The spacetime Penrose inequality and enclosing area

Proof. Fix the datum and \epsilon. Apply the quoted Riemannian theorem to the actual exterior X_N furnished by Theorem 2.2. Its metric is smooth through the compact frontier; its closure is complete and its open exterior is connected. The end has compact complement and decay exponent n-2>(n-2)/2. Its scalar curvature is nonnegative and integrable with the required pointwise decay. The whole frontier is outer minimizing and locally perimeter minimizing, hence has minimal regular part and singular dimension at most n-8. These verify each receiving hypothesis. We obtain

First let N\to\infty; exponential decay beats the fixed polynomial. Then let \epsilon\downarrow0. These are limits of numbers. The exhaustion constructing a smooth solution at each fixed N is completed before this argument, and no metric at N=\infty is required. Since A_0>0, this also proves E_0>0.□

Deduction of Theorem 1.3. Proposition 3.1, proved in the next section, supplies prepared data with energies E_j\to m=\sqrt{E^2-|P|^2} and enclosing infima A_j\ge(1-o(1))A_{\min,g}(S). Apply Theorem 2.3 to each prepared datum, completing all its fixed-data limits first. Then

No estimate uniform in the preparation index is used.

For sharpness, let M>0, r_M=(2M)^{1/(n-2)}, and take the static Schwarzschild–Tangherlini slice

Proper distance from the horizon makes this a smooth complete metric with minimal compact boundary. Its scalar curvature is zero, and its ADM vector is (M,0). Radial projection to the sphere of radius r_M has (n-1)-Jacobian at most one. More explicitly the pullback of that sphere’s area form is closed with comass at most one; Stokes’ theorem on an end truncation of any full outer domain shows that its integral over the entire cut is \omega r_M^{n-1}. Thus every full cut has area at least that number, while the boundary itself attains it. Consequently A_{\min}=\omega(2M)^{(n-1)/(n-2)} and equality holds.□

Proof of Corollary 1.4. If instead E\le |P|_\delta, choose \varepsilon_j>0 tending to zero and put

Lemma 3.6, proved in the final subsection of Section 3, gives, separately for each j, data satisfying the same one-ended strong-decay, constraint and trapping hypotheses, with

Their ADM vectors are future timelike, so Theorem 1.3 gives

The right side is fixed and the left side tends to zero, a contradiction. The theorem is applied afresh to each datum; no estimate for its end preparation is required to be uniform near the null cone. Thus E>|P|_\delta, and Theorem 1.3 applies to the original data.□

Proof. The exact isotropic metric is

Its leading perturbation of Minkowski space is 2M|y|^{-p}(db^2+p^{-1}|dy|^2). On T=0 put

The nonzero relevant leading components of the spacetime perturbation \mathsf H are \mathsf H_{11}=2MAf, \mathsf H_{aa}=2MBf, and \mathsf H_{01}=2MCf. Moreover

Thus the energy flux vector Q_i=\partial_j\mathsf H_{ij} -\partial_i\mathsf H_{jj} satisfies

Since p(A+1)=k\gamma^2, these expressions give the pointwise identity

The chosen sign of the second fundamental form gives, to first order,

Direct substitution gives

Their Euclidean trace is M\gamma^4v(p+v^2)x^1\rho^{-n}. Consequently, for the leading Euclidean trace reversal \pi_{ij}=K_{ij}-(\tr_\delta K)\delta_{ij},

Using the exact normal and the exact metric in the trace reversal changes these formulas by O_d(r^{-2p-1}). The energy remainder has first derivatives of this same order. Their sphere integrals are O(r^{-p}), and therefore vanish for every n\ge3.

The ellipsoid \{|\operatorname{diag}(\gamma,1,\ldots,1)x|<1\} has volume \omega/(n\gamma). Polar integration therefore gives

Equations (3.6) and (3.7), divided by 2k\omega and k\omega, give E=M\gamma and P_1=M\gamma v. The integrand of a transverse momentum row is a constant times x^1x^a\rho^{-n}/r, whose integral vanishes by reflection.□

Proof. We use the compact scalar divergence inverse on a connected open set: a mean-zero smooth function supported in a fixed compact set has a compactly supported vector primitive, with one Sobolev derivative gained; the support and Sobolev mapping statements are given, for example, by the regularized Bogovskii operator in [15, Section 3.1, Equation (3.13), Theorem 3.2, Corollaries 3.3–3.4, Remark 3.5, and Proposition 4.1(ii)]. Here is its local construction and the support issue. On a ball, choose a smooth unit-integral function supported in a smaller concentric ball and use the Bogovskii integral

Differentiation in distributions gives \divg\mathcal Bf=f-\eta\int f=f; the segments occurring in the integral remain in the ball. For compact input support and compact \supp\eta, their union is compact in that ball. The kernel is of order 1-n, and its differentiated kernel has the cancellation of a Calderon–Zygmund kernel. The resulting W^{d,a} estimate is the usual boundedness of that singular integral and its commutators with derivatives. Cover the prescribed support by finitely many balls compact in \mathcal A, connect them by a finite chain of overlapping such balls, and use a partition of unity. Transfer the means along the chain using unit-integral bumps in the overlaps. Each resulting ball source has mean zero; the sum of its ball primitives proves the stated inverse on \mathcal A, with fixed support and constants.

Apply it first to a scalar f with its affine moments zero, obtaining \partial_iv_i=f. Compact support gives \int v_i=-\int z_i f=0. Apply the same inverse to each v_i to obtain \partial_jT_{ij}=v_i. Then Q=(T+T^{\mathsf t})/2 has \partial_i\partial_jQ_{ij}=f. The tensor h=Q-k^{-1}(\tr_\delta Q)\delta has trace reversal Q, so \mathsf S(h)=f with the asserted two-derivative gain.

For a vector source F, first solve \partial_jB_{ij}=F_i row by row. Translation moments permit this solve. Rotation moments give

Put A_{ij}=(B_{ij}-B_{ji})/2 and solve \partial_lD_{ijl}=-A_{ij}, choosing D_{ijl}=-D_{jil}. Set

Index interchange gives E_{ij}-E_{ji}=-2A_{ij} and \partial_jE_{ij}=0: the first two double derivatives cancel after interchanging j,l, and the last vanishes by skew symmetry in those indices. Thus C is symmetric and \partial_jC_{ij}=F_i. Taking b=C-k^{-1}(\tr_\delta C)\delta proves the assertion, including the derivative count.

Finally take a nonnegative smooth bump supported in an open ball of \mathcal A. The Gram matrix of the affine functions, or of the vector Killing fields, with this weight is positive definite: a polynomial in either finite-dimensional space vanishing on that ball vanishes identically. Multiplying the basis elements by the bump and the inverse Gram matrix produces fixed dual moment bumps. Subtraction removes all moments with the stated bound on coefficients.□

Proof. Expansion of the constraint formulas about (\delta,0) gives

The remainder estimates follow, respectively, from hD^2h, (Dh)^2, and K^2, and from hDK and Dh\,K. The volume densities of g and \delta are uniformly comparable on the tail. Hence integrability of \mu,J, together with 2q>n-2, gives the claim for the original data. The model is vacuum with faster decay, so its flat sources are integrable too.

For any integrable tail function f, even without an integrable first absolute moment,

Indeed divide by L and split at a fixed large radius R_1. The inner integral divided by L tends to zero, whereas the outer integral divided by L is at most 4\int_{r>R_1}|f|, which tends to zero as R_1\to\infty.

Write Q=H-(\tr_\delta H)\delta. For an affine function a(x) define its Green flux by

Integration by parts between spheres yields \mathcal B_a(R)-\mathcal B_a(R_0) =\int_{R_0<r<R}a\,\mathsf S(H). Charge matching gives \mathcal B_1(R)\to0, and Equation (3.10) gives \mathcal B_{x^a}(R)=o(R). For the scalar commutator C_L, supported where the cutoff varies, one consequently has

This is o(1) for a=1 and o(L) for a=x^a.

For momentum put Q_{ij}=(K-K^{\mathrm b})_{ij} -\tr_\delta(K-K^{\mathrm b})\delta_{ij}. For a Euclidean Killing field Y, symmetry of Q gives Q_{ij}\partial_jY_i=0. Its Green flux is thus simply \int_{S_R}Y_iQ_{ij}\nu^j_\delta. For constant Y its limit is zero by equality of the momenta; replacing geometric trace reversal by Euclidean trace reversal changes the sphere flux by O(R^{n-2-2q})\to0. For rotational Y the same Green identity and Equation (3.10) give o(R). Equation (3.11) holds for F_L with this flux and pairing.

Under x=Lz the fields are h(Lz) and LK(Lz), and both flat sources are multiplied by L^2. A constant moment is therefore multiplied by L^{2-n} and a first moment by L^{1-n}. The preceding o(1) and o(L) bounds prove the same o(L^{2-n}) bound for all rescaled moments.□

Proof. 1. Continuation on a truncation. Write a=\varrho^{-k} and b=\varrho^{-n-\beta}. The positive regularizing term a^2 makes the right side smooth as a function of d\phi, with first gradient derivative of norm at most B. Choose a large coordinate sphere enclosing S and a compact core. On a larger truncation \Omega_R impose \phi=0 on S_R, and multiply B in the equation by t\in[0,1]. The inner Neumann and outer Dirichlet boundary components are disjoint smooth hypersurfaces.

At t=0 this mixed Laplace problem is solvable. Indeed its weak formulation is coercive on H^1 functions of zero outer trace by Poincare’s inequality; the Neumann datum is a bounded boundary functional by the trace theorem. Lax–Milgram gives the weak solution and local boundary regularity gives a smooth solution. The linearization at any solution for 0\le t\le1 is

It has a bounded drift, is uniformly elliptic, and has no zeroth order term. Its homogeneous kernel is zero by the strong maximum principle and the Hopf boundary lemma: a nonzero maximum or minimum cannot occur in the interior or on a zero-Neumann component [21, Chapter 3]. The same conclusion follows for weak limits with merely bounded drift. The drift is a compact perturbation of the mixed Laplace isomorphism in Hölder spaces, so its index is zero and this injectivity gives surjectivity. The implicit function theorem gives openness of the continuation set.

2. Bounds independent of the truncation. We detail the estimates giving closedness and exhaustion. All solutions are nonnegative. An interior negative minimum contradicts -\Delta\phi>0. At a minimum on S the Hopf lemma requires a negative outward derivative, whereas the outward derivative of \Omega_R is -\partial_\nu\phi=1. The outer boundary value is zero.

