(s). If a(s)\le0 for any sufficiently small s>0, replacing this one frontier component by that outer graph enlarges the bounding region while keeping every free component at expansion at most c. This contradicts maximality. The claimed bounds on the leaf parameter follow from positive velocity and compactness.□

Proof. Let \phi support E_z at x', and choose a nearest point x\in E, joined to x' by a length-z minimizing unit geodesic \gamma. Choose z below the injectivity radius of the fixed compact neighborhood. Since the ball of radius z centered at x is inside E_z, the support normal at x' is \dot\gamma(z). Let P_t denote parallel transport along \gamma.

For X\in T_xM, solve the Jacobi boundary-value problem

In a parallel frame its integral equation is

For small z, absorption gives \sup_t\lvert j_X(t)\rvert\le2\lvert X\rvert and \lvert A_X\rvert\le Cz\lvert X\rvert. Moreover \langle J_X,\dot\gamma\rangle is affine with equal endpoint values. Hence A_X is perpendicular to \dot\gamma(0). We can prescribe the first jet of a local unit vector field V by

To realize this jet, use a frame obtained by radial parallel transport from x, choose affine coefficients with the prescribed first jet, and normalize their length. The perpendicularity just proved means that normalization preserves the first derivative. The second derivatives of this unit field have a uniform bound on a small ball.

For F_z(y)=\exp_y(zV(y)), the Jacobi construction gives

For the last estimate, differentiate the exponential map twice, including the bounded jets of V. At z=0 the map is the identity and its covariant second derivative is zero; the first derivative in z is uniformly bounded on these jets. This construction never uses derivatives of the supporting function.

Every y\in E near x is moved a distance z, so F_z(y)\in E_z. Thus \psi=\phi\circ F_z supports E at x. Equation (4.7) makes its gradient length exactly that of \phi at x', and its normal is \dot\gamma(0). The covariant chain rule reads

Trace over \dot\gamma(0)^\perp and divide by the common gradient length. The first term is exactly the normalized tangential Hessian of \phi; the second costs Cz. Parallel transport changes the tangential trace of Q by at most Cz. Therefore \lvert\mathcal E_Q[\psi](x)-\mathcal E_Q[\phi](x')\rvert\le Cz. The assumed bound for \psi proves the result. In particular there is no error proportional to z\lvert\Hess\phi\rvert.□

Proof. Take u>0 sufficiently small that all new distance frontiers are in a fixed compact interior neighborhood, avoid \Gamma_+, and satisfy c+Cu<c' in Lemma 4.6. The collar of \Gamma_- ensures that a nearest segment ending on a new distance frontier starts on the interior frontier of E; no extension across the required boundary is needed.

Uniformly approximate \operatorname{dist}(\,\cdot\,,E) by a smooth function with error below u/32, using a finite coordinate cover, mollification, and a partition of unity. A regular value between u/3 and u/2 gives a smooth relative neighborhood U with

Fill any component of its complement that meets no designated outer face. This may enlarge its interior beyond E_{3u/4}, but only deletes free frontier components; all remaining frontier points still have distance between u/4 and 3u/4 from E.

Choose a smooth \eta, equal to one on that compact free frontier, with 0\le\eta\le1 and support in \{\operatorname{dist}(\,\cdot\,,E)<u\}. For a sufficiently large finite A>0, set

The free frontier of U is strictly negative for Q_A, while the outer faces remain strictly positive. Apply the strict-barrier part of Lemma 4.2 to each component of the complement of U. Each meets an outer face by filling and has nonempty inner boundary by connectedness of M and U\ne\varnothing. The resulting smooth frontier \Sigma encloses U and satisfies

If \Sigma enters the support of the modification, let z=\min_\Sigma\operatorname{dist}(\,\cdot\,,E)<u. We have z\ge u/4>0, and E_z lies on the enclosed side of \Sigma. Indeed, a point at distance less than z from E can be joined to E by a path of length less than z; crossing \Sigma would give a point of \Sigma at smaller distance. At a minimizing point, a defining function of \Sigma is therefore an exterior support of E_z. It has expansion at most c+Cz<c', contradicting Equation (4.8). Thus \eta=0 on all of \Sigma, giving the desired expansion. The inclusion E_{u/4}\subset U makes the enclosure strict. Filling remaining components without an outer face again only deletes whole smooth boundary components.□

Proof. Fix a compact set containing the common radial range of all the negative-threshold regions and \supp(\tr K). On this compact set M_d has a positive minimum. Restrict c_b,c_w sufficiently close to zero so that \max\{\lvert c_b\rvert,\lvert c_w\rvert\}\lvert\tr K\rvert uses less than one quarter of this minimum. The two successive right-continuity choices remain possible by Lemma 4.3. Fix the regions and their collars.

On each collar take a smooth cutoff \kappa(s), equal to one near the face and zero near its far end, and choose k>0 such that 1-ks>0 throughout that collar. A black contribution a_0\kappa(s)(1-ks) and a white contribution -a_0\kappa(s)(1-ks), extended by zero and summed over disjoint collars, give the required C with C_b=a_0, C_w=-a_0, and k_B=a_0k. The C^1 norm tends to zero with a_0, after the collars have been fixed. Since

choose a_0 small enough for the first two terms of Equation (4.11) to be at most M_d/2. They vanish outside a compact set. Finally multiply a fixed smooth positive symbol weight, equal to r^{-3-\delta} on the end, by a small constant so that \rho\le M_d/2 everywhere.□

Proof. Differentiate D=1+l^2|\dd f|^2, use \dd l=pl\,\dd t, and substitute H^f=l\Hess f/\sqrt D and w=l\nabla f/\sqrt D. This gives (4.26). Differentiating the last formula for w, and using 1-v=d, gives

Since A_d=I-w\otimes w^\flat, this is (4.27). Equation (4.28) follows from u=e^{2t}l\sqrt D. Taking the trace of (4.27) and adding \langle\nabla\log u,w\rangle gives

Now \tr H^f=F+(1-\chi)q_1-\tr K, proving (4.30).

From Z=t+\frac18\log D, Equation (4.26) yields

Its image under A_\chi has axial and transverse components

because \chi(4+pv)=4d. On the other hand,

The same two components of p_Q(w,\cdot)+4A_d\nabla t+wF are -aP_0+4dx+aF and a(M+pa y)+4y. Equation (4.25) identifies these with the preceding display, proving (4.29). Smoothness follows either from the original definitions or from

In particular no choice of e is needed at w=0.□

Proof. The stationary calculation. On a product with coordinate s_0, introduce the stationary Lorentz metric

The s_0-slices have induced metric g, shift \beta=l^2\nabla f=Uw, lapse U=l\sqrt D, and unit normal n=U^{-1}(\partial_{s_0}-\beta). We choose the sign k(X,Y)=\mathbf g(\mathbf\nabla_Xn,Y). Since the slice metric is independent of s_0,

Here \operatorname{sym}T=(T+T^{\mathsf T})/2.

For completeness, put \vartheta=\tr k. The contracted Gauss equation and the normal expansion equation, with this sign convention, read

The lapse term in the second formula follows from the acceleration \mathbf\nabla_n n=\nabla\log U; its divergence contribution is \Delta_g\log U+|\nabla\log U|^2=U^{-1}\Delta_gU. Stationarity gives Un(\vartheta)=-\beta(\vartheta), while \operatorname{div}\beta=-U\vartheta. Combining these equalities gives

In the coordinate s=s_0-f, the same metric is -l^2\dd s^2+\bar g. Direct contraction of its warped-product curvature gives R_{\mathbf g}=R_{\bar g}-2l^{-1}\bar\Delta l. The determinant and inverse metric of \bar g give, for a scalar \varphi,

Moreover, Equations (4.26) and (4.27) imply

For example, after division by U its left side is

Substituting into (4.33) therefore yields

The constraint substitution. The symmetric part of \nabla\beta is -Uk, so (4.24) gives

Use Q=K-k, p_Q=p_K-p_k, and [Q,Q]=[K,K]+[k,k]-2[K,k] in (4.34). The result is

The conformal calculation. In dimension three,

Replacing the weight U in the last flux of (4.35) by u=e^{2t}U adds 2p_Q(w,\nabla t). Writing \bar\Delta t in the preceding divergence form, and using \log(u/\sqrt D)=2t+\log l, gives

By (4.29), the two fluxes obtained so far sum to V-uwF. Its divergence contributes

It follows that the quadratic term remaining after the displayed -F\tr K in (4.32) is

The next calculation identifies it with \mathcal T, completing the proof.□

Proof. Using P_0=F-\chi q_1, direct contraction gives

Consequently (4.36) is

Completion of its axial and trace squares is exact because

This proves the asserted formula for \mathcal T.

For smoothness, rewrite (1-\chi)q_1=\tr S-F in (4.36), and ax=\langle w,\nabla t\rangle. Every term is then an invariant smooth expression in S,w,\dd t,p, with F=\tr_{A_\chi}S and A_\chi given by (4.31). At w=0 its value is

This also defines the direction-independent value at a critical point of f.

To prove (4.37), denote the translated variables in the axial, trace, and mixed squares by

The square formula bounds |P^{\mathrm{tf}}|^2,|m_F|^2, (p+1)|y|^2, (p+1)dz^2, and r_F^2 by absolute multiples of \mathcal T. Its last coefficient satisfies

so dq_1^2\le C(N+4)^2\mathcal T. Since \chi\le d\le1, one has

These estimates control x,F, and |P|^2=|P^{\mathrm{tf}}|^2+\frac12(F-\chi q_1)^2. Finally,

They prove (4.37).

If \dd t=0, Equation (4.39) becomes

Using \frac12|\chi Fq_1|\le\frac14F^2+\frac14\chi^2q_1^2 and 0<\chi\le1 proves both bounds in (4.38).□

Proof. Consider the graph of f in the Riemannian warped product g+l^2\dd s^2. Its induced metric is \bar g, its unit normal has horizontal component -w, and its vertical component has length \sqrt d. Write T=l^{-1}\partial_s for the unit vertical vector. The ambient curvature formulas are

with zero mixed Ricci components. Hence half the ambient term R_{\mathrm{amb}}-2\Ric_{\mathrm{amb}}(n,n) in the graph Gauss equation is

At the contact \dd l=0; the covariant components of the graph second form are therefore H^f, up to the harmless common sign coming from the choice of graph normal. Tracing with the graph inverse metric A_d gives

Indeed the axial square cancels, the mixed contribution is -d|A_{e\perp}|^2, and the transverse contribution is \frac14(\tr_\perp A)^2-\frac12|A_\perp^{\mathrm{tf}}|^2. Finally, at \dd t=0 the conformal scalar formula contributes -4\tr(A_d\Hess t). We have thus proved the exact formula

Both final terms are nonpositive, proving (4.40).□

Proof. Subtracting \mathcal S(S) from (4.39), with \dd t=0, gives

The identities

make its parameter dependence explicit. For the part with cross coefficient -\chi, Young’s Inequality gives

The excess cross term satisfies

It follows from (4.38) that

Because H^f=S-K, expansion of the quadratic polynomial \mathcal S gives

Here we used \tr P=F-\chi q_1 and \chi\le d. The estimate for d^2/\chi above and (4.38) bound this by

Combining with (4.43) proves the claim.□

Proof. The error weights. Since \delta+2\gamma<3, the end decay implies

Young’s Inequality therefore gives

Any bounded error supported in E is bounded by C\rho_0, because \rho_0 has a positive minimum there.

The exact tensor-drift derivative. Put H=H^f, k_0=K_{ee}, m_0=K_{e\perp}, and P_H=H|_{e^\perp}. At the point in question,

For c=(p+4)/(4+pv), Equation (4.31) reads

Its exact differentiated form is

To see the cancellation explicitly, use a normal orthonormal frame at the point. The entries of \nabla w=H A_d are

The coefficient of H_{ee}k_0, after adding \langle\nabla\log u,A_\chi K(w,\cdot)\rangle, is

The mixed coefficient is 1+d-2cv+v=2\chi. The remaining transverse terms are the last two terms of (4.49); differentiating K produces its first line. This proves the formula.

In particular every axial Hessian term in this divergence contains a factor \chi. Since H=S-K and \tr P=F-\chi q_1, the second line of (4.49) has absolute value at most

Here \chi|q_1|\le\sqrt{\chi}|q_1| and (4.38) were used. The first line is bounded by Ca|\nabla K|. Equation (4.48) proves (4.46), without an N-dependent constant.

The trace derivative in each stage. At fixed (x,w,p), the height derivatives in (4.44) are respectively

so each is at least one. At \dd t=0, the chain rule and h=\tau f give

There is no \partial_p\mathfrak F term because \nabla p=0.

The compact corrections have uniformly bounded first derivatives. For the only noncompact correction, use

The bounds

give, on the stated height interval,

The base derivative here also includes the bounded derivative of b(x) inside D_0. The second term of (4.50) is consequently bounded below by -C(1+N)^2(\rho_0+v\rho_1). For its third term, the comparison (4.38) and d^2/\chi\le(N+4)/4 yield

For example, replace H by S-K, use weighted Young Inequalities against |M|^2+\chi q_1^2, and absorb the remaining K-terms into the same right side. Thus one may take m=5, after enlarging the constant. The weight observations at the start of the proof and \partial_h\mathfrak F\ge1 establish (4.45).

Finally, outside E, all three right sides have the form

with 0\le\zeta\le1, where \zeta=0 in the first two stages. The eigenvalues of A_\chi lie in (0,1], so |F|\le |h|+C|K|\le Cr^{-\gamma} if the stated end height bound holds. On E, F is uniformly bounded by the compact hypotheses. This proves (4.47).□

Proof. The constant face value gives \Hess f|_{TB}=(\partial_\nu f)\mathrm{II}_g, and hence \tr_{TB}H^f=w_\nu H. Using Equation (4.29), or taking the normal component of the original flux directly, gives

This is the first identity. The normal in \bar g is \sqrt d\,\nu; its mean curvature is \sqrt d\,H. The latter follows also by subtracting the connection change from the tangential Hessian above. The conformal boundary formula for \hat g=e^{4t}\bar g yields the second identity. At \dd f=0 it follows by smooth continuation, or by the same direct computation with d=1.□

Proof. Choose a smooth j:\mathbb R\to[0,1], zero for t\le0 and one for t\ge1. For 0\le\lambda\le1, define

Until the last scaling stage, the inner scalar condition is

We specify the two fixed collar corrections. Choose \eta>0 with 6\eta<b_+-b_-. On black collars take a directional cutoff \chi_-(w\cdot\nu_s), supported where w\cdot\nu_s<-1/2 and equal to one at -1, and a nonincreasing height cutoff \eta_-(h), equal to one for h\le b_-+\eta and zero for h\ge b_-+2\eta. On white collars use \chi_+, supported where w\cdot\nu_s>1/2 and equal to one at 1, and a nondecreasing \eta_+, zero for h\le b_+-2\eta and one for h\ge b_+-\eta. Multiply by disjoint smooth collar cutoffs \kappa_B, equal to one on the faces, and set

Thus \partial_hD_0\ge0, D_0(x,0,h)=0; it is nonpositive at low heights h\le b_-+\eta, nonnegative at high heights h\ge b_+-\eta, and zero in every compatible black outward or white inward direction, for all lengths 0\le\lvert w\rvert\le1. All its derivatives on bounded height ranges are independent of N.

At the assigned face height, the fully corrected expression F=h+C+D_0, evaluated at the limiting normal directions w=\pm\nu, satisfies

These evaluations use \lvert w\rvert=1 and \chi=0 in the smooth direction-free coefficients. Equation (4.12) gives equality in the compatible direction. In the opposite direction the correction of magnitude A_0 gives the inequality.

Take a smooth compactly supported extension b(x), constant at its assigned face value on each smaller collar. Put

The collars are disjoint. Thus D_1(x,0)=0 and D_1(x,\pm\nu)=\pm A_1 on every face. The family is as follows.

1. (i)

   For 0\le\xi\le1, put \lambda=1-\xi, retain F=h+C and h|_B=b, and use \operatorname{div}V_\lambda=u\Xi with Equation (4.52).

2. (ii)

   For 1\le\xi\le2, put \lambda=0, \alpha=\xi-1, retain h|_B=b, and set F=h+C+\alpha D_0(x,w,h). Keep the same scalar equation and boundary condition.