For 0<\alpha<\min(\beta,q) and sufficiently large r_0, the function

is positive and satisfies

In fact -\Delta_\delta W=\alpha(p+\alpha)r^{-n-\alpha}, the metric error is O(r^{-n-q}), and B|dW|=O(r^{-n-q}). Since b+Ba=O(r^{-n-\beta}), a sufficiently large multiple C(A+1)W, where A=\sup_{S_{r_0}}\phi, is a supersolution outside r_0, dominates \phi on S_{r_0}, and is nonnegative on S_R. Here we used \sqrt{|d\phi|^2+a^2}\le|d\phi|+a and the fact that this gradient dependence is Lipschitz. Subtracting a supersolution therefore gives a scalar linear inequality with bounded drift, to which comparison applies. Hence

uniformly in R and t.

We claim that the suprema of all these solutions are uniformly bounded. Otherwise divide a sequence by its suprema M_i\to \infty. Estimate (3.23) ensures that the normalized functions have a positive supremum on a fixed compact region. Local W^{2,a_0} estimates, for any fixed a_0>n, apply uniformly, including the inner Neumann boundary: the equation has right side bounded by a fixed multiple of |d\phi|+1, and first derivative interpolation absorbs the gradient term in the second derivative estimate. The same argument applies after division by M_i. Compactness gives a nonnegative, nonzero limit u with u\to0 at infinity and

If the outer radii instead have a finite limit, the limit has zero data at that limiting sphere. In either case its positive maximum is attained. The strong maximum principle, or the Hopf lemma at S, contradicts that maximum. These principles apply because B|du| equals a bounded measurable drift paired with du, defining that drift to be zero where du=0. This proves the uniform supremum estimate.

3. Exhaustion and uniqueness. The W^{2,a_0} estimates now give uniform C^{1,\gamma} bounds on fixed patches. Their right sides are then Hölder continuous, and the scalar Schauder estimates give C^{2,\gamma} bounds. Differentiation and the strictly positive regularizer a give all higher local bounds. For each fixed truncation these estimates close continuation to t=1. Letting R\to\infty gives a smooth solution of Equation (3.19); the tail bound gives the specified limit. It is strictly positive by the same minimum principle. The difference of two decaying solutions has zero Neumann data and solves a homogeneous equation with bounded drift, by integrating the gradient derivative of the square root along the segment between their gradients. Its positive and negative extrema are excluded as above, proving uniqueness.

4. End decay and flux. On dyadic end annuli rescale x=Lz and \Phi(z)=L^p\phi(Lz). Estimate (3.23) bounds \Phi. The rescaled equation has uniformly controlled metric coefficients and the form

where the coefficient denoted O_5(L^{-q}) is nonnegative and smooth. The same interpolation, scalar estimates and differentiation give uniform bounds for six derivatives of \Phi. In the compactly supported B case the equation is just a Poisson equation on the distant end, and all available metric derivative orders give corresponding derivative bounds. This proves the asserted decay. It also gives B\sqrt{|d\phi|^2+r^{-2k}}=O(r^{-n-q}), so \beta<q proves the Laplacian ratio in Equation (3.20).

The source is integrable. Integrating the equation on \Omega_R, remembering that its inner outward normal is -\nu, gives

This proves the claimed finite negative flux.

5. Strict inequalities and charges. Finally |d\phi|^2\le Cb globally: on the end the quotient has order r^{-(n-2-\beta)}, which is bounded because \beta<1\le n-2, and on the core b is positive. Equation (3.19) implies, for every unit-ball vector v, -\Delta\phi+K(\nabla\phi,v)\ge b. Use Equation (3.2) with s=\delta\phi; for sufficiently small fixed \delta>0 the negative square term is at most half the positive k\delta b term. This proves the strict DEC margin, and e^{\delta\phi}\theta_{+,\delta}=\theta_+-k\delta<0. The conformal energy vector changes by -2k\delta\,d\phi+o(r^{1-n}). The geometric and Euclidean fluxes of d\phi agree in the limit, since their difference is O(r^{-q}) after integration. Division by 2k\omega gives the energy formula in Equation (3.21) exactly. The momentum trace reversal is multiplied by e^{\delta\phi}; its flux change is O(r^{-q})\to0. Positivity and boundedness of \phi give the last metric statements.□

Proof of Proposition 3.1. Apply Lemma 3.5 to the repaired data (g_L^*,K_L^*), choosing B compactly supported and choosing \delta_L\downarrow0 so that \delta_L\|\phi_L\|_\infty\to0 and \delta_L|\mathfrak f_{\phi_L}|\to0. There is no need for constants in this solve to be uniform in L: this choice is made after the entire replacement at that L. Its scalar curvature has the required decay, since the ultimate initial end is scalar flat and

there. Compact support of K_L^* is preserved, and all metric derivative orders have the stated asymptotics. The strict margin and expansion are those of Lemma 3.5. The core remains smooth and the tail is uniformly Euclidean comparable, so completeness with S included is preserved. The constraints are integrable on the core and have integrable decay on the tail.

By Equation (3.16) and the choice of \delta_L, the final energy tends to M=m>0, and the final momentum is zero. Strictification only increases the metric, so Equation (3.17) and the full-cut comparison remain valid; in particular, every resulting full-cut infimum is positive. Pass to a sequence along which the comparison errors are bounded by numbers \epsilon_j\downarrow0. These data satisfy Definition 2.1 and every assertion in Equation (3.1).□

Proof. For U>0 put a=U^{2/p} and (g_U,K_U)=(a^2g,aK). Substitution of s=2p^{-1}\log U into Equation (3.2) gives, as scalar and covector identities,

Indeed, the scalar terms combine as -k\Delta s-\frac{kp}{2}|ds|^2=-\frac{2k}{pU}\Delta U. Taking the new-metric covector norm introduces another factor a^{-1}, so

The boundary identity is

Thus a factor constant near S preserves its expansion sign, and -\Delta_gU\ge |K|_g|dU|_g preserves the dominant energy condition.

Choose 0<\xi<\min(q,1). In a sufficiently distant part of the coordinate end set

There T>0 and T'<0. Since p=n-2, \Delta_\delta r^{-\alpha}=\alpha(\alpha+2-n)r^{-\alpha-2}. The original metric and tensor estimates therefore give

after increasing the radius. Choose a nondecreasing smooth switch \beta from zero to one in this region, constant near its endpoints, and put

Extend G constantly through the interior. Then G\ge0, it is constant near S, and G=T sufficiently far out. Because dG=\beta\,dT,

Use U_\lambda=1+\lambda G. Equation (3.25) and the boundary formula prove the required signs for every fixed \lambda>0, without a smallness condition.

The factor is bounded for each fixed \lambda and is at least one, so completeness with S included is preserved. The manifold and its unique coordinate end are unchanged. On the tail G=O_j(r^{-p}) at every fixed derivative order. Since q<p, the added metric tail is faster than the original decay, and

Moreover,

These terms are integrable against dV_g\asymp r^{n-1}dr\,d\omega. The scalar and covector identities above, and the bounded conformal factors for fixed \lambda, show that the new constraint densities are integrable against dV_{g_\lambda}.

For the charges write f_\lambda=U_\lambda^{4/p}. Then

In the metric flux of Equation (1.2), the leading change is -k\partial_i(f_\lambda-1). Terms involving g-\delta or its first derivatives have integrated flux O(R^{-q}), and the nonlinear remainder has flux O(R^{-p}). Since

the current normalization gives

If \pi=K-(\tr_gK)g, its transformed tensor is exactly \pi_\lambda=U_\lambda^{2/p}\pi. The additional momentum flux in Equation (1.3) is O(R^{-q}), because U_\lambda^{2/p}-1=O(r^{-p}) and \pi=O(r^{-1-q}). Thus P_\lambda=P, and the new ADM limits are finite.

Finally, the admissible domains in Definition 1.2 are unchanged. For their entire intrinsic boundaries,

This comparison includes every component and boundary-coincident portion. Taking the infimum proves the claimed full-cut area inequality.□

Proof. We derive the curvature formula before completing any squares. On \R_z\times\mathcal M_N consider the stationary Lorentz metric

The constant-z slices have lapse U, shift Uw=l^2\nabla f, future unit normal \mathbf n=U^{-1}\partial_z-w, and second fundamental form

This sign follows directly from \partial_zg=2Ub+\mathcal L_{Uw}g=0 and the prescribed future-normal convention. Write B_0=\tr b. Contracted Gauss and the normal lapse equation give

Since B_0 is stationary, \mathbf n(B_0)=-w(B_0), and taking the trace in Equation (4.7) gives \divg(Uw)=-UB_0. Substitution proves

In coordinates z'=z-f, the same metric is the static warped product -l^2(dz')^2+\bar g. Its only mixed connection terms are \boldsymbol\Gamma^{z'}_{iz'}=\partial_i\log l and \boldsymbol\Gamma^i_{z'z'}=l\bar g^{ij}\partial_jl; contracting their curvature contributions yields \mathbf R=R_{\bar g}-2l^{-1}\Delta_{\bar g}l. Also \bar g^{-1}=A_d, dV_{\bar g}=\sqrt D\,dV_g, and therefore

Differentiating U=l\sqrt D and using Equation (4.4) proves the vector identity

Consequently Equation (4.8) becomes

Set Q=K-b=L+p(dt\otimes w+w\otimes dt). For any symmetric C, the product rule and Equation (4.7) give the useful exact contraction

Expanding \mathsf B(K-b,K-b) and applying this formula to C=K transforms Equation (4.9) into

For example, the terms involving K on the right cancel to R_g/2-\mathsf B(b,b)/2+(\divg P_b)(w), as they must.

The conformal scalar law in dimension k+1 is

The volume formula gives

Changing the divergence weight from U to u=e^{(k-1)t}U now yields

In particular the last change of weight adds k(k-1)|dt|_{A_d}^2 to the preceding coefficient k[p-(k-1)/2].

There are two remaining first-order algebraic identities:

For the first, P_Qw+kA_ddt has transverse and axial components

whereas V/u has components a_wM+(k+pv)y and \chi a_wj+kd x by Equation (4.4). Their difference is Fw. For the second, use \divg(Uw)=-U\tr b and change the weight. Thus the smooth invariant definition of the claimed quadratic form is

This proves smoothness and Equation (4.5). To obtain its displayed block form, the blocks of Q are T, M+pa_wy, and j+2pa_wx. The transverse terms are

The terms containing F a_wx cancel, while their remaining axial cross term is 2s_p\chi ja_wx. Finally |T|^2=|T^0|^2+(F-\chi j)^2/k gives Equation (4.6).

For completeness this derivation retains derivatives of p. In the flux of Equation (4.11) the part explicitly containing p is p\{v\nabla t-w(t)w\}; its divergence contributes p'v|y|^2. Thus the scalar side contains -p'v|y|^2. In Equation (4.5) the redistribution of this term can also be checked directly. Holding t,dt,df,H^f,K fixed when taking the partial derivative in p, one has \chi_p=-\chi v/(k+pv). The terms containing p' in w(F) and u^{-1}\divg V are respectively

Their difference is exactly -p'v|y|^2. No derivative of the cutoff has been discarded.□

Proof. The entire axial part of Equation (4.6) is exactly

Indeed completing first the x square and then the F square leaves

The last coefficient is at least c_kd/(1+N)^2, and

These inequalities control |T^0|^2,|y|^2,dj^2 and the two completed squares. Recovering M from M+s_pa_wy costs at most C_k(1+N)^2. Recovering \sqrt d\,x costs the already controlled \sqrt d\,j, since a_w/(k+pv)\leq1/k. Recovering F costs no more because \chi^2\leq d. The relation \tr T=F-\chi j then controls |T|^2. These observations give Equation (4.14) with the stated single quadratic polynomial; they do not remove the factors d.