3. (iii)

   For 2\le\xi\le3, put \lambda=0, \zeta=\xi-2, prescribe h|_B=(1-\zeta)b, and set

   F=h+(1-\zeta)\bigl[C+D_0(x,w,h+\zeta b(x))\bigr] +\zeta\bigl[\tr_{A_\chi}K+D_1(x,w)\bigr]. \tag{4.54}

   Keep the same scalar equation and boundary condition.

4. (iv)

   For 3\le\xi\le4, retain this trace equation at \zeta=1, and multiply both \Xi and the right side of Equation (4.52) by s_\xi=4-\xi.

The prescriptions agree at their endpoints. The height derivatives are 1, 1+\alpha\partial_hD_0, and 1+(1-\zeta)\partial_hD_0, respectively. At a stage-(iii) face, h+\zeta b=b, so its limiting directional value is the convex combination of the earlier corrected value and P_B+D_1(\pm\nu). Equation (4.53) therefore persists.

At \zeta=1, the trace equation becomes \tr_{A_\chi}H^f=h+D_1(x,w), with zero boundary values. At a positive interior maximum of f, w=0, A_\chi=I, and D_1=0, so l\Delta f=\tau f>0, a contradiction. A negative minimum is excluded similarly. Hence f=0 and t=Z throughout the last stage. At \xi=4 the scalar equation is \operatorname{div}(4L(Z)\nabla Z)=0, with zero inner flux and zero outer Dirichlet value. Multiplication by Z gives Z=0. For a frozen-coefficient scalar solve at this endpoint, the output is also identically zero for every input Z.□

Proof. The corrections D_0,D_1 vanish at w=0. At an interior extremum of f, the first two stages give \tr K+l\Delta f=h+C, and the third gives \tr K+l\Delta f=h+(1-\zeta)C+\zeta\tr K. The maximum and minimum inequalities bound h by \lVert\tr K\rVert_\infty+\lVert C\rVert_\infty and the boundary heights. Boundedness of the corrections and \lvert\tr_{A_\chi}K\rvert\le3\lvert K\rvert then bounds F. No floor has been used.

Choose r_0 outside the supports of C,D_0,D_1,b and \tr K. The last stage has h=0. In the other stages the tail equation takes the form

where \theta=1 in the first two stages and 1-\zeta in the third. Since \tr K=0,

For the radial test \psi=r^{-\gamma}, with its radial gradient axis, Equation (4.1) gives uniformly in 0<\chi\le1

after increasing r_0.

Consider \phi=C(r^{-\gamma}-R^{-\gamma}), choosing C large enough to dominate C_0 at r=r_0 for R\ge2r_0. In the part r\le R/2, its height is at least C(1-2^{-\gamma})r^{-\gamma}; since \gamma<1<2, this dominates the bounded adverse tensor term O(r^{-2}) after fixing r_0 sufficiently large.

In the part r\ge R/2, evaluate coefficients at a possible comparison contact, so that the given input Z and the test gradient agree with the solution’s. Put z=l\lvert\dd\phi\rvert/\tau, noting \lvert\dd\phi\rvert\asymp C r^{-1-\gamma}. If z\ge1, the magnitude of the negative Hessian term is at least c r^{-1}, whereas the adverse tensor term is at most C_Kr^{-2}. If z\le1, the former is at least c(l/\tau)C r^{-2-\gamma}, while the latter is at most

Their ratio is at most C(N+4)z/r\le C(N+4)/r. Thus a lower bound R_{\mathrm{ht}}(N), enlarged after r_0,C have been fixed, makes \phi a strict upper barrier. Its negative is a lower barrier by the same absolute estimates. Height monotonicity allows comparison with the actual input Z. This proves the first bound in Equation (4.55). Tangential derivatives vanish on S_R; comparison at their common outer value gives the stated normal derivative bound.

Finally, Z=0 on S_R, and the map t\mapsto Z(t,\dd f) is increasing. Its value at t=-\epsilon is negative whenever

The boundary slope bound makes this true once

This is a permissible further lower bound on R at fixed N,\epsilon.□

Proof. At the contact \dd t=0, \Hess t\ge0, p=N, and \dd p=0. Equation (4.32) and Lemma 4.12 give

The constraint gives exactly

Lemma 4.13 retains \tfrac1{16}\mathcal T, with an error C_N(\lvert K\rvert^2+vF^2). The first-derivative estimates for the current trace and the omitted flux, from Lemma 4.14, retain \tau a\sigma and each cost an arbitrarily small fixed fraction of \mathcal T. They have polynomial remainders.

For clarity, every coefficient remainder has the following global control, with constants independent of N,R,\xi:

The first bound uses \delta<1; the second follows from Lemma 4.18 and the tail expression F=h+\zeta\tr_{A_\chi}K. The trace term has compact support. The curvature bound follows from \Ric=O(r^{-3}). Finally, on the end \lvert\nabla K\rvert+\lvert J\rvert=O(r^{-3}) and

Weighted Young’s Inequality gives the last displayed estimate. All compactly supported coefficient derivatives are bounded by a constant times \rho_0, which has positive minimum on their fixed compact support.

Choose the two small derivative fractions so that their sum is less than 1/32, and subtract (1-\lambda)\operatorname{div}(uA_\chi K(w,\cdot))/u. This proves Equation (4.57), after renaming the positive universal coefficient. No fixed-N regularity constant enters this ledger.□

Proof. Enlarge R_{\mathrm{floor}} to include the outer-boundary requirement of Lemma 4.18. Thus an outer contact is impossible.

At an inner contact in one of the first three stages, j(t)=0. Equation (4.52) cancels the omitted drift on both sides and gives V_\nu/u=\mathcal B_F. Lemma 4.16 implies

This contradicts the nonnegative inward derivative at a boundary minimum.

At an interior contact, Equation (4.57) has right side at least 2\delta_0\mathcal T+\tau a\sigma-C_NW_N. Choose the polynomial \Pi_N so large that

This is possible since W_N\ge\rho_0 and \rho/\rho_0 is bounded. At the floor m_0=1; comparison with the prescribed source would then give

a contradiction.

In the last stage f=0, t=Z, u=L, and v=0. At an interior floor minimum, \mathcal T=\frac12(\lvert K\rvert^2+(\tr K)^2). Enlarge the same polynomial so that \delta_0\mathcal T+\rho-\Pi_N\rho_0<0 at every such point. For s_\xi>0, the scalar equation then has left side 4L\Delta t\ge0 and strictly negative right side. At an inner floor minimum its boundary condition gives 4\partial_\nu t=s_\xi(-H-N_B)<0. Both are impossible. If s_\xi=0, multiplication of \operatorname{div}(4L(t)\nabla t)=0 by t, with zero outer value and zero inner flux, gives t=0.□

Proof. For the full Hessian array the Fourier multiplier D^2\Delta^{-1} has L^2 operator norm one, since \sum_{i,j}\xi_i^2\xi_j^2/|\xi|^4=1. Its finite L^p bounds and interpolation give constants C_p with C_p\longrightarrow1 as p\downarrow2. In Euclidean coordinates the Frobenius norm of I-A is exactly 1-b. A sufficiently small perturbation of the metric therefore gives |I-A|\le1-c_0/2. Choose p_0\in(2,3) so that C_{p_0}(1-c_0/2)<1. For a compactly supported v,

Absorb the last term. Christoffel symbols add a bounded first-order term. Applying this inequality to cutoffs on nested balls and using \|Dv\|_p\le\varepsilon\|D^2v\|_p+C_\varepsilon\|v\|_p gives Equation (5.3); the usual nested-radius absorption removes the outer Hessian term. The same argument at p=2 proves the last assertion. All coefficient comparisons are pointwise, so no derivatives of b or of its axis have been used.□

Proof. Rotate so that a=\partial_3 and put v=Y_3. Differentiating Equation (5.8) in the sense of distributions gives

This is a scalar uniformly elliptic divergence equation, even when b depends on all three variables. The scalar De Giorgi–Nash estimate gives a C^{0,\alpha_1} bound for v on a smaller ball, controlled by \|v\|_2. Caccioppoli’s Inequality, applied after subtracting v(x), then gives for balls contained in that smaller region

Now \Delta Y=F=(1-b)v_3. By Cauchy–Schwarz,

Take a cutoff equal to one on B_{1/3}, supported where these estimates hold, and let V be the Newton potential of the cut-off F. Its gradient kernel is bounded by C|x|^{-2} and the derivative of that kernel by C|x|^{-3}. If d=|x-y|, the contribution to |DV(x)-DV(y)| from distances at most 2d is bounded, by dyadic annuli and Equation (5.11), by C\sum_{2^{-j}\le 2d}(2^{-j})^{\alpha_1}\le Cd^{\alpha_1}. At larger distances the bound is Cd\sum_{2^{-j}>d}(2^{-j})^{\alpha_1-1}\le Cd^{\alpha_1}. The same bounds hold for the cut-off source, with an additional harmless large-scale term. Thus DV is C^{0,\alpha_1}. The Newton kernel is locally in L^2 in dimension three, so \|V\|_\infty is controlled by \|F\|_2. Finally Y-V is harmonic on B_{1/3}; its ordinary interior derivative estimates complete Equation (5.9).□

Proof. Otherwise take errors \varepsilon_j\to0 and k_j\uparrow1 with neither alternative, and subtract z_j(0) in interior balls. Lemma 5.2, together with the gradient bound, bounds z_j in W^{2,p_0} on smaller balls. Thus Ds_j is bounded in L^{p_0}. Failure of Equation (5.12) gives \inf_{B_{1/6}}s_j\to0. The weak Harnack Inequality with bounded divergence and scalar errors implies s_j\to0 in measure on B_{1/3}; one may use Theorem 5.2 and Corollary 5.4 of [30], with the radii 1/6, 1/3, and 5/6.

We need more than this convergence in measure. Test Equation (5.7), after dropping its nonnegative Hessian term, with \eta^2(h-s_j)_+, where \eta is supported in B_{1/3}. Ellipticity and Young’s Inequality give, for fixed h>0,

On \{s_j\ge h\}, convergence in measure and the uniform L^2 bound for Ds_j give convergence to zero of its L^1 norm. On the complementary set, Equation (5.14) and Cauchy–Schwarz bound the L^1 norm by Ch+o_j(1). First let j\to\infty, then h\downarrow0. We obtain Ds_j\to0 in L^1(B_{1/4}). Testing the full Equation (5.7) with a nonnegative cutoff now yields

Since g_j\to I and Dg_j\to0, the same is true for D^2z_j. Poincaré’s Inequality and the uniform Lipschitz bound show, after passage to a subsequence, uniform convergence on smaller balls to an affine function. Its slope has length one, because s_j\to0 in measure and the gradients converge strongly in L^2. This contradicts failure of Equation (5.13). At a boundary center the functions are odd, so their affine limit has zero constant and tangential terms. Increasing k_* if necessary ensures k_*>r_*.□

Proof. Set Y=(z-L_0)/h and hold the original coefficients of the z-equation fixed when applying Lemma 5.2 to Y. The covariant Hessian of L_0 is -\Gamma DL_0, so that lemma gives \|Y\|_{W^{2,p_0}(B_{7/8})}\le C. Put a=DL_0/|DL_0| and A_0=I-(1-b)a\otimes a. The difference between the original axis projector and a\otimes a obeys

This remains valid at a zero gradient, since then h|DY|=|DL_0|; metric changes have been included in the second term. For q=2p_0/(p_0-2)>p_0 and \vartheta=(p_0-2)/2>0,

Consequently Hölder’s Inequality, with 1/2=1/q+1/p_0, gives

Here F_* includes the matrix error times D^2Y, the term G/h, and A^{ij}\Gamma_{ij}^k(D_kL_0/h+D_kY). Thus neither DY nor any derivative of b has been assumed pointwise bounded.

There is a small zero-Dirichlet correction for Equation (5.16). On the convex ball B_{3/4},

For smooth functions this follows by integrating twice by parts: \int((\Delta u)^2-|D^2u|^2)= \int_{\partial B_{3/4}}H_{\partial B_{3/4}}(\partial_\nu u)^2\ge0, with the outward convex convention. Density proves the stated version. The map

is therefore a contraction, in the Hessian norm, with factor 1-c_0. Its fixed point u satisfies A_0:D^2u=F_* and \|u\|_{W^{2,2}}\le C\|F_*\|_2. In dimension three the embedding W^{2,2}\hookrightarrow L^\infty gives

The function Y_0=Y-u solves the fixed-axis equation and has a uniform W^{2,2} norm. Lemma 5.3 applies.

Choose \alpha_2<\alpha_1, then r_2 so small that the Taylor error Cr_2^{1+\alpha_1} is at most \tfrac14r_2^{1+\alpha_2}. Next choose \eta_0, and finally h_0, so that C(h_0^\vartheta+\eta_0) is at most the same quantity. With L_1=L_0+h(Y_0(0)+DY_0(0)\cdot x), these bounds prove Equation (5.15). If the center is on the boundary plane, a is its normal, b is even, and Y and F_* are odd. Uniqueness of the correction on the symmetric ball makes u odd. Hence Y_0 is odd and its Taylor polynomial has only a normal linear term, as required.□

Proof of Theorem 5.1. Choose the constants of Lemma 5.5 first, and then decrease h_*>0 so that h_*\le h_0 and Ch_*\sum_{j\ge0}r_2^{j\alpha_2}<1/4. Use this h_* in Lemma 5.4. Set M=L+Q; the case M=0 is immediate. Normalize coordinates at each center as above. Choose a uniform initial radius \rho_0 so small that, after dividing z by M and scaling this radius to one, all errors required by Lemma 5.4 are at most \varepsilon_*, and the forcing and first-order metric errors are also at most h_*\eta_0.

As long as the drop alternative holds, the successive physical radii and gradient amplitudes are

The forcing scales by \rho_j/M_j, which decreases because r_*<k_*. The geometric errors decrease with \rho_j. In a reflected patch the plane density is recomputed from Equation (5.6) with the newly normalized gradient; it is again O(\rho_j). Thus the class is preserved at every drop, including at centers close to, but not on, the plane.

If drops continue indefinitely, Dz(x)=0, and the constant affine approximations on these balls have errors at most C\rho_jM_j. Otherwise let m be the first index where the affine alternative holds. At physical radius S=\rho_0r_*^{m+1}, divide by SM_m to obtain affine approximation with error h_* on the unit ball and slope in [1/2,2]. Apply Lemma 5.5 repeatedly. At step j the radius is r_2^j and the flatness is h_j=h_*r_2^{j\alpha_2}. The gradient amplitude M_m is kept fixed during this iteration; the spatial dilation alone is changed. The forcing and metric errors decrease as r_2^j, so their ratios to h_j decrease as r_2^{j(1-\alpha_2)}. The total change of the normalized slopes is less than 1/4, preserving the required slope range. At boundary centers every affine approximation is odd.

Choose

The drop scales give M_j\le M(\rho_j/\rho_0)^\alpha. After the first affine scale,

Before that scale the constant approximations have the same last bound. Passing from the discrete radii to arbitrary radii loses only a fixed factor. Every center therefore has affine approximations with error CMr^{1+\alpha}. Comparing consecutive approximations on the same ball bounds their slope differences by CMr^\alpha; comparing overlapping balls gives the same bound between the limiting slopes at two centers. Smoothness identifies each limiting slope with Dz. This proves the first estimate in Equation (5.2), also for the reflected function.

For the second estimate take an affine approximation L_{x,2r} on B_{2r}(x) and apply the exponent-two form of Lemma 5.2 to z-L_{x,2r}. Its source is G plus A^{ij}\Gamma_{ij}^kD_kL_{x,2r}, and its slope is bounded by CM. Scaling that estimate back gives

The connection term converting D^2z to \Hess z has squared integral at most CM^2r^3. This proves the claimed Hessian bound. Restricting the reflected estimate to the original side proves the boundary assertion.□

Proof. Write \mathcal L=-\divg(aD), choose k\ge\max\{1,2C_0/\lambda\}, and set V_s=e^{skZ} for s=\pm1. The exact distributional chain rule is

It is valid for the present weak equation: the exponential is locally Lipschitz and can be approximated uniformly and in W^{1,2} in the test, so the measure term converges as well. The inequality |B\cdot DZ|\le(\lambda/2)|DZ|^2+B_0^2/(2\lambda) and Equation (5.20) imply

In particular F_s is bounded and \nu is a positive measure with the same admissible growth, in addition to a bounded density.

The correction estimate. We record the small correction estimate on B_R:

For the bounded vector term, testing the Dirichlet equation with (w-t)_+ gives, for p>3,

Sobolev’s Inequality then yields

The exponent on the right is greater than one. Iteration, also for -w, gives \|w\|_\infty\le CR^{1-3/p}\|F_s\|_p\le CR.

For the measure term the positive scalar Dirichlet Green function satisfies G_{B_R}(x,y)\le C|x-y|^{-1}. This is the scalar measurable-coefficient Green bound: extend a elliptically to \mathbb R^3, use the fundamental-solution bound in Theorem 3.1 of [14], and compare the Dirichlet Green function by the weak maximum principle. The scalar De Giorgi–Nash estimate supplies the local Hölder hypothesis of that theorem. Dyadic annuli and Equation (5.20) give

This also constructs the needed weak correction without assuming measure solvability. Approximate the restricted positive measure by smooth positive measures, preserving the ball-growth bound. The resulting potentials have the displayed uniform supremum bound, and their energy is bounded by their supremum times the total mass. A weak W^{1,2}_0 limit solves the measure equation and retains the bound. Adding the vector and measure corrections proves Equation (5.23).

Oscillation decay. Let M_R=\sup_{B_R}Z, m_R=\inf_{B_R}Z, and set

These functions are nonnegative weak supersolutions of \mathcal LH_\pm\ge0. Each exponential deficit is bounded above and below by positive constants, depending on kM, times M_R-Z or Z-m_R, respectively. One of the two midpoint sets

has at least half the volume of B_{R/3}. Apply weak Harnack to the corresponding H_\pm. In Theorem 5.2 of [30], take the inner radius R/6: its averaging ball is B_{R/3} and its required nonnegativity ball is B_{5R/6}. Thus the infimum on B_{R/6} is at least c(M_R-m_R). Removing the correction in Equation (5.24) costs at most 2\varepsilon_R. It follows that either the upper endpoint decreases or the lower endpoint increases by c(M_R-m_R)-C\varepsilon_R, whence

Iteration gives any sufficiently small positive exponent below both \beta and -\log(1-c)/\log6.

Boundary reflection. Flatten a boundary face and put S=\operatorname{diag}(1,1,-1). For an even extension of Z, extend a^-=Sa^+S and B^-=SB^+. If q is the third component of the original total flux on x_3=0, the reflected equation has the additional density 2q\,\delta_{\{x_3=0\}}. It is bounded by the prescribed total conormal bound, after the coordinate volume factors are included. For a constant Dirichlet value, extend Z minus that value oddly and use a^-=Sa^+S, B^-=-SB^+ and the odd bulk source. The normal flux is then continuous, with no plane term. The reflected matrix remains measurable and uniformly elliptic. A bounded plane density has ball mass O(r^2), which is allowed in Equation (5.20) because \beta\le1. This proves the boundary assertion.□

Proof. Theorem 5.1 gives a uniform C^{1,\alpha} bound for f and

Coordinate connection terms do not affect this estimate. Apply Lemma 5.6 to Equation (5.26) with \mu=C|D^2f|^2dx and \beta=\min\{2\alpha,1\}. At a flux boundary include the bounded reflected plane density. We obtain a uniform Hölder bound for Z. Smooth composition in the bounded variables now makes the leading matrix and source of the f-equation Hölder. The interior and constant-Dirichlet Schauder estimates give f\in C^{2,\theta} for some uniform \theta>0.

Expand Equation (5.26) in nondivergence form. Its leading matrix is uniformly elliptic and C^{0,\theta}. Differentiating its coefficients produces terms at most quadratic in DZ, because D^2f is now bounded. Thus its right-hand side is bounded by C(1+|DZ|^2). Fix p>3. Linear local W^{2,p} estimates, with either constant Dirichlet or the normal data in Equation (5.27), give on nested patches U_x\Subset V_x\Subset V_x^+

The patch size here is chosen from the already established Hölder modulus of the leading matrix. The normal boundary operator is uniformly oblique; the local a priori estimate is, for example, a smooth-boundary specialization of Theorem 2.3 of [13], followed by localization. Only its estimate is used, not solvability for a pure oblique problem. The boundary norm is controlled by

Indeed, extend the smooth expression for H through the collar, differentiate it there, use the already bounded first two derivatives of f, and apply the trace theorem. This explains the last term in Equation (5.29) without assuming a gradient bound for Z.

We give the absorption of the quadratic gradient term. On balls or half-balls of radius h, the scaled interpolation estimate is

One way to obtain it is to subtract a constant, integrate \int\eta^{2p}|DZ|^{2p} by parts against Z, and use Hölder and Young Inequalities on the terms containing D^2Z and D\eta. At the boundary use a bounded extension operator on a half-ball, bounded also in L^\infty; its lower derivative terms are absorbed by the same interpolation. Scaling gives exactly the second power of h displayed in Equation (5.31).

Write [Z]_{C^{0,\theta}}\le H_0. Cover V_x by radius-h patches of bounded overlap, with their doubles in V_x^+. Take the \ell^p sum of Equation (5.31). There are O(h^{-3}) patches, so

Also, for every \eta>0, \|Z\|_{W^{1,p}(V_x^+)} \le\eta\|D^2Z\|_{L^p(V_x^{++})}+C_\eta on a further uniformly enlarged patch. Such enlarged patches are covered by a bounded number of the U_y. On a fixed smooth finite-domain solution let T=\sup_y\|Z\|_{W^{2,p}(U_y)}, which is finite before making any uniform assertion. Equations (5.29) and (5.32) imply

The radius for the linear estimate was fixed first. Now choose the auxiliary covering radius h, and then \eta, so that the coefficient of T is less than 1/2. This proves a uniform W^{2,p} bound, and hence Z\in C^{1,1-3/p}.

The source in the nondivergence Z-equation is now Hölder; the normal data have the corresponding first derivative regularity. Schauder estimates give Z\in C^{2,\theta'}. The f-equation then gains another derivative. Repeating this argument gives every finite order supported by the data. All coverings and constants were uniform, so this conclusion has the asserted independence from the exhaustion radius.□

Proof. Consider any sequence N_i\to\infty, R_i\to\infty of the indicated solutions. Lemma 4.18 gives, with fixed constants,

Extend each h_i across every white face by the constant b_+, and extend the fixed metric and tensor smoothly into those short collars. These extensions of h_i are continuous; no uniform boundary modulus is required. Let

This is a bounded lower semicontinuous function satisfying the limiting end bound.

Choose b_-<k'<0 so near b_- that the stage-two collar term is nonpositive at every height at most k'. Its positive white part vanishes in this fixed low-height range. Suppose \phi is a smooth lower test of \underline h at x, where \underline h(x)\le k' and d\phi(x)\ne0. For now exclude the black faces. Subtracting a small positive multiple of \operatorname{dist}(x,\cdot)^4 makes contact strict without altering the two-jet. Minimization on a small closed ball supplies points x_i\to x and constants tending to zero for which the translated tests touch h_i from below and h_i(x_i)\to\underline h(x). They cannot touch the boundary of the small ball. They cannot touch a white face or its extended side either, because h_i=b_+>k' there. Thus these are interior contacts, even if x itself lies on a white face.

At these contacts df_i=\tau_i^{-1}d\phi, whence

The positive Hessian matrix in the trace equation gives

Here l_i/(\tau_i\sqrt{D_i})=a_i/|d\phi|. The bound on the collar addition and C\le C_b therefore give

The left side is the expansion of the level set, oriented toward increasing \phi. Removing the fourth-order localization proves the same assertion for a non-strict lower test.

To pass from functions to all supports of a closed set, fix b_-<k<k' and put E_k=\{\underline h\le k\}. Define

The lower relaxed limit of v_j is zero on E_k and one off it. At an equality point this follows from q_j(k)=(1+e^j)^{-1}\to0; off E_k, lower semicontinuity gives a neighborhood whose heights exceed k by a fixed positive amount. Let \psi be a nonzero-gradient lower test of this limit at a zero value. Strict localization gives lower tests of v_j at points y_j\to x with v_j(y_j)\to0. Their original heights are below k' for large j, since q_j(k')\to1. For each fixed such j the smooth inverse q_j^{-1} turns this into a lower test of \underline h with finite nonzero gradient. Apply (6.3) to that inverse test, taking the solution-sequence limit with j fixed, and only then let j\to\infty. An increasing change of defining function preserves the level expansion: the additional Hessian term is a multiple of d\psi\otimes d\psi and has zero tangential trace. Consequently every such test \psi satisfies (6.3).

A smooth local exterior support of E_k has a defining function that is zero at contact and nonpositive on E_k. Shrinking and scaling the neighborhood so its positive values are below one makes it exactly a lower test of the preceding zero–one function. Thus every exterior support has expansion at most k'+C_b. In particular E_k cannot touch a white face: the reversed face is an exterior support with expansion

whereas k'+C_b<0 after k' is chosen sufficiently close to b_-. Equivalently, in a signed white collar with s\ge0 in \Omega, a small multiple of -s is the offending lower test. This rules out boundary layers without assuming that the Dirichlet values persist in the relaxed limit. The end bound makes E_k compact.

Adjoin the entire black region and fill complementary components which contain no designated outer barrier. An exterior support at a black frontier point also supports the smooth black region, so its expansion is at most c_b<k'+C_b. At other frontier points the preceding argument applies. Filling produces no frontier point outside the old closed set; every support of the larger filled set is still a support of that old set. The support bound is therefore preserved. The original required inner collars are covered by the black region, and the outer barriers remain outside. Lemma 4.7 places the resulting set in \mathcal B_{c''} for any sufficiently close threshold c''>k'+C_b.

For a fixed compact Q disjoint from \mathcal B_{c_b}, right continuity gives \delta>0 with \mathcal B_{c_b+\delta}\cap Q=\varnothing. Choose

Then E_k\cap Q=\varnothing for every sequence considered above. Failure of a uniform lower margin would give violating points in Q, a convergent subsequence, and a relaxed limit value at most k there. This contradiction proves the lower assertion. Apply the argument to (-h,-K,-C) for the upper assertion. In the white problem, filling retains all reversed black outer faces. If the white region is empty, a nonempty violating sublevel would still provide the seed for Lemma 4.7; right continuity at the empty region rules out such seeds at arbitrarily close higher thresholds. Multiple components and bounded complementary pockets cause no change.

For the truncation quantifier start with any R_{\min}(N)\to\infty satisfying the end estimates, and enlarge it as needed. Failure for arbitrarily large N and R\ge R_{\min}(N) would provide exactly a sequence excluded above. Only finitely many fixed compact sets will be used below, so their thresholds can be chosen simultaneously.□

Proof. Let s be a fixed outward leaf parameter in a black collar, and let n_s=\nabla s/|ds|. For h_0=b_-+\psi(s) with \psi'>0, evaluate the operator at a comparison contact, where the gradients of the test and solution agree. The floor yields

The trace on the test is exactly

where P_s=\tr_{T_s}K. In particular it is the leaf expansion H_s+P_s, an O(d) error, and the displayed logarithmic-slope term. This calculation differentiates neither the input Z nor an unknown coefficient. For a negative slope the Hessian terms have a replaced by -a.

Since H_s+P_s\ge c_b and C=C_b-k_Bs, choose small positive slopes with \psi(0)=0 and \psi(s)\le k_Bs/2. Lemma 6.1 orders the test below the solution at the outer end of a fixed short collar. The collar addition vanishes in this compatible direction. The residual of the trace equation is at least

Choose a nonnegative smooth \beta_N equal to AN on [0,N^{-1}], supported in [0,2N^{-1}], and with integral at most 2A. Set

A fixed large A makes the final term dominate Cd on the first interval, by (6.5). Beyond that interval k_Bs/2 dominates Cd\le C\ell for large N. A sufficiently small fixed m>0 ensures all height and outer-edge requirements, since the slope multiplier is at most e^{2A}. This is a strict lower barrier with its h-slope in a fixed positive interval.

For an upper barrier use \psi'(s)=M\exp(-\int_0^s\beta_N). A fixed large M makes its height gain dominate the outer-edge height bound and all O(s) variations of the leaf expansion and offset. Its logarithmic-slope term has the negative sign and dominates the O(d) errors initially; the linear height margin dominates them thereafter. At a negative minimum of h-h_0 the Hessian of h dominates that of h_0, their gradients and Z agree, and their implicit t,l,A_\chi agree. Strict height monotonicity contradicts a positive lower-barrier residual. At a positive maximum the signs reverse. This proves comparison and the black estimates. The sign-reversed construction proves the white estimates.

In stage three start at its prescribed face value and use a large positive-slope upper barrier and a large negative-slope lower barrier, both with logarithmic absolute-slope derivative -\beta_N. The directional inequalities in Proposition 4.17 give the initial residual signs. Formula (4.54) evaluates its shifted height at the original face value, so these signs persist along the interpolation. At the unshifted face height all errors are O(s+d); changing the height supplies a margin at least the absolute height change. Large fixed slopes absorb the O(s) errors and order the outer edge, and the same logarithmic correction handles the initial O(d) errors. This proves the absolute upper slope bound without using separation in stage three. Finally combine the own-color lower collar bounds with Lemma 6.1 on the rest of a fixed compact set. The opposite bound on each collar follows from that lemma as well. A finite cover proves the asserted height range.□

Proof. Set W=l\bar\nabla f=l\nabla f/D=\sqrt d\,a e. Its \bar g-norm is at most one, and uW=Lw. For a covector \zeta,

Direct differentiation gives

Both are bounded by P_N(1+\sqrt{\mathcal T}) using Lemma 4.11. In particular their axial Hessian coefficient is d^{3/2}, and their axial dt coefficient is at most (2N+2)\sqrt d; the other Hessian terms are transverse. No factor \ell^{-1} occurs. At a black or white face, respectively, uW_\nu=La or -La, and a\ge1/2 by (6.4). Apply the divergence theorem with the corresponding sign, the collar cutoff, and \phi. Replacing u by \sqrt D gives the second inequality. The formulas also hold at zero gradients by continuity.□

Proof. On the penalty band -\epsilon\le t\le-\epsilon+N^{-1}, l\asymp\ell and D=e^{8(Z-t)}. Thus a\sigma tends uniformly to infinity as Z\to\infty on that band. The penalty weights are bounded on the entire exterior, so a sufficiently large M_0(N) ensures

Outside the penalty band this is immediate. Testing the first stage divergence equation by j_0(Z) has no outer boundary term, since the outer value of Z is zero. The drift term is bounded by

The latter integral is bounded by C_N: the floor and the bounded Z strip bound u there, while K=O(r^{-2}) has finite squared integral on the three-dimensional AF end.

At either color of face, the matching signs give -H+w_\nu(F-P_B)=-(1-a)H. The omitted-drift term has normal component bounded by C\chi\le Cd, and the scalar boundary multiplier lies in [0,1]. Hence, uniformly in the first stage,

Integration and (6.11) give the left side of (6.9), up to fixed positive coefficients, bounded by C_N+C(\tau/\ell)\int_B L j_0. Use (6.7). On the fixed collars the positive minimum of \rho implies

Thus the terms with j_0 absorb for N large, by (6.1). The derivative term is bounded by

Its second summand has a bounded integral on the compact collar and the bounded Z strip. This proves (6.9).