When dt=0 the axial expression instead completes as

Its second parenthesis is bounded above and below by positive constants, and \chi^2\leq\chi. This proves Equation (4.15) independently of N.□

Proof. At dt=0, direct block contraction gives

Subtracting from Equation (4.6) gives the full difference, including its mixed term:

If (1+p)v\leq\zeta_k with \zeta_k>0 sufficiently small, then |d-1|+|\chi-1|\leq C_k\zeta_k. At d=\chi=1 the expression is

Perturbation of this finite-dimensional positive form, and Equation (4.15), therefore bound Equation (4.20) below by a fixed positive multiple of \mathcal T.

In the other case 1/v\leq(1+p)/\zeta_k. The last coefficient in Equation (4.20) is at least \chi, since \chi\leq d. The absolute value of its mixed coefficient is at most d. Hence Young’s inequality bounds the difference below by

Here

It follows, again using Equation (4.15), that the desired fixed multiple of \mathcal T is obtained after subtracting at most C_k(1+N)^2vF^2/\zeta_k.

To replace L by H^f=L-K, polarize \mathcal S in the A_d\otimes A_d norm. The preceding stationary coercivity gives

Moreover |K|_{A_d\otimes A_d}\leq|K|. Therefore

for every fixed \zeta>0. Taking \zeta below half the positive constant already obtained proves Equation (4.19).

At an interior minimum of t, dt=0 and \Hess_g t\geq0. To estimate curvature from above, view \bar g as the metric on the graph of f in the Riemannian warped product g+l^2dz^2. At that point dl=0 and l^{-1}\Hess l=p\Hess t. Its ambient scalar curvature is R_g-2p\Delta t, its horizontal Ricci tensor is \Ric_g-p\Hess t, and its unit vertical Ricci component is -p\Delta t. The graph unit normal is D^{-1/2}\partial_{z,\mathrm{unit}}-w and its second form with this orientation is -H^f, which has the same quadratic Gauss term as H^f. The Gauss equation followed by the conformal law consequently gives the exact minimum-point formula

When v=0, terms using e vanish. In particular all p'(dt)^2 terms vanish at this point.

Combining Equations (4.5), (4.19), and (4.21), and using F=\eta f, yields

Here are the weights in this estimate. The curvature-independent constraint contribution is \mu-R_g/2=(\tau^2-|K|^2)/2. This, J(w), F\tau, and the |K|^2 error are uniformly bounded on the enlarged core and vanish on the far end, so they cost C\rho_0 pointwise. On the end, v|\Ric_g|\leq Cv\varrho^{-n}\leq Cv\rho_1 because 2b_1<n; the height comparison gives vF^2\leq Cv\rho_1. The same statements hold uniformly for aK, 0\leq a\leq1. No use of the dominant energy condition in the filling has entered this calculation.

At t=-\epsilon one has m_0=1. Subtracting the right side of Equation (4.18) from the lower bound in Equation (4.22) gives

Choose C_N>\mathcal P_N+c_*+1. The expression is strictly positive, contradicting the equation. In the second homotopy stage the trace comparison gives f=0: a positive maximum or negative minimum contradicts \tr_{A_\chi}H^f=\eta f. Thus t=Z, and at a floor minimum the left side of the divergence equation divided by u is k\Delta t\geq0, whereas its right side is s(\rho-C_N\rho_0)<0 for s>0. At s=0, the equation \divg(u\nabla Z)=0 with zero outer data gives Z=0 by the maximum principle. This finishes the pointwise exclusion.□

Proof. For fixed N, the end hypotheses and Equation (4.4) give

The notation O_N here permits arbitrary fixed-N constants. Since N\epsilon>2 and t\to0, m_0=0 sufficiently far out. There \mathcal T=ks_p|dt|^2+O(e^{-c_N'r}), so every term in u\Xi is integrable: the gradient term is O_N(r^{2-2n}), and \rho=O(r^{-n-\beta}). Smoothness gives integrability on the compact filled core. The divergence theorem on exhausting domains has no inner boundary and proves the first equality and existence of the flux limit. It also proves existence of \lim\int_{S_r}\partial_{\nu_g}t directly from \Delta_gt\in L^1.

The graph part l^2df^2 contributes zero to the ADM flux by its exponential decay. The conformal perturbation is

The linear ADM integrand of 2t\delta is -2k\,\partial_i t; the remainder has integrated flux O(r^{2-n})\to0. Euclidean and g-normal fluxes differ by the same vanishing order. Thus

It remains to bound the negative part of the integral independently of those end constants. It is supported in the penalty band -\epsilon<t<-\epsilon+1/N, where t<0 and \ell\leq l\leq e\ell. Put \sigma=|df|. Directly from the definitions, with constants independent of N on this band,

For the second estimate use l\sqrt D\,v=l^3\sigma^2/\sqrt{1+l^2\sigma^2}\leq l^2\sigma. The nonnegative terms u\delta_0\mathcal T and u\rho may be dropped, while the favorable gradient term absorbs the linear \sigma terms. More explicitly Young’s inequality gives

Consequently

All integrals use dV_g on \mathcal M_N. They are bounded by C(1+N): on the end \rho_0 is integrable and 4b_1>n makes \rho_1^2 integrable, while on the core the weights are uniformly bounded and its volume is O(1+N). Since \ell^2/\eta=\ell^{1/2}, this proves the polynomial flux estimate. For fixed \epsilon>0 the resulting error tends to zero as N\to\infty.□

Proof. At a positive maximum of f, df=0, A_\chi=I, and

The negative minimum gives the reverse bound. The tensor fields in the filling have a common pointwise bound, including along the product necks, so this proves the first assertion uniformly. If K=0, both comparisons give f=0.

On the end K_a=0. At a point of contact with H(r)=C(r^{-b_1}-R^{-b_1}), considered as a test for h=\eta f, the two gradients agree. Thus the axial direction of A_\chi is the radial direction. In the Euclidean metric the Hessian contraction is

The error from g-\delta=O_2(r^{2-n}) has absolute value at most C' r^{2-n}Cb_1r^{-b_1-2}, uniformly for 0<\chi\le1. Since b_1+1<k, the contraction is strictly negative for r\ge r_1, with r_1 independent of N. The height equation is

Consequently H and -H are respectively an upper and a lower comparison function. For R\ge2r_1, their constant C can be chosen to dominate the already established height bound on S_{r_1}. The maximum principle then proves the displayed end estimate.

Taking the one-sided derivative at the outer zero boundary gives

Tangential derivatives vanish there. For all real t we have l(t)\le e. The identity Z=t+(2k)^{-1}\log(1+l(t)^2|df|^2) therefore gives, on S_R,

Choosing R_{\min}(N) sufficiently large proves the assertion.□

Proof. Write \sigma=|df|, B=K_a(w,\cdot), A=A_\chi, and

The floor already implies

Only the upper bound for Z remains to be proved.

High levels. On the support of m_0, the floor implies -\epsilon\le t\le-\epsilon+N^{-1} and l\asymp\ell. In particular,

The weights are bounded above uniformly, and C_N is polynomial. It follows from D=e^{2k(Z-t)} that there is a positive polynomial threshold Z_0=P(N) such that

For example, it suffices that e^{kZ_0}\ge C(1+C_N)\eta^{-1}, after increasing the fixed constant; its logarithm has polynomial size. Away from the support of m_0 the inequality is immediate.

Choose a smooth nondecreasing \zeta that vanishes on (-\infty,Z_0] and equals one on [Z_0+1,\infty), with 0\le\zeta'\le C. Testing the divergence equation with \zeta(Z) gives

The integration has no boundary term since Z=0<Z_0 on S_R. On the last strip u=l e^{kZ-t}\le\exp(P(N)). The support of K consists of a fixed exterior compact set and the enlarged filling; it has volume O(1+N). Thus

This estimate uses no upper bound for \sigma. It controls the high levels of Z with exactly the volume and slope weights needed on the graph. We next convert it into the initial integrability for an iteration on fixed base patches.

Fix 0<\alpha\le k-1, independent of N. Cover the enlarged core and a fixed extra end annulus by base coordinate patches of uniformly controlled size and geometry. The number of patches is O(1+N), and \rho has a uniform positive lower bound on them. We claim that on each such patch Q,

Below Z_0+1, Equation (5.6) proves this directly. Above that level, when D\le2 use

When D\ge2, use instead

The ratio of the second expression to the first is at most C\eta^{-1}e^{(\alpha-k+1)t}\le\exp(P(N)) because 1/2+\alpha/(2k)\le1 and t\ge-\epsilon. Equation (5.8) proves the claim.

Sobolev inequality on the product graph. We use the ordinary graph of f in (Q\times\R,g+dz^2) so that its ambient geometry depends only on the fixed background patch. The warp l(t) has no a priori derivative bound yet. We compare the ordinary graph’s measure and inverse metric with d\mu_D and A, and use \mathcal T to control its mean curvature. The ordinary graph has measure and inverse metric

The ratios of D and 1+\sigma^2 lie between \exp(-P(N)) and \exp(P(N)), even when \sigma is unbounded. The axial eigenvalues d and \chi=kd/(k+pv) differ by at most a factor 1+N/k. Thus these graph norms compare with d\mu_D and A with factors \exp(P(N)).

Its mean curvature is (1+\sigma^2)^{-1/2}\tr_{A_*}\Hess f. Write K_a+H^f in its transverse and axial blocks. The transverse contraction is controlled by |T|, and the axial coefficient of A_* is at most e^2d. Also \sqrt D/(l\sqrt{1+\sigma^2})\le\exp(P(N)). Proposition 4.2 therefore gives

Extend the metric of a slightly larger fixed base patch to a closed manifold and use Nash’s smooth isometric embedding theorem [35]; then take its product with the line. The additional second fundamental form is uniformly bounded. Such embeddings can be chosen from finitely many fixed models for the cap, transitions, and product necks. The Michael–Simon inequality [34], in the formulation of [42, Chapter 4, Section 6, Theorem 6.7], applied to compactly supported functions on this smooth n-dimensional graph, then gives, after its usual L^1 to L^2 conversion,

To see the conversion explicitly, apply the L^1 inequality to |\varphi|^{2(n-1)/(n-2)} and use Cauchy–Schwarz on its derivative and mean-curvature terms. The resulting common L^{2n/(n-2)} factor cancels. No bound for the height or area of the graph is required.

Exponential testing. The graph inequality still contains \mathcal T on its right side. The divergence equation controls that term together with the gradient of an exponential of Z; this is the estimate that closes the iteration. Globally \Xi_a\ge\delta_0\mathcal T-P(N). Let \psi be a base cutoff compactly supported in Q and test the equation with \psi^2e^{2qZ}/W. Put \beta_W=d\log W=(k-1+p)dt. Since A\le A_d, scalar coercivity gives |\beta_W|_A^2\le P(N)\mathcal T. Integration by parts yields exactly

Young’s inequality bounds the \beta_W\cdot A\,dZ term by \delta_0\mathcal T/8+P(N)|dZ|_A^2, the \beta_W\cdot AB term by \delta_0\mathcal T/8+P(N), and the qB\cdot A\,dZ term by a fixed fraction of kq|dZ|_A^2+C q. The cutoff terms cost a further fixed fraction of that gradient term and P(N)(\psi^2+|d\psi|^2). Consequently there is a polynomial q_*(N)\ge\max\{1,\alpha\} such that, for q\ge q_*(N),

Only dt occurs in this test; it does not differentiate p.

Apply Equation (5.10) with \varphi=\psi e^{qZ} and then use Equation (5.11). For nested base patches Q'\Subset Q separated by a coordinate distance b\le1, this gives

All norms here use d\mu_D. Iterate with q_i=\kappa_n^iq_* and geometrically decreasing patch gaps. The sums of q_i^{-1} and i q_i^{-1} converge, so

where C is independent of N.

Finally interpolate the norm on the right with Equation (5.9):

Choose expanding patches Q_j, all inside one fixed larger patch, with gaps comparable to 2^{-j}. The logarithms M_j=\log\sup_{Q_j}e^Z satisfy M_j\le\theta M_{j+1}+P(N)+Cj. Iteration gives

The last polynomial is justified by (1-\theta)^{-1}=2q_*/\alpha\le P(N). For each individual smooth solution the suprema on the largest patch are finite, so the first term tends to zero. We have proved Z\le P(N) on the enlarged core and the drift support. A larger maximum outside that set would have K_a=0, dZ=0, and, by Equation (5.7), \Xi_a>0; at the same point u^{-1}\divg(kuA\nabla Z)=kA:\Hess Z\le0, a contradiction.

For the second homotopy f=0. A positive maximum above the penalty band has right side s\rho>0 if s>0, which is impossible. When s=0 the solution is zero. This proves its upper bound as well. Equation (5.6) and the height bound prove the remaining bounds in Equation (5.1). In particular,

The same bounds give positive upper and lower bounds for u at each fixed N. Thus both A_\chi and kuA_\chi are uniformly elliptic, with constants independent of the truncating radius and the homotopy parameter. These are the ellipticity bounds used in the following scalar estimates.□

Proof. We use only the scalar linear De Giorgi–Nash and weak Harnack estimates, the Krylov–Safonov interior estimate for strong nondivergence solutions, and the Euclidean Hessian estimates of Calderón and Zygmund; see [21]. The two structural arguments needed to apply these results are proved below. All coordinate norms of matrices use the Frobenius norm unless indicated otherwise.

1. A Hessian estimate with a strict absorption margin. Normalize the metric at the center of a chart. The coordinate principal matrix is

If g=I, then |I-a|=1-\chi\leq1-c. Consequently, on a sufficiently small chart, depending only on c and the metric modulus,

This remains true for the reflected metrics used below, since they are continuous and uniformly Lipschitz. In particular this assertion places no condition on the oscillation of e or \chi.

Let \mathcal R=D^2\Delta^{-1} on \R^n, with its matrix-valued range given the Frobenius norm. Plancherel’s identity gives \|\mathcal R\|_{L^2\to L^2}=1. Its L^4 norm is finite by the Calderón–Zygmund theorem. Interpolation therefore supplies s_0>2, depending only on n,c, such that

For a compactly supported v the identity \Delta v=a:D^2v+(I-a):D^2v now gives, by absorption,

Cutoffs, first-derivative interpolation and absorption on nested balls give the local version

For completeness, let \psi be a smooth cutoff for the nested balls. Applying the compact-support estimate to \psi v first produces an additional (t-\rho)^{-1}\|Dv\|_{L^s(B_t)} on B_\rho\Subset B_t. The interpolation estimate on a slightly larger ball bounds this by \varepsilon\|D^2v\|_{L^s} plus C\varepsilon^{-1}(t-\rho)^{-2}\|v\|_{L^s}, with the scale factors absorbed into \varepsilon. Choose the coefficient of the first term smaller than the square of the geometric ratio of the successive gaps, and sum over increasing nested radii. The resulting geometric series is exactly Equation (5.16). These estimates initially apply to the individually finite Sobolev norms of the functions under consideration and then, by approximation, to W^{2,s} functions.

2. The exact flux and the constant-value reflection. Write P=\nabla_g u, H=\Hess_g u, and q=|P|_g^2/2. The equation implies \Delta_g u=(1-\chi)H(e,e)+G. Since \nabla_g q=H^\sharp P, it gives the pointwise identity

Both sides vanish when P=0, so no choice of axis at those points affects this identity. For smooth u, commuting third derivatives gives

For C^2 functions the same formula holds distributionally: approximate u in C^2 on compact subsets of each original smooth side. The flux and the right side converge uniformly, so integration against a smooth compactly supported test function passes to the limit. This argument establishes only the geometric divergence identity; the algebraic Equation (5.17) is applied to the original solution, and no approximant is required to solve the equation. Neither identity differentiates \chi.

Put \gamma=1-(1-c)^2>0. Young’s inequality, with the gap \gamma/2, gives

Thus, when |P|_g\leq1 and s=1-|P|_g^2\geq0,

We record precisely how this inequality is used at the face. Subtract the constant boundary value and take Gaussian coordinates g=dt^2+h_{ab}(y,t)\,dy^a dy^b on t\geq0. Extend u oddly and h evenly across t=0, and extend \chi evenly and G oddly. The reflected equation holds almost everywhere. Tangential derivatives of u vanish on the face, while the normal derivative agrees from both sides. Therefore u is C^1 after reflection and belongs locally to W^{2,s} for every finite s individually. The metric is Lipschitz, smooth on either side, and s is a continuous W^{1,2} function.

Let J=\sqrt{\det g} and define the face mean curvature using the normal +\partial_t by

On the upper side, H_{ab}=\tfrac12(\partial_t h_{ab})p and hence \mathcal F^t_+=-\kappa p^2. On the lower side the signs of these tangential Hessians reverse and \mathcal F^t_- =\kappa p^2. Since J is continuous across the face, the density-valued flux J(A\nabla_gs+2GP)=-2J\mathcal F has normal jump

Adding the coordinate vector

cancels that interface measure exactly. Indeed its coordinate divergence is -j(y)\delta_{\{t=0\}}; no derivative in y occurs, because the vector has only a t component. Its L^\infty norm is at most C\|\kappa\|_\infty under the gradient normalization. All these statements are local, so the coefficient j(y) only needs to be extended constantly in the normal direction over the chart.

On a normalized unit ball let the metric oscillation, Christoffel symbols, smooth-side curvature, face second fundamental form if present, and forcing be at most \varepsilon. The reflected and interior cases then both give the distributional inequality

where B_1^i=2JG P^i+B_{\rm face}^i and \gamma_1>0 is fixed. The matrix a_1 is symmetric and uniformly elliptic with fixed constants. The face can be absent or can be a plane at any position in the ball.

3. Gradient-norm drop or approximation by a nonzero affine function. Fix r_*\in(0,1/32). For every b_*>0 there are k_*\in(r_*,1) and \varepsilon_*>0 with the following property. Suppose that |\nabla_g u|_g\leq1 on B_1, the normalized geometric and forcing errors just listed are at most \varepsilon_*, and u(0)=0. Then either

or there is an affine L such that

Here is the compactness argument with its energy step. Otherwise choose a sequence with errors tending to zero, no approximation in Equation (5.23), and gradient supremum tending to one on B_{r_*}. Its functions are uniformly Lipschitz and vanish at zero. Equation (5.16), applied to

gives uniform local L^2 bounds for D^2u, hence for H and Ds. The weak Harnack inequality applies to the nonnegative weak supersolution s in Equation (5.21) after discarding the negative Hessian term. It gives, for some p_0>0,

The infimum tends to zero because B_{r_*}\subset B_{1/3} and the gradient supremum there tends to one. Thus s\to0 in measure locally.

To recover the Hessian energy, take a compactly supported cutoff \psi in B_{1/3} and test the same inequality with \psi^2(b-s)_+, where 0<b<1 is fixed. Its weak form is

Dropping the nonnegative Hessian term, expanding the test derivative and using Young’s inequality proves

On a fixed compact subset, the integral of |Ds| over \{s<b\} has limsup at most Cb. Its integral over \{s\geq b\} tends to zero by the uniform L^2 bound for Ds and convergence in measure of s. Letting b\downarrow0 yields Ds\to0 locally in L^1. Testing Equation (5.21) with a fixed nonnegative smooth cutoff now yields H\to0 locally in L^2. Since \Gamma\to0 and Du is bounded, also D^2u\to0 locally in L^2.

Uniform Lipschitz compactness produces a locally uniform limit. Its distributional Hessian vanishes, so it is affine. Poincaré’s inequality and D^2u\to0 give convergence of the gradients to its slope in local L^2. Since s\to0 in measure and g\to I, that slope has norm one. This contradicts the failure of Equation (5.23) and proves the dichotomy. Increasing k_* if necessary ensures r_*<k_*<1.

Repeated gradient-norm drops will give decay directly. To handle the other alternative, we next improve approximation by a nonzero affine function. Its fixed slope determines the direction to freeze; the axial eigenvalue remains measurable throughout this argument.

4. The frozen-axis linear equation. Let e_0 be a fixed Euclidean unit vector. We first prove the quantitative statement

for Y_0\in W^{2,2} and arbitrary measurable c\leq\chi\leq1. Rotate so that e_0=\partial_n and put w=\partial_nY_0\in W^{1,2}. Differentiating the displayed equation only in the sense of distributions gives exactly

This is a scalar divergence equation for a W^{1,2} weak solution with a measurable uniformly elliptic matrix. The De Giorgi–Nash estimate gives w\in C^{\alpha_1} on smaller balls, for a fixed 0<\alpha_1<1, quantitatively in its L^2 norm. Testing with \psi^2(w-w(x)), where \psi is a cutoff on the ball, and using that Hölder bound gives, uniformly for balls in a smaller fixed ball,

The ordinary Laplacian of Y_0 is f=(1-\chi)\partial_nw. By Cauchy–Schwarz,

Here is why this controls every component of DY_0. Cut f off on an intermediate ball and form its Newton potential V. For two points at distance r in a smaller ball, the part of DV within distance 2r of either point is bounded by

For distances R\geq2r, the difference of the two gradient kernels is at most CrR^{-n}. Summing over dyadic annuli and using Equation (5.28) gives

The farther part is bounded by the total L^1 norm of f and contributes O(r). The same estimates show that V,DV are bounded. The difference Y_0-V is harmonic on the smaller ball, so interior harmonic estimates control its gradient and its gradient Hölder norm by its L^2 norm. Together with the initial W^{2,2} bound this proves Equation (5.25). Taking slightly nested balls in this argument yields the stated B_{1/2} estimate.