Now constants may have arbitrary fixed-N dependence. The measures dV_\Gamma=\sqrt{1+\sigma^2}\,dV_g and \sqrt D\,dV_g are comparable because \ell\le l\le e. Moreover

The middle bound follows by separating l\sigma\le1 and l\sigma>1, on which a\ge1/\sqrt2. On a fixed compact Q, its positive minimum of \rho and (6.9) control the high-level part of (6.10). On Z\le M_0+1 the floor bounds D, \sigma, and e^{2Z}, so the remaining graph volume is bounded by a constant times the fixed base volume. This is a local weighted estimate, with no global graph-volume assumption.□

Proof. Put d_o=(1+\sigma^2)^{-1}. The matrices A_{d_o},A_d,A_\chi are uniformly comparable at fixed N, without an upper bound on Z. Here |\nabla_\Gamma\omega|=|d\omega|_{A_{d_o}}. Extend a fixed base patch to a compact smooth enlargement and isometrically embed it in Euclidean space by Nash’s Theorem [23]. Its product with the line immerses the ordinary graph isometrically. The base embedding contributes a bounded amount to the graph mean curvature vector. Its mean curvature in the product is

The prefactor is bounded by C_N and d_o/\sqrt d\le C_N. Coercivity therefore gives |H_\Gamma|\le C_N(1+\sqrt{\mathcal T}). The Michael–Simon Inequality [22], in the form with an ordinary boundary integral stated explicitly by Brendle [7, Theorem 1], says

for nonnegative supported \phi. The graph over a compact smooth patch is embedded by the fixed Nash embedding. Enlarging the ambient Euclidean space if needed puts its codimension at least two, as in Brendle’s Theorem. That theorem applies first to \phi+\varepsilon>0; letting \varepsilon\downarrow0 and using \sqrt{a^2+b^2}\le a+b gives the displayed form. The boundary form follows also by extending each smooth immersed graph slightly across its boundary and letting a cutoff collar shrink: its derivative integral tends to the boundary integral and its added curvature integral tends to zero. This is done for each smooth graph before taking the uniform estimate. On an inner face f is constant, so dA=dA_g. Equation (6.8) bounds that boundary integral by C_N\int_\Gamma[(1+\sqrt{\mathcal T})\phi+ |\nabla_\Gamma\phi|]\,dV_\Gamma. Taking \phi=|\omega|^4 and using Cauchy–Schwarz proves

For a compact base cutoff \eta test the scalar equation by q=\eta^2e^{2kZ}/L. Its differential is

The principal exponential term contributes 8k\eta^2e^{2kZ}|dZ|_{A_\chi}^2 against the measure \sqrt D\,dV_g. Coercivity gives

For k\ge k_*(N) the second term absorbs. The drift terms obey

The cutoff terms have the same bounds with a remainder C_Nk e^{2kZ}(\eta^2+|d\eta|_g^2). The penalty is bounded on the entire exterior and contributes to this remainder. Before the last inequality of (6.12), the boundary term is at most C\eta^2e^{2kZ}\sqrt d\le C\eta^2e^{2kZ}. Using (6.8) with \phi=\eta^2e^{2kZ} bounds it by

Young’s Inequality absorbs the \sqrt{\mathcal T} term and the term linear in the gradient; the latter costs C_Nk times the weighted volume. After comparison of the graph measures we have

Its constants are independent of k and R.

For nested base patches U_r\subset U_{r'}, combine (6.13) with (6.14) applied to \eta e^{kZ}:

Iterate with k_j=3^j k_* and geometrically decreasing gaps. The sums \sum k_j^{-1} and \sum j/k_j converge, giving

Increase k_* to at least two. The local L^2 estimate (6.10) and interpolation give S(r)\le C_N(r'-r)^{-A_N}S(r')^{1-1/k_*}. For any \vartheta>0, Young’s Inequality turns this into

Use radii increasing geometrically to a fixed outer patch and choose \vartheta<2^{-A_Nk_*-1}. The resulting series converges. The final remainder tends to zero because each individual smooth solution has a finite supremum on that closed outer patch. The bound obtained is independent of this individual supremum. A finite cover proves the lemma, including at inner faces.□

Proof. It remains to extend the first-stage compact bound to the tail; the tensor K need not be compactly supported. Everywhere, not only at a critical point of t, differentiation gives

Also

In differentiating A_\chi, the derivatives of v have the form 2a d\,H^f(\cdot,e)+2p d a^2dt. Thus every axial H^f(e,e) or axial dt term in u^{-1}\operatorname{div}_g(uA_\chi K(w,\cdot)) has a factor d or \chi. More explicitly its absolute value is bounded by

For the d\log u term this follows from A_\chi K(w,\cdot)(e)=\chi aK(e,e); for the other terms it follows directly from (6.15) and the displayed derivative of the scalar coefficient. Coercivity and Young’s Inequality imply

Since K=O(r^{-2}), \nabla K=O(r^{-3}), and \tau a\sigma=\tau a^2\sqrt D/l\ge(\tau/e)a^2, the second term on the right is, sufficiently far out at fixed N, at most

Indeed the cross term is absorbed by a fixed fraction of \tau a^2 with remainder C_Nr^{-6}, while r^{-4}=o(r^{-3-\delta}) because \delta<1. At an interior maximum of Z above M_0, the principal divergence contributes 4uA_\chi:\Hess Z\le0. Equations (6.11) and (6.16) contradict the scalar equation. The compact upper bound and outer value zero therefore give a global first-stage bound.

In stages two and three the drift is zero and the source is positive above a fixed high level, so an interior maximum there is impossible. At an inner face

Consequently high boundary Z implies t>1, where the boundary multiplier makes the conormal derivative zero. The boundary point principle on the high superlevel excludes that maximum. These maximum principles apply to each smooth solution with its positive definite coefficients, so they require no prior uniform ellipticity bound. Stage four has f=0 and t=Z. Its positive scales have the same high-level argument; at scale zero its homogeneous mixed problem gives Z=0. This is uniform as the scale tends to zero.

Finally D=e^{8(Z-t)} and l\ge\ell give

The height bound was already proved. The finite set of compact separation and collar choices, followed by the polynomial absorption, fixes N_0. Enlarge R_{\min}(N) to contain all fixed patches and the end thresholds. The preceding arguments apply to floor-contact solutions as well.□

Proof. On the bounded variable range l,u,D are bounded above and away from zero and \chi\ge c_N>0. The normalized trace equation is

Its right side is bounded. Theorem 5.1 gives Df Hölder and

The scalar principal coefficient in coordinates is 4\sqrt{\det g}\,u A_\chi^{ij}; the drift is bounded. By the smooth implicit change t=t(x,Z,Df), Dt=t_ZDZ+t_{Df}:D^2f plus controlled coordinate terms. The source, including every homotopy modification, is therefore bounded in absolute value by C_N(1+|DZ|^2+|D^2f|^2). The prescribed inner flux is bounded and the outer value of Z is zero. Reflection contributes at most a bounded plane density, which has mass O(r^2). Lemma 5.6 now gives a Hölder estimate for Z.

All smooth bounded-range coefficients of the trace equation are then Hölder, so its Dirichlet Schauder estimate gives f\in C^{2,\theta}. At a face Df is normal, and the scalar condition solves for

with G_\xi smooth on the bounded range: A_\chi\nu=\chi\nu. The expanded scalar equation has Hölder leading matrix and right side bounded by C_N(1+|DZ|^2). Proposition 5.7 applies. Its interpolation after subtracting a constant uses small Hölder oscillation of Z to absorb the quadratic gradient term in local W^{2,p} estimates. The normal datum’s trace norm is controlled by a lower-order W^{1,p} norm; the derivatives of f entering that datum are already bounded. Subsequent Schauder estimates give every finite order. The data have uniform smooth end-patch bounds, and the large outer spheres have uniform boundary-chart constants. The local supremum-of-patch-norm argument in Proposition 5.7 therefore makes all these estimates independent of R.□

Proof. Take a smooth extension of M and a distance collar radius smaller than a normal injectivity radius and half the separation of distinct boundary components. At a face use the barriers given by its value plus or minus

At slopes q=\psi'\ge1, the upper barrier’s axial contribution is at most -ckq/2, whereas its tangential curvature and source cost at most C(1+q). Choose k large, then a collar radius r_0 sufficiently small, and finally B large with Br_0\ge1 and \psi(r_0)>2\operatorname{osc}f. Then q\ge1/(2kr_0) throughout the collar. The lower barrier has the reverse sign. It suffices to check their residuals at possible comparison contacts, where their heights are in the fixed solution range; height monotonicity gives comparison. These barriers exclude a positive maximum of

with one endpoint on \partial M. Their value at r_0 excludes the other edge of this two-point domain.

At a positive interior maximum, the endpoint gradients have common length q=\psi'(r) and are parallel transports along the short joining geodesic. Simultaneous endpoint variations in parallel transverse directions, using the second variation of distance, give

The transverse eigenvalues of both operators are exactly one. Separate axial variations give f_{e_xe_x}(x)\le\psi''(r) and -f_{e_ye_y}(y)\le\psi''(r). Therefore

which contradicts the choices of k and r_0. No derivative or continuity modulus of b_x-b_y was used. Interchanging the points and then letting their distance tend to zero yields the global Lipschitz bound |df|\le\psi'(0).

Interpolate the normalized equation by A_s=(1-s)I+sA_b and Q_s=(1-s)c_*f+sQ. The axial lower bound, source growth, constant barriers, and strict height sign persist. The preceding gradient estimate is uniform in s, so the equations are uniformly elliptic. Theorem 5.1 first gives C^{1,\theta}, then Schauder gives C^{2,\theta}. On this range Df and the prescribed Z are Lipschitz; the undifferentiated coefficients therefore have bounded C^{b'} norms. Schauder upgrades to C^{2,b'} and one further differentiation gives C^{3,b'}. The linearization has positive definite principal matrix and negative zeroth-order coefficient -\partial_fQ_s. The Dirichlet maximum principle and linear elliptic theory give its inverse. The implicit function theorem supplies openness of continuation from \Delta_g f=c_*f; the uniform estimates and compactness supply closedness. Comparison at a maximum of the difference proves uniqueness, because the gradients agree and the height source is strictly increasing. The implicit function theorem, or compactness and uniqueness, gives the asserted dependence and uniform bounds.□

Proof. The implicit definition (4.15) is smooth and strictly increasing in t at fixed df. The height estimates use only the trace equation, including its strict height sign, so they hold without a floor. For bounded Z and \sigma\to\infty, necessarily t\to-\infty, p=N, and l=e^{Nt}. The exact relation is

Uniformly on bounded Z ranges,

Since N\ge4, the axial factor is at least c/(1+\sigma). After division by l/\sqrt D the source is

All collar terms are bounded on the height range, and Q_f=(\sqrt D/l)\tau(F_\xi)_h>0. The principal matrix is independent of f itself. The normalized interpolation to \Delta f=\tau f retains these conditions and the constant barriers. Apply Proposition 6.8.□

Proof. The fixed set in \overline{\mathcal O} is compact by the joint compactness and continuity, and it lies in \mathcal O by hypothesis. For each parameter, surround its compact fixed-point set by a bounded open neighborhood whose closure is contained in all sufficiently nearby sections. All nearby fixed points of \overline{\mathcal O} lie in this neighborhood: otherwise a sequence outside it would converge to a fixed point at the original parameter outside the neighborhood. Excision identifies the section degree with the degree on this one neighborhood, and ordinary homotopy invariance makes it constant locally. At a parameter without fixed points the same compactness argument makes all nearby degrees zero. A finite covering of the parameter interval proves the assertion. These are the compact-perturbation degree and excision properties of Leray–Schauder [19].□

Proof. Choose N_0\ge\max\{4,\lceil2/\epsilon\rceil\} large enough for Proposition 4.20 and all the preceding estimates. Enlarge R_{\min}(N) to tend to infinity and to enforce the outer height and gradient barriers. Explicitly the outer slope has the bound \sigma\le C_{\mathrm{out}}\tau^{-1}R^{-1-\gamma}. It suffices also to require

At outer Z=0 the implicit expression for Z evaluated at t=-\epsilon is then strictly negative, so its monotonicity excludes an outer floor contact. The choices may depend arbitrarily on fixed N; there is no restriction requiring a polynomial outer radius.

Fix N,R and let

For an input Z, solve f=S_\xi(Z) by Corollary 6.9 and compute all auxiliary variables and source terms from this pair. Denote the frozen drift summand by D_\xi, the full frozen bulk source by E_\xi, and the prescribed inner flux by q_\xi. Define H_\xi(Z)=\widetilde Z by

Only the displayed principal gradient uses the output. On bounded input sets, 4uA_\chi is positive definite and C^{1,b'}, the drift is C^{1,b'}, the bulk source is C^{0,b'}, and the boundary datum is C^{1,b'}. In particular Dt involves DZ,D^2f, and no second derivative of the input Z. The nonempty outer Dirichlet face gives the Poincaré Inequality, so the mixed linear problem is uniquely solvable by coercivity. The inward-normal convention changes only the sign of its boundary functional. Linear regularity gives \widetilde Z\in C^{2,b'}. The compact embedding into X and continuous dependence make H a continuous compact homotopy on bounded sets, including its concatenation points. Its fixed points satisfy the original equations and bootstrap to smoothness.

After the trace solve set

This is relatively open because the trace solve and implicit change are continuous. Choose M larger than the fixed-point bound supplied by Propositions 6.6 and 6.7, using their higher estimates if necessary to reach the chosen exponent b'. A fixed point in \overline{\mathcal O} is smooth and has t\ge-\epsilon, so these estimates exclude its norm boundary. Equation (6.19) and Proposition 4.20 exclude its floor boundary. Thus Lemma 6.10, with the parameter interval rescaled, applies. Notice that arbitrary inputs need not satisfy the floor, and the map need not preserve \mathcal O_\xi.

At \xi=4 the unique trace solution is f=0, the drift vanishes, and both bulk source and inner flux in (6.20) are zero for every input. Its output is identically zero. The zero input has t=0> -\epsilon and lies inside the norm cutoff. Hence the final degree is one, and so is the physical degree. A physical fixed point exists in \mathcal O_0. The asserted radius-independent local bounds are Proposition 6.7.□

Proof. Apply Proposition 6.11 to any R_i\to\infty. Uniform classical estimates and a diagonal subsequence give smooth convergence on each compact subset, including the fixed inner faces. The equations and boundary conditions pass to the limit, as does t\ge-\epsilon. We prove the end bounds already on the truncations so that their normalization does not escape to infinity.

Where C=0 and \tr_gK=0, the trace equation is

Indeed \tr_{A_\chi}K=-(1-\chi)K(e,e) and

At fixed N the equation has uniformly elliptic principal coefficient, bounded drift, and c_f\ge\tau/e>0. For a sufficiently small c_N>0, the positive function \exp[-c_N(r-r_0)] is a supersolution of its negative-principal operator beyond a fixed large r_0: the positive height term dominates the O(c_N^2) radial and O(c_N/r) tangential Hessian terms and the bounded drift term. Multiply it to dominate the inner sphere data; it also dominates the zero outer data. Comparison for both signs gives exponential decay uniformly in the truncation. The already uniform smooth coefficient bounds, followed by the homogeneous linear estimates for (6.23), give exponential decay of every fixed derivative. The corresponding zero-Dirichlet estimates apply near the outer spheres. Thus Z-t and graph errors are exponentially small with derivatives.

Using tracefreeness of K on the tail, the quadratic form is now

and A_\chi=I+O(e^{-c_Nr}), u=L(Z)+O(e^{-c_Nr}). The drift uA_\chi K(w,\cdot) is exponentially small as well. After decreasing c_N to absorb polynomial factors, the scalar equation becomes

All fixed-order local derivatives of the exponential error are controlled. The right side is O(r^{-3-\delta}) on the bounded solution range, even before the penalty disappears. Define the increasing smooth function

Its derivative is bounded above and below on that range, and \Delta_g\Phi_N(Z)=O(r^{-3-\delta}). For 0<\delta'<\delta<1,

at large radius. Comparison of both signs, with zero outer values, gives |\Phi_N(Z)|\le C_Nr^{-1} uniformly in truncation, and hence Z=O(r^{-1}).

In the limit, sufficiently far out t> -\epsilon+N^{-1}, so the penalty vanishes. Rescale comparable-radius annuli to unit size. The equation for \Phi_N(Z) has a bounded rescaled source and uniform smooth metric coefficients. Local W^{2,p} estimates first give the scaled gradient bound. Its source is a smooth function of Z times the symbol weights \rho, |K|^2, plus the controlled exponential errors. Schauder estimates then give two symbol derivatives, proving (6.21). Since L(0)=1, u=1+O(r^{-1}), and the graph and drift errors are exponential, the first assertion of (6.22) follows. The divergence u\Xi is integrable: its polynomial tail terms are O(r^{-4}) and O(r^{-3-\delta}), and its remaining terms are exponential. The divergence theorem on annuli proves the flux limit.□

Proof. At the physical endpoint F=h+C and w(h)=\tau a\sigma. The scalar identity (4.32) therefore gives

The physical height range from Lemma 6.2, |w|\le1, and the preparation inequality for C,\rho make this nonnegative. That preparation is needed only on the compact support of \tr K and dC; on the rest of the end it reduces to \rho\le8\pi(\mu-|J|). The boundary identity and prescribed flux give H+4\partial_\nu t=-N_B. Since f is face-constant,

The pointwise inequality \widehat g\ge e^{-4\epsilon}g proves completeness, with the compact smooth boundary included.