5. Improvement near a nonzero affine function. Fix a universal gradient bound, say |\nabla_g u|_g\leq4. There are \alpha_2\in(0,\alpha_1), \lambda\in(0,1/8) and small b_0,\delta>0, depending only on n,c, with the following property. If L is affine and

then there is an affine L' such that

The metrics may again be smooth or reflected.

Set Y=(u-L)/b. Its coordinate equation is

whose right side has absolute value at most C\delta. Equation (5.16) and interpolation give

Put e_0=DL/|DL| and a_0=I-(1-\chi)e_0\otimes e_0. Since Du=DL+bDY and |DL|\geq1/4, the direction projections obey

Indeed the map taking a nonzero vector to its unit-direction projection has the estimate |\Pi(v)-\Pi(q)|\leq C\min(1,|v-q|/|q|) when q\ne0. At v=0 any assigned unit projection satisfies the same estimate after enlarging C. The change from Du to its metric gradient contributes O(\|g-I\|_\infty). Hölder’s inequality, with 1/2=1/s_0+1/p and interpolation of the bounded matrix difference in L^p, therefore gives

We remove this residual in W^{2,2}, without any Sobolev embedding into C^0. On the convex ball \mathcal B=B_{3/4}, integration by parts for a zero-trace function gives

The omitted boundary term is the nonnegative integral of the sphere’s mean curvature times (\partial_\nu v)^2; the identity extends by density to W^{2,2}\cap W^{1,2}_0. Thus the Dirichlet solution operator \Delta_D^{-1} satisfies \|D^2\Delta_D^{-1}f\|_2\leq\|f\|_2. The map

is a contraction, with factor at most 1-c, on W^{2,2}(\mathcal B)\cap W^{1,2}_0(\mathcal B) equipped with the Hessian norm. Its fixed point satisfies

Consequently Y_0=Y-v satisfies the homogeneous frozen equation and has bounded W^{2,2}(\mathcal B) norm. By Equation (5.25), \|Y_0\|_{C^{1,\alpha_1}(B_{3/8})}\leq C.

Independently, Y has a uniform C^\theta bound on B_{1/2} by the Krylov–Safonov estimate applied to its unfrozen equation: the matrix a is measurable and uniformly elliptic, its source is bounded, and \|Y\|_\infty\leq1. The function Y is individually W^{2,n} (also across the face), so the strong-solution version of that theorem applies. Hence v=Y-Y_0 has uniformly bounded C^\theta norm on B_{3/8}. Its small L^2 norm now implies

To see the first inequality for small \|v\|_2, a value |v(x)|=m forces |v|\geq m/2 on a ball of radius (m/(2C))^{1/\theta}; integrating its square gives the stated exponent.

Choose \alpha_2<\alpha_1, then \lambda so small that C\lambda^{1+\alpha_1}\leq\tfrac12\lambda^{1+\alpha_2}, and finally \delta,b_0 so small that the right side of Equation (5.33) is at most \tfrac12\lambda^{1+\alpha_2}. The affine function

then satisfies Equation (5.30).

6. Iteration, including its transition between the two alternatives. Decrease b_0 further so that Cb_0/(1-\lambda^{\alpha_2})\leq1/8, where C is the slope constant in Equation (5.30). Apply the dichotomy with b_*=b_0. At each center, first normalize g there and choose a fixed sufficiently small radius R_0. Divide heights by K R_0, where K\geq\max(1,\|\nabla_g u\|_\infty), and subtract the center value. On the rescaled unit ball the gradient is at most one. The forcing is multiplied by R_0/K, the connection and face second form by R_0, and smooth-side curvature by R_0^2. Metric oscillation from its center value is O(R_0). Choose R_0 uniformly so that these errors satisfy both the dichotomy threshold and b_0\delta.

Denote this normalized function by u_0. After j successive norm drops, the normalized function is

It again has gradient at most one. Its forcing has acquired the factor (r_*/k_*)^j, and its metric, connection and face errors have acquired at least the factor r_*^j. All required smallness conditions persist because r_*<k_*. If this branch continues forever, then

If the affine alternative first occurs after j norm drops, rescale its ball once by r_*, retaining its current gradient normalization. The resulting function U has a unit-ball affine error at most b_0 and an initial slope in [1/2,2]. Iterate the affine improvement using

Only constants are subtracted from the nonlinear unknown if a renormalization of its value is needed; the affine functions are used solely as comparison functions. In particular the original gradient-aligned equation remains its equation at every step. The relative error threshold remains valid: geometric and forcing errors acquire a factor \lambda^i, and

The gradient remains bounded by its entry normalization, and the affine slopes change by at most Cb_i. Their total change is at most 1/8, so all slopes remain within [1/4,4], as required by Equation (5.29).

Take 0<\alpha\leq\min(\alpha_d,\alpha_2). Before affine entry the constant approximations above have error Cr^{1+\alpha}. At entry the original radius is r_e=r_*^{j+1} and the gradient normalization is a_e=k_*^j. At the subsequent radii r=r_e\lambda^i, the affine errors in the original u_0 are bounded by

because a_e r_e^{-\alpha}=k_*^{-1}r_e^{\alpha_d-\alpha}\leq k_*^{-1}. Thus the constants remain uniform across every possible entry scale. Intermediate radii are handled by restricting the approximation from the next larger radius, whose ratio is one of the two fixed numbers r_*^{-1} and \lambda^{-1}.

We have proved that at every center and every sufficiently small radius there is an affine approximation with uniform error Cr^{1+\alpha}. The elementary affine Campanato criterion gives the gradient estimate: comparison of the affine functions on concentric radii r,r/2 bounds their slope difference by Cr^\alpha; summation identifies the limiting slope with Du at the center. Comparing the approximations at two centers in a common ball of radius comparable to their distance then bounds the difference of these limiting slopes by that distance to the power \alpha. This proves the first estimate in Equation (5.14), also on the reflected face, with constants uniform on the smaller original chart.

Finally let L_x(y)=u(x)+Du(x)\cdot(y-x) in fixed coordinates. The gradient estimate gives \|u-L_x\|_{L^\infty(B_{2r}(x))}\leq Cr^{1+\alpha}. Since

Equation (5.16) with s=2 yields

The bounded connection term converting D^2u to \Hess_g u has L^2 norm O(r^{n/2}). Squaring the estimate and using \alpha<1 proves the Hessian estimate in Equation (5.14). Restricting the reflected estimate to the original half-ball proves the face version.□

Proof. If a Dirichlet face is present, first subtract its constant boundary value. This changes neither the equation nor its source bounds. The reflection creates no measure on that face. Flatten it as \{x_n=0\} and write J=\operatorname{diag}(1,\ldots,1,-1). For x_n<0 extend the four quantities by

The zero trace makes Z^e a W^{1,2} function, and ellipticity is preserved. To verify the equation across the face, test the original equation on the upper half with \varphi(x)-\varphi(Jx), which has zero trace. Changing variables in the reflected half gives precisely Equation (5.34) for the extended quantities. Equivalently, the normal component of the full flux ADZ+B agrees in the weak flux sense on the two sides. Thus there is no boundary flux jump to estimate. A different constant Dirichlet value is handled by subtracting that constant first. Reflect the nonnegative density 1+F_0 evenly. The ball-mass bound changes by at most a fixed multiplicative factor. The reflected coefficients need not be smooth.

Exponential absorption and potentials. Use \nu=(1+F_0)\,dx, with the even extension just described where needed. We give the two-sign exponential argument for Equation (5.34). Put Q_\varepsilon=e^{\varepsilon LZ} for \varepsilon\in\{-1,1\}, and let \mathcal L=-\partial_i(A^{ij}\partial_j). The weak chain rule gives

Indeed it follows directly by testing the equation for Z with \varepsilon LQ_\varepsilon\varphi. Young’s inequality bounds the second term on the right by \tfrac12\lambda L^2Q_\varepsilon|DZ|^2+C L^2Q_\varepsilon. Taking L\ge 2C/\lambda absorbs also the quadratic part of \varepsilon LQ_\varepsilon S. Since Z is bounded and L is now fixed, we obtain, for both signs,

This uses no derivative of B, A, or the density of \nu.

On a ball B_r in the resulting full patch, let v_\varepsilon solve

The density of \nu is bounded for each individual classical solution, so this solve and the following tests can first be made in W^{1,2}. Only the uniform mass bound in Equation (5.35) enters the estimate. The part with right side \divg B_\varepsilon has absolute value at most Cr\|B_\varepsilon\|_\infty. This is the scaled bounded-divergence-source estimate: testing by positive and negative level truncations gives \int|D(v-k)_+|^2\le C\|B_\varepsilon\|_\infty^2 |\{v>k\}|, and Sobolev level iteration on the unit ball proves the asserted bound after rescaling.

For the measure part, the scalar Dirichlet Green function of a uniformly elliptic divergence operator in dimension n\ge3 satisfies 0\le G_r(x,y)\le C|x-y|^{2-n}. For bounded measurable coefficients this follows from the whole-space Green-function comparison [30, Section 7]: extend A uniformly elliptically off the ball, and compare the whole-space and zero-Dirichlet Green functions by the maximum principle. Dividing B_r into annuli about any x\in B_r gives

We work sufficiently far inside the coordinate patch that the mass bound applies to the balls in this sum. Consequently

Oscillation decay. Here is the oscillation reduction, including the role of the two signs. Let m=\inf_{B_r}Z, M=\sup_{B_r}Z, and \omega=M-m. By Equations (5.38) and (5.39), the functions

are nonnegative weak supersolutions of \mathcal L W\ge0 on B_r. At least one of the sets

has at least half the measure of B_{r/2}. On the first set e^{LM}-Q_+\ge c\omega, and on the second e^{-Lm}-Q_-\ge c\omega, because the exponential and its inverse are uniformly Lipschitz on the fixed bounded range of Z. For the corresponding sign the weak Harnack inequality, with its fixed exponent q>0, yields

Subtracting v_\varepsilon+E_r, whose absolute value is at most 2E_r, and using the same Lipschitz comparison with Z, improves one of its two extremes. Hence, for a fixed \theta\in(0,1),

Iteration gives a uniform C^b bound for any fixed

Applying the same conclusion to the reflected equation proves the estimate up to the constant Dirichlet face.□

Proof. 1. The trace estimate and divergence form. Fix N and a member of the homotopy. Proposition 5.3 gives uniform bounds for |f|, |df|, and |Z|, and uniform ellipticity on coordinate balls of a fixed small radius. The radius and constants may depend on N, but can be chosen independently of the outer truncation. Lemma 5.4, applied to the trace equation, gives some a\in(0,1) such that

Here and below the boundary version uses a smooth half-ball with constant Dirichlet data; coordinate and covariant Hessians differ by a bounded quantity. Each occurrence of a ball in a mass estimate allows all centers in the coordinate patch, not just one fixed center.