Proposition 6.12 gives the metric decay. The conformal formula and exponential graph error give

Equation (6.24) gives \Delta_g t=O(r^{-3-\delta}), proving the required scalar falloff. Exponential graph errors have zero ADM contribution. The conformal ADM formula and t=O_2(r^{-1}) give

Replacing the normal and area by their Euclidean versions costs o(1), and the O(r^{-3}) error in (6.22) has vanishing sphere integral. This gives the stated normalization and sign.□

Proof. The inward normal on B gives

Both integrals are finite by the end estimates. At a physical face, V_\nu/u=-(1-a)H-N_Bd\ge-Cd. The signed slopes yield \sqrt d\le C\tau/\ell, so its negative integral is at most C(\tau/\ell)\int_B L. Apply (6.7) with a constant test and a fixed collar cutoff. On those collars \rho has a fixed positive minimum, and therefore this loss is at most

For N large it absorbs in one quarter of those positive terms.

On the penalty band l,L\asymp\ell, with constants independent of N, and the exact definitions give

Thus, increasing only a polynomial constant,

The three remaining weight integrals are finite. Their end orders are r^{-3-\delta}, r^{-6-2\delta}, and r^{-4\gamma}, respectively, with 4\gamma>3. After inserting both estimates in (6.25), positive fractions of all three bulk terms remain. Dropping them proves the bound. Finally \ell^2/\tau=\tau/\ell=\ell^{1/2}, which decays faster than every fixed polynomial in N.□

Proof. For each fixed large N, the metric of Lemma 6.13 has a strictly mean-negative inner boundary and strictly mean-positive large coordinate spheres. Apply the three-dimensional trapped-region theorem with zero tensor to obtain its full outermost minimal enclosing frontier. It is a compact smooth surface, possibly disconnected, and bounds the connected exterior containing the end. It is outer area-minimizing: its minimizing enclosure exists by the direct method between the fixed barriers; minimal-obstacle regularity and the maximum principle make that enclosure a smooth minimal bounding frontier. A strict enlargement would contradict maximality of the full minimal trapped region. These are also the outermost/outer-minimizing horizon conventions in the Riemannian Penrose Theorem [5, 6].

Let this full cut be \Gamma_N. Its exterior is complete with boundary, has one AF end of order one through two derivatives, and has nonnegative scalar curvature with pointwise falloff O(r^{-3-\delta}). These satisfy the boundary version of the Riemannian Penrose Theorem [6, Theorem 1.4]; no extension across the original boundary is needed. Every cut in the prepared exterior is a full enclosing cut of the original flat obstacle. The pointwise metric inequality and the two-dimensional area scaling therefore give

Lemmas 6.13 and 6.14 imply

First the end was exhausted at fixed N. Now let N\to\infty with \epsilon fixed, and then let \epsilon\downarrow0. Only numerical estimates are passed to the last two limits; no smooth limit of the comparison metrics as N\to\infty is used. This completes Theorem 4.1.□

Proof. Let \chi_R equal one on r\le R and zero on r\ge2R, with uniform differentiated bounds in \log r. Interpolate the metric to b by q_R=b+\chi_R(g-b), keeping the original metric on the compact part. Let P=K-(\tr_gK)g. On the exterior of a fixed sufficiently large sphere r=L, solve

The vector Laplacian acts on one-forms. On this fixed exterior the Ricci endomorphism is bounded above by a negative multiple of the identity, uniformly for large R. The source has C^{0,\alpha}_{\tau'} norm O(R^{\tau'-\tau_{\rm data}}). Kato’s Inequality compares |X_R| with the scalar operator -\Delta+2+o(1). For a radial power,

The coefficient is positive for a<1+\sqrt3, so r^{-\tau'} is a common supersolution after enlarging L.

For clarity, existence in this argument follows on finite annuli from the coercivity of -\Delta-\Ric with zero Dirichlet values. The maximum comparison just given and local boundary estimates are independent of the outer annulus radius. Dirichlet exhaustion, followed by local Schauder estimates, gives

On r>2R the metric is exactly b and the source vanishes. The same comparison with r^{-\kappa}, and differentiation on hyperbolic balls, gives X_R=O_\infty(r^{-\kappa}) there, with constants allowed to depend on R.

Choose a fixed cutoff equal to zero near r=L and one beyond a slightly larger sphere, and add the cutoff of

to \chi_RP. Denote the resulting stress by P_R, and let K_R be its inverse trace reversal in q_R. The identity

shows that

where E_R is supported in a fixed annulus and tends to zero in every needed compact norm. The data are unchanged near S. The family K_R is bounded by a common smooth weight of order -\tau', and its exact tail has order -\kappa.

Here is the error estimate needed for the energy condition. Let w_0>0 be smooth and equal to r^{-3-\delta} on the tail. There is a number e_R\to0 such that

Indeed the expression with Q_{q_R}(K_R) replaced by \chi_RQ_g(K), and with the last norm in g, is nonnegative by the original dominant energy condition. Its errors are the fixed-annulus term E_R, products involving X_R and K, the quadratic cutoff error (\chi_R^2-\chi_R)Q_g(K), and the change of norm between g and q_R. The noncompact terms are bounded by a coefficient tending to zero times r^{-2\tau'}. Since 2\tau'>3+\delta, these are o(1)w_0. The fixed-annulus terms have the same conclusion by compact positivity of w_0. This argument uses no extra pointwise decay of the original density \mu.

Let D\ge |K_R|_{q_R} be a common smooth positive weight of order -\tau', and let w_2>0 be a smooth weight equal to r^{-\tau'} far out. Choose \eta_R\to0 sufficiently slowly that e_R+\varepsilon_R=o(\eta_R). Solve

The scalar forcing satisfies

by the differentiated cutoff estimates above. Here and below \nu at an inner boundary points into the exterior. This is a scalar equation with a bounded first-order drift and a positive zeroth-order coefficient. A complete continuation argument is as follows. On a finite truncation impose zero outer values and multiply the right side and the inner normal derivative by a parameter a\in[0,1]. Subtract a\eta_R times a fixed collar function with normal derivative -1; the remaining function has homogeneous Neumann data. At its extrema the gradient term is bounded by the fixed collar derivatives and the regularizing source. The Neumann maximum principle therefore gives a uniform O(\eta_R) bound. On the end, r^{-\tau'} is a supersolution for the equation with any of its bounded decaying drifts, because 3+2\tau'-(\tau')^2>0. Hence

The estimates of Lemma A.1, first absorbing the bounded first-order term by interpolation and then applying the one-sided Schauder estimate, give closedness of continuation. Its linearization has positive zeroth order and the same mixed boundary conditions, so the maximum principle and linear elliptic theory give openness. Exhaustion now gives the displayed solution. On the exact tail the sources decay faster than r^{-\kappa}; comparison improves the bound to that power. The regularizer w_2 bounds all differentiated coefficients on rescaled balls. Bootstrap gives the stated estimates of all orders at fixed R.

Set g_R'=e^{4v_R}q_R, K_R'=e^{2v_R}K_R. Direct conformal calculation gives

The norm in the first equation contributes another factor e^{-2v_R}. Equations (7.5)–(7.8), D\ge|K_R|, and 2\tau'>3+\delta imply strict dominant energy with a positive multiple of \eta_Rw_0. At the boundary

Moreover, the four terms following \chi_R(R_g+6) in Equation (7.8) are O(\eta_R)w_0.

We include the mass argument because uniform metric convergence alone would not prove it. The original constraint and the weighted integrability assumption imply \int V_0|R_g+6|\,dV_g<\infty. For any static potential V_a, its scalar-linearization flux has divergence V_aDR_b(g-b). The difference between DR_b(g-b) and R_g+6 is quadratic in the metric error and its first two derivatives. Its weighted radial integral is bounded by

Equation (7.8) makes the curvature tail uniformly small, and the quadratic error has the same uniform integrable bound. Compare the fluxes on a fixed large sphere, where the data converge, and then send that sphere to infinity. Since |V_a|+|dV_a|\le C V_0, this proves convergence of all four components. It also proves their existence for each approximating metric. The improved tail bounds and the constraints give all the weighted integrability requirements for the new data. If the original data have only the stated finite regularity, the constructed data are already smooth outside a compact set: there the vector equation is homogeneous on b, and all coefficients and sources of the scalar equation are smooth after bootstrap. On the remaining compact set, smooth the metric in C^2 and the tensor in C^1, with a cutoff leaving that smooth tail unchanged. The strict energy margin and strict boundary expansion have positive minima in absolute value on their respective compact sets. Continuity of scalar curvature in C^2, momentum divergence in C^1, and boundary expansion in these norms therefore preserves both inequalities if the smoothing error is sufficiently small. Choose that error to tend to zero with R. This gives smooth approximants with unchanged asymptotic estimates and mass components.

Finally, uniform bilinear ratio convergence bounds every cut area between factors tending to one, and therefore bounds their infima by the same factors.□

Proof. First smoothly replace r^2 by a constant on a sufficiently large compact set. On a fixed compact region, taking A_0 large makes all derivatives of the logarithm of the resulting square root small, proving Equation (7.10) there. For the exact hyperbolic metric on the tail, the tangential and radial Hessian eigenvalues divided by \lambda are

Each is at least

Their gaps are positive multiples of r^{-2} for A_0>1. Also

The errors from g-b=O_2(r^{-\kappa}) are smaller than these gaps because \kappa>2. This proves the assertions after enlarging the fixed compact transition. The last smallness condition follows by making s=0 throughout a sufficiently large compact set: on its complement r^{-\kappa}=o(r^{-(3+\delta)/2}).□

Proof. We first derive the graph identity, keeping the full spatial dependence of l. On a product with coordinate s, consider the stationary Lorentz metric

The s-slices have induced metric g, shift \beta=l^2\nabla f=Uw, and lapse U=l\sqrt D. For the future normal U^{-1}(\partial_s-\beta), with the convention k(X,Y)=\mathbf g(\mathbf\nabla_X n,Y), stationarity gives

The contracted Gauss equation and the normal Ricci equation for these stationary slices give

For completeness, this sign can be fixed directly by combining

with the normal expansion identity

Indeed n(\operatorname{tr}k)=-U^{-1}\beta(\operatorname{tr}k) by stationarity, and \operatorname{div}_g\beta=-U\operatorname{tr}_g k. Substitution is exactly Equation (7.22).

In the coordinate s-f, the same spacetime metric is -l^2\mathrm d(s-f)^2+\overline g, so

The inverse metric and volume element of \overline g give

Differentiating U also gives

It follows from Equation (7.22) that

Let

Since the symmetric part of \nabla\beta is -Uk,

Using p_k=p_K-p_Q and R_g=16\pi\mu-6+[K,K], we obtain

The conformal scalar-curvature formula in three dimensions is

Replacing U by u=e^{2t}U in the first divergence in Equation (7.23) contributes 2p_Q(w,\mathrm dt) after division by two. For the conformal Laplacian term, the equality \mathrm d\log(e^{2t}l)=(p+2)\mathrm dt+A yields

The cosmological terms are -3+3e^{4t}. Consequently, if V_{\mathrm{pre}}=u(p_Qw+4A_d\nabla t), then

We next incorporate the trace equation into the flux. Direct differentiation gives

Since u=e^{2t}l\sqrt D, these identities imply

One way to see the cancellation of A is to first obtain

and then use \operatorname{tr}_gS =\operatorname{tr}_gK+\operatorname{tr}_gH^f+2A(w) =F+(1-\chi)q_1. Define V=V_{\mathrm{pre}}+uwF. The divergence of the added term contributes

to Equation (7.24).

We verify the resulting quadratic expression explicitly. Write z=\operatorname{tr}_{e^\perp}P=F-\chi q_1. The three blocks of Q are P, M+pa y, and q_1+2pa x_t, so

and

Adding the terms in Equation (7.24) and Equation (7.26), and setting aside -F\operatorname{tr}_gK, gives

Completing the x_t-square leaves the coefficient

in front of q_1^2. Completing the F-square then gives exactly Equation (7.19). Equivalently, \mathcal T is the coordinate-free expression

The apparently directional terms in this last expression are smooth: (1-\chi)q_1=((p+4)/(4+pv))S(w,w) and a x_t=\mathrm dt(w). This proves the asserted extension through zero gradient.

Finally, Equation (7.25) gives

Using (1-\chi)(4+pv)=(p+4)v, the preceding definition of V reduces to

This proves both the stated flux formula and Equation (7.21).□

Proof. For a fixed spatial function \phi, write

Differentiating A_d=I-w\otimes w with Equation (7.25) gives

where the drift vector is

To check every contraction, the coefficient of \nabla\phi before substituting S is

Now

which proves Equation (7.32).

In applying this computation to \nabla_xG(t(x),x), there is one additional differentiation in t. The defining integral gives

It follows that

Thus the factor -4 in the primitive correction cancels exactly the prescribed 4\langle A,\mathrm dt\rangle_{A_d}. The A-part of \mathfrak q produces E_G. The remaining \mathrm dt-dependent terms are

Use

These terms become

as asserted. The last correction in V_* contributes 4\Delta_gG_c/u=4\sqrt d\,\Delta_gG_c/(\lambda L). This establishes Equations (7.29)–(7.31). On a collar with spatially constant \lambda,n, both \nabla_xG and \nabla G_c vanish. The boundary assertion follows.□

Proof. The first two terms in Equation (7.19) are nonnegative. The last two terms, keeping the X_1-square intact, combine to

Completing their square gives the exact identity

Since (9-d)/12\ge2/3, the result follows.□

Proof. We first show that the nonlinear terms have no mass flux. Since D=(1-|w|_g^2)^{-1}, the graph correction is exactly

Consequently, with differentiated bounds through order one,

For the four background static potentials,

Thus a metric error O_1(r^{-j}) contributes at most O(r^{3-j}) to its flux. The assumptions in Equation (7.35) make all three contributions in Equation (7.40) tend to zero.

Write \mathbb U_b(V,e) for the one-form

appearing in the definition of the metric mass. In three dimensions,

Indeed \operatorname{div}_b(4tb)=4\mathrm dt, \operatorname{tr}_b(4tb)=12t, and (4tb)(\nabla_bV,\cdot)=4t\,\mathrm dV. Combining Equation (7.41) with Equation (7.40) proves the asserted comparison of the finite-radius flux integrals.

We establish existence of the limiting fluxes before taking their difference. Put \widehat e=\widehat g-b and \nu=\min(\kappa,s,2\beta)>3/2. Then \widehat e=O_2(r^{-\nu}). The scalar-curvature linearization at the curvature -1 background is

Since \nabla_b^2V_a=V_a b and \Delta_bV_a=3V_a, differentiating the defining one-form gives exactly

The Taylor remainder in scalar curvature has the estimate

The constant is uniform on a sufficiently distant tail, where \widehat g and b are uniformly equivalent. As

the scalar-curvature contribution and the Taylor remainder in the right side of Equation (7.42) are absolutely integrable there:

The divergence theorem on annuli therefore makes each family of flux integrals a Cauchy family. All four limits exist and are finite. Taking the difference from the given fluxes of g now proves Equation (7.37).

It remains to identify the corrected vector field with the time potential. On the ultimate tail, set

The constant n is fixed here, and

Taylor expansion at zero gives

On the same tail A=\mathrm d\log\lambda, so

Substituting these bounds into Equation (7.20) and Equation (7.28) yields the explicit error ledger

All three terms are o(r^{-2}) by Equation (7.35). In particular their integrated normal fluxes tend to zero.

Finally,

Replacing V_0 by \lambda in the conformal flux integral costs O(r^{1-s})=o(1). Replacing \nu_b,\mathrm dA_b, and the background gradient by their g-counterparts costs O(r^{3-s-\kappa})=o(1), since s+\kappa>3. Equation (7.39) follows from Equations (7.37) and (7.38). The proof treats the four static potentials separately; it does not assume a causal character for the mass covector of \widehat g.□

Proof. Since D=(1-|w|^2)^{-1}, we have Z-t=O_2(r^{-2\beta}). The assumed end equation therefore gives

First apply the conformal scalar-curvature formula to e^{4t}g:

The gradient term is O(r^{-2s})=O(r^{-3-\delta}). The difference \widehat g-e^{4t}g=e^{4t}D\,w^\flat\otimes w^\flat is O_2(r^{-2\beta}). The local scalar-curvature formula, on the uniformly equivalent end metrics with bounded curvature, then gives R_{\widehat g}-R_{e^{4t}g}=O(r^{-2\beta}). This proves the required decay.□

Proof. At an interior extremum of f, w=0, so A, D_0, and D_1 contribute nothing to the trace equation. In stage three it reads

In the first two stages the same formula has \zeta=0. Since A_R\ge1, the maximum principle and the boundary values give

No derivative of a collar correction has been used. The necessary correction amplitudes are uniformly bounded: on a black face H=c_b-\tr_TK, and the white face has the analogous bound. All supports can be kept in a fixed compact enlargement by shortening the individual collars.

Take r_0 outside that enlargement and the fixed lapse transition. Compare f with

on a truncation at r=T. At a contact the axis is radial. Put k=\beta+\partial_{\log r}\log A_R, so \beta\le k\le\beta+1<4 and its derivative is nonnegative. The Hessian term, together with the radial part 2\chi A(w) coming from s=d\log\lambda, has upper bound

for the positive decreasing test. This is bounded above by a fixed negative multiple of a, uniformly for 0\le\chi\le1. The derivative of k has favorable sign.

The only additional spatial-profile term occurs where n varies. Positive t helps, because pt\,d\log n points outward. For negative t put X_0=-nt. The adverse term is at most C\chi aX_0. If X_0>C_1\log(r/\tau), then l_0=e^{-X_0} and the tested gradient gives

by taking the fixed C_1 large. In the complementary regime \chi a\le C/\sqrt n, so

uniformly on the varying region because q>2\beta. There the shift at T is negligible after T is sufficiently large at fixed profiles.

The tensor term is O(r^{-\kappa}), dominated by the positive test height since \beta<\kappa, except near T. Near T>2R, n is constant, and on the test a\ge c_Nr^{-\beta} whenever the tested slope is small: \lambda/A_R and l_0 have positive lower bounds at fixed profiles. Thus the negative Hessian contribution dominates the tensor term there as well. For larger slope it is stronger. The negative test gives the reverse inequality. Choose B_1 to dominate the already proved bound on the fixed sphere r_0; this uses no collar derivatives. Height monotonicity proves Equation (7.48). At the outer face the barrier derivative tends to zero in the quantity l|df|, at fixed profiles. The relation Z=t+\log(1+l^2|df|^2)/8=0 then excludes t=-\epsilon, once T is sufficiently large.

We now specify the compact choices without circular dependence. Use this common B to choose a compact set Q outside which Br^{-\beta}|\tr K|\le\rho. This is possible because \beta+\kappa>3+\delta. Choose the two negative thresholds sufficiently close to zero, then their collars, and finally sufficiently small offset amplitudes, so that

The constants controlling derivatives of the now fixed collars need not be uniform in this choice. The next lemma proves precisely the required physical height interval on Q. Outside Q the end bound proves Equation (7.49), after making the support of C lie in Q. Finally enlarge the fixed constant-n region so that Equation (7.48) rules out heights near b_- and b_+ beyond it. This radius is fixed before N and R.□

Proof. We verify the extension of Lemma 6.1 rather than assume an asymptotically flat hypothesis. A putative failure supplies N_i\to\infty and outer radii tending to infinity. On the relevant compact range A_R=1,n=N, and l/\tau\ge c\ell/\tau=cN^{B_0/4}\to\infty. At a fixed nonzero-gradient lower test \phi of a lower relaxed height limit,

The last term is the only new trace term on this compact range, where A=s is bounded. The limiting inequality is therefore

at heights below k', with b_-<k'<0 close to b_-. The low-height D_0 term is nonpositive. The lower relaxed passage, increasing logistic reparametrizations, and Lemma 4.7 now apply exactly under the hypotheses established in Lemma 6.1. In particular each height is extended constantly as b_+ across a white face before taking the relaxed limit. Low test contacts are therefore interior, even if a boundary layer is present. The reversed white face has expansion -c_w>0, and excludes contact of the resulting low closed sublevel set. The end bound makes that set compact. After adjoining the full black region, right continuity of the black threshold family excludes its meeting any compact set away from that region. Reversing signs proves the upper separation, including the empty-white-region case.

On an actual boundary collar A=0, n=N, A_R=1, and \lambda=\lambda_0 is constant. For a test h=b_-+\psi(s) of fixed positive slope,

The trace residual is the leaf expansion plus O(d) and the term a\chi|ds|(\log\psi')'. Use

where \gamma_N is a fixed large multiple of N on [0,1/N], vanishes beyond 2/N, and has bounded integral. The first term dominates O(d) on the short interval. Outside it the offset margin is a fixed multiple of s and dominates d when B_0>2. Compact separation orders the lower barrier on the far end of the collar; fixed large slopes order the upper barrier. The third-stage directional inequalities give the same upper slope barriers of both signs. This verifies all hypotheses and all changed small parameters in the proof of Lemma 6.2; it does not require the special flat choice \tau=\ell^{3/2}.

For the trace inequalities use the signed field \lambda_0Lw, whose boundary flux has magnitude at least \lambda_0L/2. Direct differentiation on these collars gives

With the L weight removed, the same formula has only dp\,a x_t in its gradient term. Coercivity of Equation (7.15), equivalently Lemma 4.11, bounds these expressions by CN^m(1+\sqrt{\mathcal T}). The gradient action is bounded by |d\phi|_{A_d}, and A_d\le(1+N/4)A_\chi. Integration with fixed collar cutoffs proves Equation (7.50). No inverse power of \ell occurs in this calculation.□

Proof. At an interior contact dt=0,\Hess t\ge0. The Gauss equation of the graph in the Riemannian warped product g+l^2ds_0^2 gives

where

To obtain this formula, the spatial lapse derivatives contribute w\otimes A+A\otimes w to the graph second form, and its vertical connection contributes d^{-1}A(w)w\otimes w. The remaining p\Hess t lapse curvature and conformal Hessian terms are nonpositive at a contact. The rank-one term changes \mathcal S by only vA(w)\tr_\perp(S-K).

We give the derivative ledger that controls the additional terms. At a floor contact, p=n, d_xp=dn, and

Positive-order derivatives of \log n are bounded. Thus \Hess_xl/l, A, and the needed spatial derivatives have fixed-degree polynomial bounds in n, uniformly in position. The factor \epsilon<1 introduces no larger growth. Moreover

at contact. Therefore the derivatives of A_\chi always carry a d factor in the uncontrolled axial Hessian direction. In d\log u, that direction contracts with \chi in the flux. The same is true for the new drift A_\chi(A(w)w)=\chi A(w)w. A spatial derivative of A in this drift is contracted axially and retains \chi.

The algebraic floor comparison of Lemma 4.13, applied to S, and Equation (7.51) therefore give, with universal c>0, small prescribed \eta>0, and fixed finite powers m, the bounds

Here |K|^2+|\nabla K|^2\lesssim\rho_0, and the height estimate bounds the squared end height by the same weight. For example a|\nabla K| is bounded by v+|\nabla K|^2. The derivative of h=\tau A_Rf gives the displayed positive term and h\,w(d\log A_R), whose absolute value is at most Ca|h|\le C(v+\rho_0) for this floor estimate. Derivatives of D_0,D_1,b,C are fixed compact coefficients; the derivative in h is nonnegative and at least one. These observations enumerate all new spatial terms beyond the flat floor calculation, including d_xp, which is not zero here.

Combining these bounds with Equation (7.18) and the constraint gives

for fixed sufficiently small \delta_0. The cosmological constants cancel in deriving this lower bound: 8\pi\mu-R_g/2-3=Q_g(K)/2. In the prescribed source the remaining explicit term 3(e^{4t}-1) is negative at the floor, where d_0=\delta_0 and m_0=1. Choose P_0>m+1 and N sufficiently large that its penalty dominates the error and 2\rho+\rho_0. Equation (7.52) then contradicts the scalar equation.

Outer contacts were excluded in Lemma 7.10. At an inner contact j=0, so Equation (7.45) restores the full flux condition. Since A=0 on the collar, its face identity is

It forces \partial_\nu t<0, contrary to a minimum. In the final scalar stage f=0,t=Z. At a floor minimum the unscaled source is negative after the same penalty is enlarged; at positive scale the inner derivative is negative as well. At zero scale the homogeneous mixed problem gives Z=0 by multiplication by Z. This proves separation for all stages.□

Proof. We first prove estimates for all smooth floor-admissible solutions of the continuation, without assuming their existence.

The principal divergence at high levels. Put B=A+p\,dt. Differentiation gives

At fixed profiles all explicit coefficients and their needed derivatives are bounded on t\ge-\epsilon. The derivatives in the axial Hessian direction carry d or \chi, as in the proof of Lemma 7.12; the same holds for p_tdt, because \partial_p((4+p)/(4+pv))=4d/(4+pv)^2. The derivative of A in the drift \chi A(w)w is contracted axially and retains \chi. Coercivity of Equation (7.15) and Young’s Inequality consequently give

for any fixed \eta>0. These statements bound contracted expressions; they do not assert a bound for the unweighted axial entry of H^f.

On a sufficiently high Z level, either t is large or D is large. In the first case 3(e^{4t}-1) absorbs the bounded errors. In the second case

does so. At fixed profiles \tau A_R/\lambda has a positive lower bound on the whole exterior. The floor penalty is bounded by a fixed-profile constant and is absorbed in the same manner. Choosing \eta below a fixed fraction of \delta_0 yields

in the first three stages. The positive c is fixed; the level M_N may depend arbitrarily on all fixed profiles.

The boundary bound before graph comparison. In stage one the compatible face equations give -H+w_\nu(F-P_B)=-(1-a)H=O(d). The normal drift has order \chi, since A=0 on the collars. Thus Equation (7.45) gives a principal normal flux of absolute value at most Cud. The signed slope bound gives

on these fixed collars. Multiply Equation (7.53) by an increasing cutoff j_0(Z), zero up to M_N and one above M_N+1. The outer boundary contributes zero. Apply Equation (7.50) to the inner boundary term. Its nondifferentiated part is at most

which is absorbed by the positive bulk if B_0>4m. The derivative term is absorbed by \int u j_0'|dZ|_{A_\chi}^2, leaving a bounded compact strip integral. We obtain

Only the polynomial native-weight trace constant was used in this smallness choice. In particular no comparison involving \ell^{-1}, no Schauder constant, and no power of n_\infty enters the inequality B_0>4m.

The local bound near inner faces. After Equation (7.54) is established, constants may depend arbitrarily on the fixed profiles. On the ordinary graph \Gamma=g+df^2, the measures dV_\Gamma and \sqrt D\,dV_g, and their inverse gradient metrics, are comparable on a fixed compact patch. Moreover

Thus Equation (7.54), including the bounded low strip, gives a local L^2(dV_\Gamma) bound for e^Z. The graph mean curvature is bounded by C_N(1+\sqrt{\mathcal T}): its axial Hessian coefficient is (1+|df|^2)^{-1}\le C_Nd, while Equation (7.15) controls the transverse and d-weighted axial entries. The graph Sobolev argument of Lemma 6.5 therefore applies, including its boundary term by Equation (7.50). It gives

For explicit control of the scalar test, multiply its first-stage equation by \eta^2e^{2kZ}/L, with compact patch cutoff \eta. The derivative of L contributes (p+2)dt+A-s. Its first term costs a small \mathcal T fraction plus C_N|dZ|_{A_\chi}^2; the spatial term is bounded at fixed profiles and has the same Young bound. To control the drift, write b_D=K(w,\cdot)+A(w)w^\flat, so \mathcal D=A_\chi b_D. On the fixed compact patch, the exact dual-metric identity gives

Thus |\mathcal D\cdot dz|\le C_N|dz|_{A_\chi}. For z=Z, Young’s Inequality bounds the test’s 2k\,dZ term by k|dZ|_{A_\chi}^2+C_Nk. For z=t, use A_\chi\le A_d and coercivity to bound the (p+2)\,dt term by \varepsilon_1\mathcal T+C_{N,\varepsilon_1}, with \varepsilon_1>0 chosen sufficiently small. The cutoff terms obey the same dual-metric estimate. With k\ge k_*(N),

At a face the remaining trace terms have only \sqrt{\mathcal T} and k|\nabla_\Gamma Z| growth. Young’s Inequality absorbs them in the left side of Equation (7.55), at cost C_Nk on the right. This step requires no new small factor \tau/\ell.

The Sobolev Inequality iterates 2k to 6k on nested patches. Interpolation with the bounded L^2 norm closes the iteration: the larger-patch supremum has exponent 1-1/k_*<1, so geometric radius gaps give a uniform smaller-patch supremum. This includes inner faces. Equation (7.53) excludes a higher interior maximum elsewhere, and the outer value is zero. Hence Z is uniformly bounded above in stage one.

In stages two and three there is no drift. The high-level source excludes an interior maximum. At an inner face the upper f-slope bound forces t>1 if Z is sufficiently large. Then j=1 and the conormal derivative is zero, contradicting the boundary point principle. In stage four f=0, and the same argument works at positive scale; at zero scale Z=0 by its energy identity. Consequently

throughout the continuation. On compact sets |df|\le C_N; on the end the same bound holds for \lambda|df|.

Classical estimates. On a hyperbolic unit patch centered at x_0, put c=\lambda(x_0) and f_{\rm rescaled}=cf. Also divide the entire scalar divergence equation, its total flux, and its prescribed conormal datum by the same constant c. In local coordinates its leading matrix is then

The ratio \lambda/c is uniformly bounded above and below on the patch, with bounded logarithmic derivatives. At fixed capped profiles, u/c and \chi have positive upper and lower bounds on the range just proved. The divided drift and source have the same normalization, and division by a constant creates no derivative term. At fixed profiles the trace coefficients are uniformly elliptic and bounded on the range just proved, and its normalized source is bounded. The rank-one estimate of Theorem 5.1 gives a positive Hölder exponent for the rescaled gradient and

After t=t(x,Z,df) is substituted in the scalar equation, its source is bounded by C_N(1+|DZ|^2+|D^2f_{\rm rescaled}|^2). The drift flux and prescribed inner conormal value are bounded. Lemma 5.6 therefore gives Hölder continuity of Z, including the disjoint Dirichlet and conormal faces. The coupled bootstrap of Proposition 5.7 applies: the trace coefficients and source are now Hölder; after solving the trace equation for two derivatives of f, the scalar equation has Hölder leading coefficients and natural quadratic growth in DZ. On a face its normal derivative is a smooth function of x,Z,f,df, because df is normal. There is no second derivative of f or derivative of Z in that boundary function. These checks give every local classical bound, uniformly in truncation.

A trace solution for arbitrary inputs. The preceding estimates apply only to prospective fixed points. To define the compact map, fix a finite truncation and a bounded input Z\in C^{1,b}, without a floor assumption. At large \sigma=|df|, the defining relation for t gives

Since n\ge N\ge4, \chi\ge c/(1+\sigma). The new coefficient A has growth O(\log(2+\sigma)), but its normalized trace contribution has size

All other terms have the same linear growth on the bounded height range. The principal matrix is independent of the height, the normalized source has strict positive height derivative, and all expressions are smooth and direction free on bounded ranges. These are exactly the hypotheses of Proposition 6.8. It gives the unique C^{3,b} trace solution, continuous in the input and continuation parameter. Its constants may depend on this finite truncation; this causes no difficulty in defining the map.

Freeze its coefficients, drift, and source, and solve the linear mixed problem for a new Z, leaving only its principal gradient unfrozen. The outer Dirichlet face gives coercivity. The output is bounded in C^{2,b} on bounded input sets, hence the map is compact in C^{1,b}. The set defined by \min t(x,Z,df)>-\epsilon and a sufficiently large norm cutoff is open in the parameter–input product. Lemma 7.12 and the preceding bounds exclude boundary fixed points. The endpoint map is zero, and zero is admissible. Lemma 6.10 therefore gives a physical fixed point. First take N sufficiently large for the finite list of polynomial absorptions and compact separations; then take each outer radius sufficiently large for all fixed-profile barriers. The a priori bounds are uniform in that outer radius, so a diagonal compact limit solves the physical system with t\ge-\epsilon.

Finally, on the physical system,

Young’s Inequality and the end height bound give

Indeed it vanishes before R. Beyond R, \tau A_R/(\lambda l_0)\ge c\tau/R, so its remainder is at most C(R/\tau)|h|^2; relative to \rho it is bounded by CR^{4+\delta-2\beta}/\tau=o(1). Equations (7.18), (7.49), and M_d>10\rho imply R_{\widehat g}\ge-6, since d_0\le1/2. The face identity gives

Also \widehat g\ge e^{-4\epsilon}g, so it is complete including its boundary.□

Proof. The profiles are fixed throughout this proof. On the ultimate tail n=n_\infty is constant and A=s, and A_R is proportional to r. The height bound of Lemma 7.10 therefore gives \lambda f=O(r^{-\beta}). On each hyperbolic unit patch rescale f by the central value of \lambda. The trace equation is uniformly elliptic there by Proposition 7.13. Its inhomogeneous term at f=0 is O_j(r^{-\kappa}); the remaining zeroth-order term is a bounded coefficient times the rescaled f. The local linear estimates for this fixed smooth solution consequently give \lambda f=O_j(r^{-\beta}), using \beta<\kappa. The equations for w and D give the next two estimates. The same argument holds on the outer truncations with uniform boundary estimates at their zero Dirichlet face.