The implicit relation defining t has derivative 1+pv/k\ge1 with respect to t. On the bounded range now under consideration it therefore defines a smooth function of (x,Z,Df), whose relevant partial derivatives are bounded. In particular,

All the expressions in the equations are smooth before the block decomposition, including at df=0. Since \mathcal T is quadratic in H^f, dt, and K, Equation (5.43) and the bounded zeroth- and first-order quantities imply the following coordinate form of the second equation:

The matrix A is symmetric. With volume densities included, A represents kuA_\chi, B represents uA_\chi K(w,\cdot), and S=-\sqrt{\det g}\,u\Xi. The same bounds hold uniformly along both stages of the homotopy; scaling \Xi towards zero improves its absolute-value bound. No continuity modulus for A is being used yet.

Apply Lemma 5.5 with F_0=|D^2f|^2. Its coefficient and source hypotheses are precisely Equation (5.44). On patches of size at most one, Equation (5.42) gives \nu(B_r)\le Cr^{n-2+a} for \nu=(1+|D^2f|^2)\,dx. The lemma therefore gives a uniform C^b bound for Z for some 0<b<a, up to the constant outer Dirichlet face as well. Both Z and Df now have the continuity needed to improve the trace equation by Schauder theory.

2. Absorbing the quadratic gradient term. The coefficients of the trace equation are smooth functions of (x,Z,Df). The preceding estimate and Equation (5.42) therefore give a uniform C^c modulus for those coefficients and the trace right side, with c=\min(a,b)>0. Interior and Dirichlet Schauder estimates give

on smaller patches of the same uniform type. This use of Schauder requires only the now established Hölder modulus of (Z,Df).

Expand the divergence equation on each original ball or half-ball. Since A=A(x,Z,Df) and B=B(x,Z,Df), their derivatives consist of bounded terms, terms linear in DZ, and terms linear in D^2f. Equation (5.45) and the formula for Dt show that the expanded equation has the form

The bound on the Hölder norm of A does not presuppose a bound for DZ. Coordinate lower-order terms are included in H.

Fix any finite s>n. Freezing the now Hölder principal matrix on sufficiently small balls gives the interior and flat-Dirichlet W^{2,s} estimates with uniformly bounded coefficient in front of \|H\|_s. On nested patches Q_r\subset Q_{2r} these give

The last constant includes the bounded height, the constant term in Equation (5.46), and the scaled cutoff costs. For a ball or a fixed-shape half-ball the scaled interpolation inequality is

where Q has size r. Indeed apply the usual second-order interpolation inequality on the unit ball or half-ball after subtracting a constant and rescaling. On a half-ball, the constant Dirichlet value belongs to the closed range of Z, so it can be used for this subtraction. The constants depend only on the uniform chart shapes, s, and the dimension.

For clarity, the absorption is local and does not lose the number of patches in a large truncation. For a fixed sufficiently small r, let

It is finite for each individual classical solution on a truncation. The uniform local geometry allows each Q_{2r}(x) to be covered by at most J_0 patches of size r, with J_0 independent of the outer radius. Thus Equations (5.47) and (5.48) imply

Choose r small enough, using the uniform C^b bound already proved, that the displayed coefficient of \mathcal M_r is at most 1/2. This gives a uniform local W^{2,s} bound up to the outer face, independently of the total number of patches. Sobolev embedding now bounds Z in C^{1,d} for d<1-n/s.

3. Bootstrap. The trace equation consequently bounds f in C^{3,d'} for some d'>0. Its coefficients now have one Hölder derivative; the expanded second equation has Hölder right side, giving Z\in C^{2,d'}. Repetition yields bounds of every fixed order, since the background data are smooth. More explicitly, once Z\in C^{j,d'} and f\in C^{j+1,d'} for j\ge1, the trace equation first gives f\in C^{j+2,d'}, and the divergence equation then gives Z\in C^{j+1,d'}. Shrinking the exponent once at the first step suffices. The argument applies initially to the C^{2,\gamma} fixed points constructed below: their individual Hessian norms are already finite, and smoothness is the conclusion of the bootstrap. All constants in this paragraph may depend arbitrarily on fixed N and the chosen derivative order.□

Proof. 1. Coefficient growth without a floor. Let b=\sqrt D/l. On a bounded trial range the coefficients satisfy, for every \sigma=|df|,

We verify the bounds also at large slope, where t has no fixed floor. The implicit relation implies t<0 when \sigma is sufficiently large. There p=N, l=e^{Nt}, and

Uniformly on a bounded z range, as \sigma\to\infty,

The exponent is less than one because N>k. Moreover b=\sigma/\sqrt v\asymp\sigma there. The remaining bounded slope range is compact in (z,\sigma), and the coefficients are smooth and positive on it. This proves Equation (5.49).

2. A global gradient bound. Continue from \Delta_g f=f by solving, for 0\le q\le1,

The scalar c_q is positive. Extrema give a common height bound \|f\|_\infty\le H, since

On this height range the right side of the equation written as A_q:\Hess f=G(x,f,df) satisfies, for a covector \xi,

uniformly in q,a. The eigenvalues of A_q are one in the transverse directions and \chi_q=1-q+q\chi\ge c/(1+|\xi|) along \xi.

Here is a gradient estimate using precisely this lower axial bound. Choose K_0 sufficiently large, then a positive distance d_0 smaller than the boundary tubular radius, the injectivity radius of a smooth extension of the compact domain, and (2K_0)^{-1}. For M>0 set

By increasing M we ensure both \psi'\ge1 on [0,d_0] and \psi(d_0)>2H. Indeed \psi'(d_0) tends to (K_0d_0)^{-1}>2, while \psi(d_0) tends to infinity.

First compare with \pm\psi(d(x,\partial\mathcal D)) in the boundary collar. At a contact, the gradient is normal and has length \sigma=\psi'. The tangential Hessian contraction costs at most C\sigma, whereas

Choose K_0 large enough to dominate the collar curvature and Equation (5.51). The positive barrier is a strict supersolution; the negative one is a strict subsolution. On the inner edge of this collar they dominate \pm f by the height bound, and both vanish at the boundary. Therefore

Consider now f(x)-f(y)-\psi(d(x,y)) for pairs with 0\le d(x,y)\le d_0, using the distance of the smooth ambient extension. It is nonpositive on the diagonal, for d=d_0, and when either endpoint lies on the boundary, by Equation (5.53) and the monotonicity of \psi. Suppose it had a positive interior maximum. The endpoint gradients are parallel along their unique short joining geodesic and have the same length \sigma=\psi'\ge1. Write e for their common parallel direction. Varying one endpoint in the axial direction gives

Varying both endpoints in parallel transverse directions and summing the second-variation inequalities gives

For completeness, the simultaneous endpoint variation is generated by a parallel vector field along the geodesic; its index form is bounded above by C d(x,y) since the ambient curvature is bounded. This is the displayed estimate after multiplication by \psi'. The transverse eigenvalues are exactly one at both endpoints, so

The right sides of Equation (5.50) differ by a quantity at least -2C(1+\sigma). A sufficiently large K_0 contradicts this. Notice that no estimate for \chi_{q,x}-\chi_{q,y} was needed: both signed axial Hessian bounds have the favorable sign. We conclude |f(x)-f(y)|\le\psi(d(x,y)) and hence |df|\le\psi'(0)=M.

3. Continuation and regularity. Equation (5.49) now gives uniform ellipticity throughout the continuation. Its normalized forcing is bounded. Lemma 5.4 gives a common positive Hölder exponent for df, including the boundary. Since the coefficients are smooth functions of (x,z,df), the Dirichlet Schauder estimate first gives f\in C^{2,\gamma_0} for a possibly smaller exponent. The resulting Lipschitz bound for df then makes those coefficients C^{0,\gamma}, so the estimate improves to C^{2,\gamma}. Differentiating the equation once gives C^{3,\gamma} estimates: the trial is C^{1,\gamma} and all differentiated coefficient and source terms are controlled. These scalar estimates are uniform along the continuation; they are the interior and smooth Dirichlet estimates of [21] on a finite coordinate covering.

The linearization in f is a uniformly elliptic scalar operator with bounded Hölder first-order coefficients and zero-order coefficient -c_q<0. Its homogeneous Dirichlet kernel is zero by the maximum principle. The scalar linear Dirichlet continuity theorem, starting from \Delta_g-1 and using the Schauder estimate, makes it invertible from C^{2,\gamma}_0 to C^{0,\gamma}. The implicit function theorem gives openness of the nonlinear continuation set; the bounds and compactness give closedness. The initial solution is f=0, hence existence follows at q=1.

4. Uniqueness and continuity. For uniqueness compare two solutions at a positive maximum of their difference. Their gradients agree there, so their coefficient matrices and their positive factors b agree; Hessian ordering and the strictly negative zero-order term give a contradiction. For continuity in the stated norm, regard the normalized trace operator as a map

It is continuously differentiable in the trial and solution variables, since its coefficient functions are smooth on every bounded jet range. At a solution its linearized principal, first-order, and zero-order coefficients are C^{1,\gamma}. The same maximum principle and the one-derivative-higher Dirichlet Schauder estimate therefore make that linearization an isomorphism from C^{3,\gamma}_0 onto C^{1,\gamma}. The implicit function theorem supplies local continuous dependence into C^{3,\gamma}; uniqueness joins these local maps. The a priori estimates already proved give the bounded-set assertion.□

Completion of the proof of Theorem 5.1. Keep N fixed and let R\to\infty. The estimates of Proposition 5.6 on compact sets are independent of R. A diagonal subsequence therefore converges smoothly on every compact subset of \mathcal M_N to a solution (f,Z). The polynomial bounds and height comparisons survive. They give t\ge-\epsilon; equality would be an interior global minimum and contradict Lemma 4.3.

We prove the remaining end estimates on the truncation solutions, uniformly in R at this fixed N; the estimates then pass to the smooth limit just obtained.

On the end K=0 and the trace equation reads

For small a_N>0 and a sufficiently large fixed r_2(N), v_N(r)=e^{-a_N(r-r_2)} obeys

In fact 0<A_\chi\le I and the end metric bounds give A_\chi:\Hess v_N\le C(a_N^2+a_N/r)v_N; choose Ca_N^2<\eta/(4e) and then r_2 large. Multiples of \pm v_N dominate f at S_{r_2} and at the outer zero sphere. The maximum principle for the operator in Equation (5.56), with its coefficients evaluated at the solution, proves exponential decay uniformly on all truncations. The already established scalar local estimates, on balls and boundary half-balls of a fixed size, then give the same decay for every derivative of f. In particular t-Z and its derivatives are exponentially small.

The second equation can consequently be written, with an error exponentially small through every fixed number of derivatives, as

To verify this formula, set f=0 in the coefficients. Then D=1, t=Z, w=0, A_\chi=I, \mathcal T=ks_{p(Z)}|dZ|^2, and V=ku(Z)\nabla Z with (\log u)'=k-1+p. Every difference from these expressions has an exponentially small factor involving a derivative of f; its other factors are bounded at fixed N by local regularity. This also proves the differentiated error assertion. The outer boundary causes no loss, since the same fixed-size local estimates hold there uniformly in R.

Define a smooth increasing function Q_N on the bounded range of Z by

Its derivative has positive upper and lower bounds at fixed N. Equation (5.57) and the chain rule give

Here C_N' denotes a constant at fixed N, not necessarily the penalty polynomial. The function

is positive for r>1 and satisfies

sufficiently far out. The correction dominates the metric error because \beta/2<n-2. Taking a sufficiently large multiple of b_* dominates both the source in Equation (5.58) and the inner boundary values. Comparison with its two signs, also valid on the outer zero sphere, gives |Q_N(Z)|\le C_N' r^{2-n} on every truncation. Monotonicity of Q_N gives Z=O(r^{2-n}), and hence the same bound for t.

Rescale successive end annuli to unit size. The metric coefficients have uniformly controlled derivatives and converge to the Euclidean ones. The right side of the equation for Q_N(Z) is a smooth function of Z times r^{-n-\beta} plus exponentially small terms. Interior Poisson W^{2,p} estimates, then Schauder estimates and differentiation, give Z,t=O_j(r^{2-n}) for each j; the constants may depend on N,j. Finally |dZ|^2=O(r^{2-2n})=O(r^{-n-\beta}), since \beta<n-2. Equation (5.57) now proves the stated \Delta_g t bound and finishes the theorem.□

Proof. Use the signed transverse coordinate s increasing from each product neck toward S and the original end. The fixed transition from product geometry to an original collar has been chosen so that its leaves have

for some fixed \delta>0. Shrink \delta when necessary. On the cap and the product part K=-L_0g, where L_0 is a fixed large constant.

It is useful to write the trace equation directly in terms of h. At a point where a test function has the same gradient as h, put a_0=\eta/l and let \chi have its value for the solution there. The operator applied to the test function v_* is

The equation is \mathcal Q(h)=h. The coefficients in Equation (6.6) are evaluated at the solution; we do not replace its value of t by one computed from the barrier. Nevertheless,

These inequalities suffice for a comparison at a positive maximum of h-v_*.

Set v_*=-\delta on the cap. On a product neck let q_N=v_*'\ge0 start from zero, increase approximately exponentially, and end at a small fixed number q_*>0. One explicit construction is to take an exponential q_*e^{s-L_N} where its size lies between a constant multiple of \eta and q_*, smooth its end to the constant q_* on an interval of fixed length, and multiply its initial part by a fixed smooth cutoff on an interval where it is O(\eta). Choose

The cutoff can be flat at its initial endpoint. This construction gives

with C independent of N. Thus v_* glues smoothly to the constant on the cap. The same construction is made separately on every neck. The cap value is common to all components.

The product Hessian has only an axial component. Equations (6.7)–(6.8) give, even where the slope vanishes,

Consequently \mathcal Q(v_*)\le-kL_0+C' on this part. Choose L_0 large enough that this is less than -2\delta. Since v_*\ge-\delta, this is a strict supersolution inequality \mathcal Q(v_*)<v_*. It also holds on the constant cap, where the trace of K is -nL_0.

Continue with the fixed slope q_* across the fixed transition and up to S. Choose q_* so small that all the increase up to S, including Equation (6.8), is less than \delta/4. Beyond S, still inside the fixed strictly trapped collar, increase the slope smoothly until v_*>C_0+1 at the outer edge of that collar. This requires only a fixed, possibly large, slope and fixed derivatives; they do not depend on N. Keep v_*'\ge q_* on the entire transition and collar. For such a function of signed distance,

Equation (6.7) shows that this converges uniformly to the left side of Equation (6.5). For example, the absolute error is bounded by C\ell/q_*^2+C\ell\sup|v_*''|/q_*^3, with fixed C. For large N it is therefore less than \delta. As v_*\ge-\delta, again \mathcal Q(v_*)<v_*.

Apply the maximum test on the compact region consisting of the cap, necks, transitions, and these outer collars. On its outer boundary h-v_*<0. At a positive interior maximum, the gradients agree and \Hess h\le\Hess v_*. Positive definiteness of A_\chi would give

a contradiction. Thus h\le v_*. Up to S the barrier is less than -3\delta/4. Continuity of the fixed collar barrier gives the same strict negative bound on a small neighborhood of S. Taking, for example, b_2=\delta/2 proves the assertion.□

Proof. The solution supplied by Theorem 5.1 has t>-\epsilon and the parameters of the scalar construction. Thus Equation (4.23) applies without a change of data or normalization. On |h|\le b_3, its constraint and trace terms have lower bound \rho/2 by Equation (6.9); the remaining terms are nonnegative and retain the positive contribution (1-\delta_0)\mathcal T.

For the new conformal factor we use Equation (4.25), with \eta=\ell^{3/2}. Its constants are polynomial in N for these fixed prepared data. The factor \sqrt d in the height Laplacian cancels the axial loss in weighted coercivity, as shown immediately after that equation. No arbitrary fixed-N estimate for unweighted second derivatives enters this bound.

The conformal scalar-curvature law in dimension n=k+1 reads

The chain rule, Equation (4.25), and the fixed derivative bounds of \zeta imply

On the support of these errors b_3/4\le|h|\le b_3/2. This region is disjoint from the filling by Lemma 6.1, and is contained in a fixed compact portion of the original exterior. There \rho\ge c_*>0. Young’s inequality bounds the error containing \sqrt{\mathcal T} by (1-\delta_0)\mathcal T/2+P_N\ell^{1/2}. All remaining errors tend to zero. Equation (4.23) therefore proves nonnegative scalar curvature there for large N. Where \zeta is constant, Equation (6.12) has no error terms and the same conclusion follows directly.

The floor t>-\epsilon, the nonnegative graph term, and \psi_N\ge0 give the metric comparison. Finally \psi_N=0 near infinity, so Equation (6.11) is exactly Proposition 4.4. In particular its sign is the upper energy estimate required for the Riemannian inequality.□

Proof. At fixed N the end is smooth and asymptotically Euclidean with the differentiated falloff in Equation (6.3). Consequently sufficiently large coordinate spheres have positive mean curvature for \widetilde g_N toward infinity. The required radius may depend arbitrarily on N.

Here is the clipping argument that gives the compact support needed for the direct method. All objects in this paragraph use \widetilde g_N. On the large end let Y=\nabla r/|\nabla r|, so |Y|=1 and \divg Y=H_r>0. For a bounded finite-perimeter competitor F and almost every large R, Gauss–Green on F\cap\{r>R\} gives

The spherical normal here is -Y. Boundedness of F eliminates an outer flux; alternatively first cut off Y beyond its support. The first boundary integral is at most P(F;\{r>R\}). The BV truncation formula therefore yields the stronger clipping inequality

The Gauss–Green and truncation formulas hold at almost every radius by the trace and slicing theorems for BV functions [3]. The notation \{r<R\} here includes the entire compact core. In particular it contains O_N once R is sufficiently large.

Choose a fixed interval of such radii beyond the obstacle. Applying Equation (6.16) to each term of a minimizing sequence, with an almost-everywhere admissible radius in that interval, confines the sequence to one compact set. BV compactness gives L^1 convergence along a subsequence, and lower semicontinuity gives a minimizer. Containment of O_N is preserved almost everywhere. The minimum is the unrestricted bounded-competitor minimum, so local variations are not subject to an artificial outer constraint.

Choose a fixed original coordinate sphere S_{R_*} so far out that |h|<b_3/4 on and beyond it for every N. It encloses O_N when N is large. On it \psi_N=0, and Equation (6.2) implies \widehat g\le e^{P_N}g. Its filled side is a competitor; increasing P_N gives Equation (6.15). The fixed sphere need not lie in the region used for clipping: the latter was needed only for compactness at fixed N.

All minimizers have a common compact support bound: if a minimizer had positive volume beyond an almost-everywhere admissible clipping radius, the strictly positive last integral in Equation (6.16) would contradict minimality. Among minimizers, maximize the enclosed volume. Compactness, lower semicontinuity, and convergence of volume show that this maximum is attained. Perimeter submodularity implies

For two minimizers both sets on the left contain O_N, so equality holds and their union is a minimizer. Maximal volume therefore implies G\subset F_N modulo null sets for every minimizer G.□

Proof. The two different density arguments are illustrated in Figure 1. Suppose p belongs to this intersection. It cannot lie in the interior of O_N, because F_N contains O_N almost everywhere. Thus \dist_g(p,C_N^h)=s_N. The complete background metric has a minimizing geodesic from some z\in C_N^h to p. In particular B^g_{s_N}(z)\subset O_N. The midpoint of this geodesic is the center of a ball of radius s_N/4 contained in both O_N and B^g_{s_N}(p). Uniform background ball-volume bounds give

This is the filled-side density, obtained from the distance obstacle.

We derive the other density from minimization. On B^g_{s_N}(p) the high conformal multiplier is constant. Remove it and write \gamma=\widehat g. Equation (6.2) gives, on this ball and every k-plane,

after changing the polynomial. No derivative of \gamma appears in this comparison.

Put U=\mathcal M_N\setminus F_N and V(r)=\Vol_g(U\cap B^g_r(p)). Since p is in the support of perimeter, V(r)>0 for every r>0: otherwise \mathbf1_{F_N} would be constant almost everywhere on some neighborhood of p. For almost every 0<r<s_N, filling the ball is an admissible competitor. Cancel the unchanged exterior perimeter and the common constant factor e^{kA_N}. The perimeter truncation formula gives

Here U^{(1)} is the measure-theoretic interior; the displayed trace agrees with it for almost every radius. The coarea formula in the smooth background metric and Equation (6.19) then yield

The absolute isoperimetric inequality on a uniformly controlled background ball, applied to the zero extension of \mathbf1_{U\cap B^g_r(p)}, includes its spherical trace:

Local Euclidean isoperimetry in a uniform coordinate ball proves this form; the smooth background metric changes its constant by a fixed factor. Since V is absolutely continuous and positive for r>0, its nth root satisfies

Integrate from a>0 to r and let a\downarrow0. The result is

with another polynomial. This proves the unfilled-side density from a one-sided comparison, without assuming that the obstacle contact is a regular point or that the deformed metric has controlled derivatives.