Here is the scalar expansion before assuming decay of Z. Since t is bounded and n_\infty is fixed, the substitution t=Z+O_j(r^{-2\beta}) has uniformly bounded differentiated coefficients. In the flux,

The tensor part of S has order O_j(r^{-\beta}). In Equation (7.15), the part quadratic in dZ is 4(p+1)|dZ|^2 to error O_j(r^{-2\beta}). Every mixed term between S and dt has an additional factor a: its two possible forms are a\langle M,y\rangle and a q_1x_t. Thus these terms, including their differentiated bounds, have order r^{-2\beta}. The tensor drift K(w,\cdot) has order r^{-\kappa-\beta}. The leading lapse term 4\langle s,dZ\rangle from u^{-1}\operatorname{div}(4u\nabla Z) cancels the explicit lapse term on the right side of the scalar equation.

The capillary term is O_j(r^{-2\beta}). Even before its support disappears, the penalty contributes O_j(r^{-3-\delta})+O_j(r^{-2\beta}), because n_\infty is fixed and all derivatives of t are bounded. Since 2\beta>4+\delta, the resulting equation is

Here p(z)=n_\infty p_*(n_\infty z) and d_0(z) denotes the corresponding composed coefficient. This expansion requires only the bounded differentiated range already proved.

Define the increasing function

On the bounded solution range both \Phi' and its reciprocal are bounded. Writing Y=\Phi(Z), Equation (7.57) becomes

The quotient is positive and bounded away from zero on this range. For \gamma=1, \Delta_g r^{-\gamma}=-r^{-\gamma}+o(r^{-\gamma}); hence a large multiple of r^{-1} bounds both signs of Y by the maximum principle, first on truncations whose outer value is zero and then in their limit. In particular Z\to0, so c(Z)\to3. For any 0<\gamma<3,

whose leading coefficient is negative. The same comparison proves Y,Z,t=O(r^{-\gamma}). Unit-patch estimates for the transformed equation, followed by differentiation and the already bounded coupled coefficients, prove the differentiated versions.

Choose \gamma>(3+\delta)/2. Lemma 7.8 now applies to Equation (7.57) and gives the scalar-curvature estimate, also after differentiation. In particular the penalty vanishes sufficiently far out. Proposition 7.7 applies with s=\gamma>3/2: all the exponent inequalities there follow from \kappa>2 and \beta>2. This proves existence of the four limits and both stated flux identities.□

Proof. Use the primitive of Proposition 7.5. For brevity write

The exact divergence is

where \mathcal R is Equation (7.30). We estimate each of its terms. Every Young constant below depends only on a chosen fixed fraction of the positive terms; it does not depend on N,R, or a classical bound for the solution.

On r\le2R first set aside the additive term 4\Delta G_c in the ordinary-volume divergence. The bounded logarithmic derivatives of n,\lambda give

Also, uniformly in this whole region,

since q>3+\delta. This estimate includes the capped transition, where n\asymp NR^q.

Positive values beyond the transition in t, r\le2R. If t\ge1/n, then p=0,A=s. Integrating the fixed profile on 0\le nt\le1 and differentiating in \log n through order two gives

For example the primitive above 1/n is \lambda\{L_\infty(e^{2t}-1)/2+c(n)\}, where L_\infty>0 is independent of n and |(n\partial_n)^jc(n)|\le C_j/n for j\le2. Division by \lambda L_\infty e^{2t} proves these bounds, including their uniformity for arbitrarily large t.

Put z=e^{2t}\ge1. Direct factorization gives

Since E\le3, the zero-derivative terms supply 36b_c^2. The principal F error obeys

By Lemma 7.6 and d_0=1/2, the available F^2 coefficient is at least 1/3. Thus strict fractions remain in both places. The K term is at most Cb_c|s||K|, and is bounded by a small fraction of the remaining b_c^2, plus C|s|^2|K|^2. The lapse smallness in Lemma 7.3 makes the latter an arbitrarily small fraction of \rho. The errors in Equation (7.60) are bounded by a further small fraction of \mathcal T, plus C/n; the terms containing p are zero. Equation (7.59) absorbs this remainder in \rho.

The nonnegative transition band, r\le2R. For 0\le t\le1/n,

The F term costs \eta F^2+C_\eta/n^2; the K term costs C/n since the prepared K is bounded. For the axial and transverse derivative terms, respectively,

Their remainders are O(1/n). The zero-derivative negative term has the same order; the cosmological term is nonnegative. Coercivity and Equation (7.59) absorb every cost.

Negative t, r\le2R. Write X=-nt\ge0. Then

The first two derivative bounds follow by differentiating \log G=\log\lambda+(n+2)t-\log(n+2) with t fixed. The same Young estimates as above, now with p=n, show that all terms other than a small fixed fraction of \mathcal T cost

This includes the cosmological term, since 1-e^{4t}\le4|t|=4X/n. For X\le C_0(\log N+\log(1+r)), Equation (7.59) absorbs the cost in \rho.

Choose the fixed C_0 larger than the exponents in 1/\tau,\lambda, and the polynomial X^2. For larger X, if v\ge1/2, the ratio of this cost to \mathfrak C is bounded by

which is o_N(1), uniformly in this region. If v<1/2, then d>1/2 and u\le C\lambda e^{-X}. After multiplication by u, the unabsorbed cost is at most C\lambda/n, and its total integral on r\le2R is O(1/N). These two alternatives account for all negative t in this region.

The capped tail, r\ge2R,\ t\ge0. Here n=n_\infty, s_G=A=s, and E_G=E\le E_0, where E_0=\Delta\lambda/\lambda\le3. Set

Since p\ge0, L'/L=p+2\ge2; therefore 0\le c\le b_c. Also L\ge1, so 0\le\gamma_c\le1/(n+2). Combining the primitive and correction terms gives

For the last inequality use the exact expression for E to bound |E-E_0|\le Cv, and 1-\sqrt d\le v.

Split the F coefficient into c+\gamma_c. The c term is bounded by 18b_c^2+(2/9)F^2, and the \gamma_c term costs a small remaining fraction of F^2, plus Cv/n^2. The K term containing c is absorbed using the lapse smallness as before. Its \gamma_c part satisfies

When p=0 there are no further terms. When p>0, necessarily t<1/n and b_1=O(1/n); the two derivative terms cost \eta\mathcal T+C_\eta v/n by the preceding shifted-square estimates. Finally, on this tail,

Thus every remaining Cv/n is absorbed by an arbitrarily small fixed fraction of \mathfrak C, using Equation (7.14).

The capped tail, r\ge2R,\ t<0. Put x=(n+2)|t| and X=n|t|. The zero-gradient potential, including the centering correction, is

If x\le1, the inequalities e^{-y}\ge1-y and e^x\ge1+x+x^2/2 give

Since 3-E_0=O(r^{-2}), this is O(r^{-4}/n), hence o_N(1)\rho because \delta<1. For x\ge1, (e^x-1)/x\ge e-1, so the same linear bound for e^{-4|t|} shows P(t)\ge0 as soon as E_0(e-1)\ge3. This holds on the whole tail for all sufficiently large N.

The difference from the zero-gradient potential is at most Ce^xv/n, because |E-E_0|+|1-\sqrt d|\le Cv. The floor gives e^x\le e^{2\epsilon}e^X\le Ce^X. The F, axial, and transverse terms, after a small fraction of \mathcal T, cost Cv/n; the K term costs a small fraction of \rho, plus Cv/n^2. All these residuals are at most Ce^Xv/n, which is absorbed by \mathfrak C=(\tau A_R/\lambda)e^Xv/\sqrt d. Again the required ratio is exactly R/(n_\infty\tau)=o(1).

We have proved that, away from the penalty, the integral of Equation (7.58) retains positive fixed fractions of u\rho,u\mathcal T,u\mathfrak C, up to an ordinary volume error with total integral o_N(1).

The penalty band. On its support,

with constants independent of N. On n<2N, which is a fixed compact region, \lambda,A_R and the weights are bounded, and l_0\le C\ell. With a fixed polynomial P(N), the definitions imply

For the last inequality use the exact identity u\mathfrak C=e^{2t}\tau A_R l^2\sigma^2 and apply Young before replacing l_0 by its upper bound. Thus the integrated remaining cost on this fixed compact region is at most

once B_0 exceeds the fixed polynomial exponents.

On n\ge2N, compare the penalty’s v term directly with the capillary term. Their quotient is at most

This tends to zero uniformly. To see the order without using n_\infty as an uncontrolled polynomial constant, put y=n/N\ge2. Before capping, \lambda/A_R\le C y^{1/q}; after capping the same bound holds with y=n_\infty/N, since \lambda/A_R\asymp R. The displayed quotient is therefore bounded by

For large N this is decreasing in y\ge2, with maximum O(N^{P_0-3B_0/4}). Increasing B_0 makes it vanish. The \rho_0 term is treated the same way where v\ge1/2, since \rho_0 is bounded. Where v<1/2, use d>1/2: its ordinary-volume cost is at most

The supremum of its final factor for n\ge2N tends to zero, and \int\lambda\rho_0\,dV_g<\infty. This proves the required o_N(1) total penalty loss.

The inner boundary. The correction leaves V_*=V on every inner collar. On a physical face,

Thus the negative boundary integral is at most C(\tau/\ell)\int_{\partial\Omega}L. Equation (7.50), with a constant test and a fixed collar cutoff, bounds it by

Here \rho has a fixed positive minimum on the collars. Since \tau/\ell=N^{-B_0/4}, choosing B_0 larger than these fixed polynomial exponents absorbs this loss in the positive fractions already retained.

The parameter order is now complete: \delta_0 and P_0 were fixed by the floor; choose B_0 larger than the finitely many compact polynomial requirements, and then choose a by Equation (7.13). No estimate used a polynomial bound for a fixed-profile Schauder constant or for n_\infty. Apply the divergence theorem on finite exterior domains. With \nu pointing into the exterior at inner faces it gives

The preceding estimates give a lower bound -o_N(1) after dropping the remaining nonnegative integrals. Lemma 7.14 supplies the limiting outer flux, so the asserted estimate follows. Its mass consequence has the stated sign by Equation (7.39).□

Proof of Theorem 7.1. Fix one prepared data set and the compact black–white exterior used in the construction. Every full enclosing cut in this smaller exterior is a full enclosing cut in the prepared original exterior. Since t\ge-\epsilon and the graph term is nonnegative, \widehat g\ge e^{-4\epsilon}g. On each tangent two-plane its area density is therefore at least e^{-4\epsilon} times the g-density. Taking the infimum over all full cuts gives

This comparison uses the entire intrinsic boundary of each cut, including the original boundary when the exterior itself is chosen.

By Proposition 7.13 and Lemma 7.14, (\widehat\Omega,\widehat g,0) has one complete AH end with the required tensor-norm decay and weighted scalar-curvature integrability, all four finite metric fluxes, scalar curvature at least -6, and strictly mean-negative boundary. Thus the time-symmetric case of Theorem 3.1 applies in the designated chart, irrespective of the causal character of its flux covector. Together with Proposition 7.15, it gives

Send N\to\infty; then \epsilon=B_0\log N/N\to0. Finally remove the strict preparation using Lemma 7.2, which preserves the four metric flux limits and the full-cut infimum. This proves Equation (7.1).□

Proof. The four static potentials form the defining Lorentz representation of the orientation and time-orientation preserving hyperbolic isometry group. The metric flux is linear in the potential. Consequently, composing the designated end chart with such an isometry acts on its four components by the corresponding Lorentz transformation. This follows directly by changing variables in the flux formula: the background metric and its connection are preserved, and the replacement sphere exhaustion has the same limit because the weighted scalar-linearization divergence is absolutely integrable. The quadratic remainder is integrable under the stated decay exponent >3/2. The tensor decay and weighted integrability conditions persist, since two radial functions related by a fixed hyperbolic isometry are comparable far out.

A future timelike covector admits a proper future Lorentz transformation taking it to (m_{\rm AH},0,0,0). Apply Theorem 7.1 in this chart. The geometry of the exterior, its future boundary signs, and its intrinsic full-cut area infimum have not changed. This proves the assertion. Only a hyperbolic change of spatial end coordinates was used.□

Proof. Large coordinate spheres have mean curvature 2+o(1), uniformly for these paths. Fix a radius beyond which this mean curvature is positive and the obstacle is absent. On a larger compact truncation, the direct method for perimeter gives an obstacle minimizer. Its boundary cannot have a maximal radius in the strictly mean-convex region: at a free maximum it is minimal and lies on the inner side of a strictly mean-convex coordinate sphere, in contradiction with the local comparison principle. Contact with the outer truncation can likewise be moved inward to lower area. Thus every such minimizer lies inside the same fixed radius. Truncations containing any prescribed smooth competitor show that their minimum is the full infimum.

The local obstacle regularity theorem [15, Regularity Theorem 1.3(iii)] applies here to pure perimeter, a smooth obstacle, a smooth metric and ambient dimension three. It gives C^{1,1} boundaries and smooth minimal free parts. Part (iii) is used for this individual regularity. For a fixed 0<\alpha_0\le1/2, the local C^{1,\alpha_0} estimates in Part (ii) are uniform: the obstacle bounds and positive metric C^1 bounds are uniform for the family g_s.

Only away from S do we need density estimates. Here is a local proof with uniform constants. In a sufficiently small ball centered at a boundary point, compare a minimizing set E with E\setminus B_\rho and E\cup B_\rho, filling any unnecessary pockets afterwards. The ball misses the obstacle. For f(\rho)=\Vol(E\cap B_\rho), minimality and coarea give, for almost every \rho,

Uniform local isoperimetry implies f'(\rho)\ge c f(\rho)^{2/3}, hence f(\rho)\ge c\rho^3. The same argument applies to the complement. Relative isoperimetry then gives P(E;B_\rho)\ge c\rho^2; the comparison above and the uniform sphere-area bound give P(E;B_\rho)\le C\rho^2. These constants are uniform on compact sets disjoint from S. Perimeter compactness now gives subsequential indicator convergence. Lower semicontinuity, together with testing a fixed minimizer, gives convergence of perimeters, including for varying metrics.

Here is the measure-continuity detail needed for differentiation. In a fixed orthonormal frame let D\chi_{E_j} be the vector normal measures. Their weak convergence and convergence of total variations imply weak convergence of their variation measures: any weak limit dominates |D\chi_E|, and equality of total masses eliminates the excess. Approximate the limiting unit normal in the square-integral norm of this measure by continuous vector fields. For such a field v, expand

Both terms converge. Continuous approximation and uniform continuity then give convergence for every continuous function of position and unit normal, and hence for functions of the unoriented tangent plane. A partition of unity proves the assertion globally. Uniform metric convergence converts varying-metric perimeters to this fixed-metric argument.

On every rectifiable cut of bounded area, the area-element expansion is uniform:

A fixed minimizing cut gives the upper one-sided bound in (8.10). For the lower bound take minimizers T_s for g_s, pass along any sequence s\downarrow0 to a member T\in\mathcal M, and use

Compactness and continuity of a'_T(H) prove the formula.

Finally, the union and intersection of two filled minimizing sets remain obstacle competitors. Perimeter submodularity gives

Both terms on the left are at least A, so both are minimizers. Different tangent planes at an intersection would give a corner in one of them, contradicting interior regularity. To check continuity on the union, take points on a sequence of minimizers converging away from the obstacle. The two phase-volume density bounds prevent disappearance of the limiting point under indicator convergence, so it lies on a limiting minimizer. The uniform local C^{1,\alpha_0} estimates and strict perimeter convergence then give tangent-plane convergence there. The common-plane property makes this independent of the chosen sequence.□

Proof. Solve (8.13) on spherical truncations with outer Dirichlet value zero. Continue both the source and the inner Neumann value from zero. The gradient term can be written b\cdot d\phi+f, where

The operator is consequently -\Delta-b\cdot\nabla+3. A fixed collar function supplies the prescribed outward Neumann derivative, which is +1; adding a sufficiently large constant dominates its bounded compactly supported derivatives and the source. The maximum principle gives a bound independent of the truncation and continuation parameter. Beyond a fixed sphere,

The decaying drift and metric errors preserve its positive leading coefficient. Comparison with a multiple of this function, using the already bounded value on the fixed sphere and zero outer data, gives |\phi|\leq C r^{-\beta}.

Lemma A.1, followed by Sobolev embedding and the one-sided Schauder estimate, gives the same weighted bound through two derivatives. At the fixed inner face use the corresponding Neumann estimates. The linearized equation has bounded drift, zeroth-order coefficient 3, homogeneous Neumann data at the inner face and homogeneous Dirichlet data at the outer face. Its kernel is zero by the maximum principle, so the usual elliptic Fredholm alternative makes it invertible. The a priori estimates close the method of continuity on each truncation. Exhaustion and local compactness then produce the asserted decaying solution, smooth on compact sets including the one-sided boundary.

The additional source is O(r^{-2\beta})=o(w), proving the ratio claim. Replacing \Delta_g by \Delta_b costs O(r^{-\tau-\beta}). Both this error and r^{-2\beta} have finite integrals with weight V_0\,dV_b, as does w. The scalar linearization identity

therefore proves existence of all the conformal fluxes. Replacing b by g in 4\phi b costs order r^{-\tau-\beta}, beyond the mass order.

For the exact conformal transformation g_s=e^{4s\phi}g, K_s=e^{2s\phi}K, direct differentiation gives

The first term vanishes on \mathcal Z. The expansion transforms as \theta_s=e^{-2s\phi}(\theta+4s\partial_\nu\phi); since \theta=0 on S, its derivative is -4.□

Proof. Compactify \mathcal Z by one point, collapsing all rays whose base points escape compact sets; if it was compact, add an isolated point. The ball fibers are compact. The functions \mathcal C'/w extend continuously with value 8a(H,P). On the disjoint compact union of this space, S, and \mathcal M, consider the linear test map

Lemma 8.2 supplies continuity on the last piece.

Suppose its image contained a function positive everywhere. Write its direction as aQ+(H_0,P_0). Positivity at the additional point gives a>0. Realize the direction by

These metrics remain positive and uniformly comparable to g. On the far end, retain the nonnegative background constraint with its positive conformal scaling factor. The remaining terms have the following bounds, uniformly over the unit ball:

For clarity, the non-strict end metric and tensor tests are O(r^{-3}), their background constraint terms are O(r^{-4})+O(r^{-3-\tau}), and their products with the strict direction are O(r^{-3-\beta}). The strict direction’s own quadratic errors are O(r^{-2\beta}). All are controlled by w. Applying the exact conformal law after the intermediate variation gives these bounds without differentiating the nonnegative original constraint into an uncontrolled error. One may compose the two isometries of unit balls in this calculation: at an active unit ray a difference of isometric identifications is tangent to the unit sphere and is annihilated by J; at a vacuum ray J=0.

On the remaining compact ball bundle, positivity of the derivative on the closed active set persists on a neighborhood of it. The original constraint has a positive minimum on the compact complement. Taylor’s formula thus gives the DEC for sufficiently small s>0 everywhere. The boundary has strictly negative future expansion. The same estimates are integrable with weight V_0, and the flux divergence identity proves differentiability of all four fluxes. The covector remains future timelike by continuity. Thus (8.2) applies to this path.

Positivity on \mathcal M, however, says

By Lemma 8.2 and (8.1), the right derivative at zero of the difference in (8.2) is negative. That difference is initially zero and is nonnegative for s>0, a contradiction.

The image of (8.16) is therefore disjoint from the open positive cone of continuous functions. Hahn–Banach separation gives a nonzero continuous functional, nonnegative on nonnegative functions and annihilating this linear image. The Riesz Representation Theorem gives finite nonnegative measures on its three compact pieces. The objective piece has positive mass: otherwise evaluation on Q, strictly positive on both other pieces, would contradict annihilation. Normalize that mass to one. Divide the finite measure on finite active rays by w and call it \mathfrak L; absorb eight times the end-point atom into z. Rearranging the identity gives (8.15).□

Proof. On a compact neighborhood of the support, choose a Gaussian Lorentzian collar

Prescribe its second time jets so that

The spatial Einstein tensor depends on the freely chosen normal derivative of K by a nonzero multiple of trace reversal. In dimension three trace reversal has eigenvalues 1 and -2; it is invertible. The remaining terms are fixed spatial jets. Thus these smooth second jets can be prescribed, and a quadratic time extension with a cutoff realizes a smooth Lorentzian metric on a sufficiently short collar. The normal and mixed components of the displayed tensor are \mu,J, by the constraints.

Set D=J\otimes J/\mu where \mu>0, and set it to zero at vacuum. The DEC gives

At a zero of the smooth nonnegative function \mu, its differential vanishes; |J_i|\leq\mu then gives dJ_i=0 there also. The displayed derivative bound proves that D is C^1 across vacuum with zero derivative there. Splitting any compact set into \{\mu\geq\delta\} and \{\mu<\delta\} proves D_\epsilon\to D in C^1.

Vary the slice by graphs t=a s(x), differentiating at a=0. The metric derivative is 2sK. At a nonvacuum active ray, \mu=|J|, v=-J^\sharp/\mu, and \zeta=n+v is null. The limiting prescribed stress has the form

The covector \gamma is nonnegative on future causal vectors. Parallel transport \zeta in any spatial direction using the spacetime connection. Its contraction with \gamma is nonnegative and has a zero, so its derivative there is zero. Consequently the spatial covariant derivatives of the limiting stress, contracted in one slot with \zeta, vanish.

Let T_\epsilon=(G_{\mathbf g_\epsilon}-3\mathbf g_\epsilon)/(8\pi). Contracted Bianchi, before taking the limit, gives

The right side tends to zero by the compact C^1 convergence just proved. Thus the required normal derivative of the constraint combination tends to zero too. Differentiating its arguments causes no additional limiting term: the first argument is paired against a tensor with \zeta in its kernel, and the second argument remains null; at activity \gamma is proportional to \zeta^\flat, so nullity kills its differentiated contraction. At vacuum the tensor and its spatial derivatives vanish, and Bianchi gives the same conclusion for every vector in the closed unit ball. All expressions use only the fixed compact jets and D_\epsilon,\nabla D_\epsilon; the convergence is uniform, including near vacuum.□

Proof. Apply Lemma 8.5 in (8.15). Mass, expansion and end-point terms are zero. The constraint measure is finite over the fixed compact support, so the limit gives

Off S, form the positive aggregate area measure \mathfrak b=\int_{\mathcal M}dA_T\,d\omega(T). Lemma 8.2 makes F(x)=\tr_{\Pi(x)}K(x), for the common tangent plane \Pi(x), a single continuous function on the union of the cuts locally away from S. The last identity is F\mathfrak b=0. Fubini on a countable compact exhaustion shows that, for \omega-almost every cut, \tr_TK=0 almost everywhere off S. Such a cut is smooth minimal there, so continuity gives H_T=\tr_TK=0 throughout its free part.

At contact, write the cut and S as C^{1,1} graphs with the same orientation. Their second derivatives agree almost everywhere on the coincidence set: apply twice the fact that the derivative of a Lipschitz function vanishing on a measurable set is zero almost everywhere on that set. The cut therefore solves H_T+\tr_TK=0 almost everywhere across contact as well as away from it. This is a uniformly elliptic quasilinear graph equation. Its coefficients are Hölder after the C^{1,1} bound; interior elliptic regularity and differentiation make the graph smooth.

If this smooth MOTS touches S, their one-sided difference satisfies a linear elliptic equation with bounded coefficients. The strong comparison principle gives local coincidence. The contact set is thus open and closed in connected S, so the cut contains S. Its full area is already A, excluding additional positive-area components. If it does not touch S, it is a smooth compact full enclosing MOTS entirely in the interior, after filling removable pockets. This contradicts outermostness. Hence almost every cut is exactly S.□

Proof. For a pure tensor variation P, integration of momentum by parts gives the distributional coefficient

Its trace gives \operatorname{div}X=-u\tau_K, and substitution gives (8.18). A simultaneous conformal test (4\psi g,2\psi K), using complementarity to cancel its background scaling term, gives

This is (8.19). Diverging the shift equation and using its trace gives

Together with the lapse equation this is an elliptic system with diagonal scalar-Laplacian principal part and smooth first-order couplings. Measures belong locally to a negative Sobolev space. The local elliptic estimate on nested compact cutoffs improves their Sobolev order by one, because the couplings have at most one derivative. Iteration and Sobolev embedding yield smoothness.

For completeness, the metric equation is computed next. Put P_K=K-\tau_Kg. The compact constraint pairing is the variation of

together with -8\pi\int H(X,J^\sharp)\,dV_g. The latter is exactly the vector-transport correction. Changing the fixed background volume density to the varying one adds zero, since its background integrand is 16\pi(u\mu_{\Lambda_{\rm c}}+J(X))=0. Integrating the shift term once gives -\int P_K^{ij}(\mathcal L_Xg)_{ij}\,dV_g. At fixed covariant K, contravariant X, and u, use

Integration of \mathcal L_XH and collection of coefficients gives

The shift equation gives

Taking the trace of (8.22) and using complementarity yields

Substitution gives (8.20).

To exhibit finite type, write B_{ij}=\nabla_{(i}X_{j)}=-uK_{ij}. Commuting covariant derivatives gives

All commutators are curvature times X. Derivatives of B use only u,du and the first derivatives of K. Equation (8.20) similarly expresses the Hessian of u in first jets. Therefore, for \mathcal J=(u,X,\nabla u,\nabla X),

for a smooth bundle-valued linear coefficient. ODE uniqueness along paths proves continuation. At a zero of u\geq0, du=0; domination |X|\leq u also gives X=dX=0. Nonzero moments consequently have positive lapse everywhere.□

Proof. The jet at p cannot be zero by continuation. On any fixed compact set, a finite family of coordinate neighborhoods and paths from p has bounded lengths and uniformly bounded coefficients in (8.24). Grönwall’s Inequality gives uniform first-jet bounds there. The equations and local elliptic estimates give bounds of every order on smaller compact sets. Diagonal compactness gives the limit. Its jet at p still has norm one, and causality is a closed pointwise condition. Lemma 8.7 supplies positive limiting lapse.□

Proof. In a smooth normal collar of S, the coefficient in (8.24) is smooth up to the face. Solve its linear ODE down each collar segment from a fixed interior surface. Smooth dependence on its footpoint gives a smooth extension; ODE uniqueness identifies it with the original field in the collar.

Subtract integration by parts of these smooth interior fields from the compact multiplier identity. Take a metric variation whose value and full first jet vanish on S, and set the tensor variation to zero. All area, expansion and integrated boundary terms vanish. The remaining boundary scalar moment tests -\partial_\nu^2\tr_SH, which is arbitrary, by choosing a prescribed s^2 tangential trace in collar coordinates. Momentum and frame terms involve only lower jets and are zero. Thus the scalar moment has no boundary part. Its domination of the vector moment eliminates that part as well.

We next derive end behavior without imposing growth on the measures. Express the first-jet system in a b-parallel radial frame, with \eta=\operatorname{arsinh}r. At (b,0) its lapse system is \Hess_bu=ub and its shift system is hyperbolic Killing transport. Their radial propagators have norm at most C e^{|\eta-\eta'|}. This follows directly from the 2-by-2 blocks y''=y and the constant blocks, or from the ambient linear functions and Lorentz generators on the hyperboloid. The actual coefficient differs by O(e^{-\tau\eta}); only two metric derivatives and one tensor derivative occur. Variation of constants gives

Grönwall therefore gives \mathcal J=O(r), uniformly in angle.

The lapse and tangential-shift radial equations now read

and the normal-shift derivative is O(r^{1-\tau}). Variation of constants, with 3/2<\beta<\min\{2,\tau\}, consequently gives continuous angular coefficients

Convergence of the leading coefficients is uniform; no angular differentiability is required at this stage. The homogeneous r^{-1} mode is included in the remainder because \beta<2.

Use first a conformal prototype \psi=r^{-3}f(\omega) in the multiplier identity and integrate to a large sphere. Its interior pairing vanishes by the adjoint equations, its S terms vanish, and its end-point coefficient is zero. All pairings converge: the moments grow at most as r, while the prototype derivatives are O(r^{-4})+O(r^{-3-\tau}) or better. The sole nonvanishing boundary expression is

Terms containing KX\psi have integrated order O(R^{-\tau}); replacing g by b has the same vanishing order. The displayed limit is 32\int_{\mathbb S^2}f_0f\,d\omega. The left side is 16\pi p'_0=32\int_{\mathbb S^2}f\,d\omega by the defining metric flux. Arbitrary f gives f_0=1.

For a mixed tensor prototype, the only boundary term is twice its trace-reversed stress flux against X. Its limit is 2\int_{\mathbb S^2}\langle W_0,W\rangle_\sigma\,d\omega, whereas its metric mass and area derivatives vanish. Hence W_0=0. The tangential trace of the shift equation on coordinate spheres now gives

Interpret this on the unit sphere. The divergence term tends to zero distributionally, since integration against a fixed test function gains the factor R^{-1}, and H_{S_R}\to2. Thus a_0=0. The already uniform limits identify it pointwise.

We have obtained the value estimates in (8.3), since V_0-r=O(r^{-1}). The coupled elliptic system for (u-V_0,X) has uniformly controlled unit-patch coefficients and source O_{0,\alpha'}(r^{1-\tau}). Local estimates applied to these value bounds yield the first and second derivative Hölder bounds in (8.3), with a smaller \alpha' if necessary. The prototype normalization makes the field nonzero; the last assertion of Lemma 8.7 gives u>0.□

Completion of the proof of Theorem 8.1. Since u>0, (8.7) is a smooth Lorentzian metric on the open exterior product. Its coefficients are independent of t, so \xi=\partial_t is Killing. Its future normal is n=u^{-1}(\partial_t-X); the future-normal convention fixed after Equation (2.5) gives

by (8.4). Thus both induced tensors are the original ones.

We verify its curvature without a spacetime existence assumption. The Gauss–Codazzi and lapse identities, obtained by differentiating the displayed metric, give for spatial slots

and

Equation (8.23) with \Lambda_{\rm c}=-3 gives R_{\mathbf g}=-12. Equation (8.6) then gives

The normal and mixed components of this tensor are the constraints, 8\pi\mu and 8\pi J_i. Complementarity and causality imply

Where \mu>0, equality forces |X|=u, |J|=\mu, and J=-\mu X^\flat/u. Where \mu=0, J=0. Substitution into all the normal, mixed and spatial components gives

Since the matter term is tracefree, this is exactly (8.8). It allows nonzero matter on the null set; its exclusion is a separate argument in the next section.

It remains to determine the boundary data. There are no boundary constraint atoms by Lemma 8.9, and the integration normal at S is -\nu. A compact pure tensor test P, arbitrary at the boundary, has integrated boundary term

Its expansion derivative is \tr_SP. Separating mixed and tangential-trace components in (8.15) gives

For a compact simultaneous conformal test (4\psi g,2\psi K), its area derivative is 4\int_S\psi\,dA_g and its expansion derivative is 4\partial_\nu\psi. Integrating the lapse equation gives

The boundary value and normal derivative of \psi can be prescribed independently. They give d\eta=2u\,dA_g, X=u\nu, and \partial_\nu u+K(X,\nu)=c/2=\kappa_h.

Finally let L_{\rm MOTS} be the normal expansion linearization at S, with principal part -\Delta_S. Extend any s\nu, s\in C^\infty(S), to a compact one-sided field Y and test with (\mathcal L_Yg,\mathcal L_YK). In the open exterior its active-ray derivative is zero: under diffeomorphism transport it differentiates a nonnegative scalar at a zero, and the difference from isometric transport is annihilated at activity. No boundary constraint measure remains. The mass derivative is zero, while the area and expansion derivatives are \int_SHs\,dA_g and L_{\rm MOTS}s. The multiplier identity therefore gives

Outer area minimization gives H\geq0, by the first area variation for arbitrary nonnegative outward speeds. The nonnegative function u|_S is consequently a supersolution of an elliptic operator with principal part -\Delta_S. The strong minimum principle applies: after reversing the operator and subtracting a sufficiently large zeroth-order constant, the usual nonpositive zeroth-order condition holds because u\geq0. On connected S, either u>0 everywhere or u=0 everywhere. This completes all assertions.□