Relative isoperimetry in B^g_{s_N}(p), together with Equations (6.18) and (6.21), now gives

Absorb s_N^{n-1}=e^{-(n-1)Q_N} into the polynomial exponential, use Equation (6.19), and restore the constant multiplier. We obtain

For fixed \epsilon>0, e^{N\epsilon/4} dominates every polynomial in N. Equation (6.22) contradicts Equation (6.15) for large N.□

Proof of Theorem 2.2. The connected exterior and its curvature region.

Choose F_N as in Lemma 6.3. Its frontier is compact and, by Lemma 6.4, has positive distance from O_N. Every sufficiently small variation of the set near a frontier point remains admissible. Thus its boundary is locally perimeter minimizing in the smooth metric \widetilde g_N.

We use the regularity theorem for codimension-one minimizing boundaries: the support of the perimeter measure is a smooth embedded minimal hypersurface outside a relatively closed set of Hausdorff dimension at most n-8. A directly applicable smooth-ambient formulation is [42, Chapter 7, Section 5, Theorem 5.8]; its hypersurface dimension is k=n-1, so the singular bound is k-7=n-8. Multiplicity one for the boundary of a set is recorded in [42, Chapter 7, Section 5, Remark 5.2]. The compactness and tangent-cone conclusions used in the minimizing-boundary regularity argument are Theorems 5.3 and 5.5 of that section.

We also specify the representative of the filled set. Since the frontier is detached from the obstacle, both insertion and deletion of a sufficiently small ball are admissible. The coarea and isoperimetric argument used in the preceding lemma, now applied to each side, gives positive lower volume densities for both F_N and its complement at every point in the support of perimeter. On a ball not meeting that support the indicator has zero distributional derivative, hence is constant almost everywhere. Let U_N be the union of those balls on which that constant is zero, and let the closed filled representative be \mathcal M_N\setminus U_N. The two density bounds identify its topological frontier with the perimeter support: every neighborhood of a support point contains both filled and unfilled points. Conversely a point outside the support has a constant-indicator neighborhood. This representative agrees with F_N up to a null set; the regularity theorem makes its frontier n-dimensional-volume null. Its perimeter and all the preceding minimizing comparisons are therefore unchanged.

Put \Sigma_N=\partial U_N. There is only one component of U_N. Indeed there is exactly one unbounded component, since F_N is bounded and the ambient manifold has just one end. Filling all bounded complementary components cannot increase perimeter. This statement can be seen directly on regular boundary charts, where it deletes the separating sheets and adds no sheet; the singular set has zero \mathcal H^k measure. Equivalently it is the component decomposition and perimeter additivity for finite-perimeter sets [3]. The resulting set is still an admissible bounded competitor. If any bounded open component existed, its positive volume would strictly increase the filled volume; minimality would either be contradicted by a perimeter decrease or preserved with a volume increase. Both alternatives contradict the choice of F_N. Thus U_N is connected. We take X_N=\overline{U_N}, with its ambient boundary included.

The exterior lies in \Omega\cap\{|h|<b_3\}, because the filled set contains O_N\supset C_N^h. In fact it has a smooth ambient neighborhood in that region: detachment gives positive separation between \Sigma_N and O_N, and compactness permits a smooth compact domain containing O_N whose boundary is still strictly inside the filled set. Remove that domain. On the remaining smooth ambient exterior, Lemma 6.2 gives nonnegative scalar curvature, also on a neighborhood of \Sigma_N.

All local variations of \Sigma_N have zero first variation. More precisely, if Y is a compactly supported smooth ambient vector field near the frontier and \Phi_s is its flow, then \Phi_s(F_N) remains admissible for both signs of small s. Differentiation of the perimeter area formula gives

This is stationarity of the full boundary varifold, including the assertion that there is no additional first-variation term at the singular set. Global outer minimization follows because every bounded superset of F_N also contains O_N. The maximal choice of F_N also makes it the outermost member among enclosures with this same minimum perimeter, although outermostness is not required below.

Comparison with full original cuts.

We verify the comparison with original full cuts in some detail. Detachment and Lemma 6.1 put \Sigma_N in \operatorname{int}\Omega, separated from a neighborhood of the entire original filling, including every component of S. There are bounded smooth open sets F_j in the filled manifold such that

and each F_j still contains a fixed neighborhood of the whole filling. Here is a way to impose the last condition in strict BV approximation. Choose a smaller compact neighborhood on which \mathbf1_{F_N}=1 almost everywhere, separated from \Sigma_N, and a compact exterior region where it is zero. Smooth the characteristic function in finitely many coordinate charts only in the remaining region, using a partition of unity and leaving those two constant regions fixed. The resulting smooth functions u_j\in[0,1] converge in L^1 with total variation converging to P_g(F_N). Choose \delta_j\downarrow0 so slowly that \|u_j-\mathbf1_{F_N}\|_{L^1}/\delta_j\to0. By coarea there is a regular value t_j\in(\delta_j,1-\delta_j) for which

The L^1 difference of this level set from F_N is at most \|u_j-\mathbf1_{F_N}\|_{L^1}/\delta_j, and hence tends to zero. Lower semicontinuity gives the reverse perimeter liminf. Thus F_j=\{u_j>t_j\} satisfies Equation (6.24). This is the strict approximation/coarea construction of [3], localized away from the two constant regions.

For each j retain the component of \mathcal M_N\setminus\overline{F_j} that contains the distant end, and let D_j be its closure. A compact smooth boundary has finitely many components; retaining this exterior component merely selects some of those entire boundary components. It preserves smoothness and can only decrease the full boundary area. Because F_j contains a neighborhood of the whole filling, D_j\subset\Omega, its interior lies in \operatorname{int}\Omega, and D_j is closed and connected with a smooth compact intrinsic boundary \Gamma_j. It is an admissible full cut in Definition 1.2. Thus

Taking the limit gives P_g(F_N)\ge a. No component of a cut has been discarded in evaluating its intrinsic boundary; the passage to D_j is an admissible choice of the exterior domain itself. There is no coincidence with S in this approximation because of the fixed neighborhood, and the original definition still allows coincidence for its other competitors.

Since \widetilde g_N\ge e^{-2\epsilon}g on \Omega, comparison on every approximate tangent k-plane now yields

The first equality uses the regularity theorem and the zero \mathcal H^k measure of the singular set.

Completeness and the end.

The filled metric is complete, since \widetilde g_N\ge e^{-2\epsilon}g and the background filled metric is complete. Its restriction to X_N is complete with the frontier included. Here completeness is understood in the enclosing-set convention of the Riemannian input: the intrinsic length distance on X_N may be extended-valued, and its restriction to each finite-distance equivalence class is complete. For the intrinsic-length formulation, a Cauchy sequence has a subsequence whose successive intrinsic distances are less than 2^{-j}. Join successive terms by curves of length less than 2^{1-j} in X_N. Their concatenation has an ambient endpoint, by ambient completeness, and that endpoint belongs to X_N because X_N is closed. Appending this endpoint gives a curve in X_N whose tail length tends to zero, proving intrinsic convergence of the subsequence and then of the original Cauchy sequence. The frontier is included in the closed-enclosing-set sense used in the Riemannian input in Section 2.

Finally the end satisfies \widetilde g_N-I=O_j(r^{2-n}). The conformal law applied to Equation (6.3) shows

If needed decrease \beta so that 0<\beta<n-2; then the displayed bound is O(r^{-n-\beta}). All graph errors decay exponentially, and \psi_N is zero on the end. Scalar curvature is integrable there and, being smooth and bounded on the compact ambient neighborhood of the frontier, integrable on the remainder of X_N. The energy estimate is Lemma 6.2. This proves all the stated properties.□

Proof. A sufficiently large coordinate sphere encloses B, so the infimum is finite. To obtain a lower bound, choose a point of B and a smooth proper embedded ray starting there, initially normal to B and pointing into \Omega. Choose the ray to be straight in the asymptotic coordinates outside a compact set. A tubular neighborhood has product coordinates (t,y)\in[0,\infty)\times D^2, with its initial disk contained in B. On the straight tail the transverse coordinate disk can have a fixed positive width.

Choose a smooth two-form \eta supported in the interior of D^2 with \int_{D^2}\eta=1. Pull it back by (t,y)\mapsto y and extend it by zero across the lateral boundary of the tube. This gives a smooth closed two-form \omega on \Omega, smooth also at B. Its pointwise comass is bounded: this follows from compactness on the initial portion of the tube and from asymptotic flatness and fixed coordinate width on the tail. Write

Orient the transverse disk so that the flux through a sufficiently large outward-oriented coordinate sphere S_R is one.

Let \Gamma be a smooth enclosing cut lying in the interior of \Omega, oriented toward the end, and choose S_R outside it. The region between \Gamma and S_R is compact and disjoint from B; all of its inner boundary components are counted in \Gamma. Stokes’ theorem gives

This calculation also applies when the cut is disconnected. If a smooth cut touches or coincides with B, push it into the interior by a smooth collar flow pointing from B into \Omega, filling the collar on the obstacle side. The resulting surfaces are enclosing cuts and their areas converge to the full area of the original cut. Thus the same lower bound holds for boundary-coincident cuts. Taking the infimum yields a_g(B)\ge C_\omega^{-1}>0. The smooth/perimeter equivalence in the definition of the full enclosing infimum transfers this bound to its filled-perimeter formulation as well.□

Proof. Apply the pointwise identities in Equation (3.2) with n=3, k=2, p=1, and s=2f. The densities in that identity are \mu_{\mathrm A}=8\pi\mu and J_{\mathrm A}=8\pi J in this section’s physical normalization. Setting the vector argument to zero gives 8\pi e^{4f}\mu_f=8\pi\mu-4\Delta f-4|df|^2; multiplication by two is Equation (7.10). The covector formula proved there gives 8\pi J_f=e^{-2f}(8\pi J+4K(\nabla f,\cdot)). Its new-metric norm introduces a further factor e^{-2f}, so the triangle inequality gives Equation (7.12) with precisely its factor 8. The boundary formula has k\partial_\nu s=4\partial_\nu f and proves Equation (7.13). Finally Equation (3.3) uses U=e^f=u here and gives the stated positive-factor form. These are pointwise identities; no decay or additional differentiated end hypothesis is used.□

Proof. Choose smooth b\ge|K|_g, positive everywhere and equal to b_0r^{-1-\beta} sufficiently far out, where b_0>0 is chosen large enough. This is possible because \beta<q. Choose a smooth positive regularizer \zeta, equal to r^{-2} on the end, and extend w smoothly and positively to \Omega. We solve

The regularizer makes this equation smooth even at critical points.

First truncate at a large coordinate sphere S_R and prescribe \phi=0 there. The Neumann and Dirichlet conditions occur on disjoint boundary components. For 0\le t\le1, replace b in Equation (7.19) by tb. At t=0, the mixed Laplacian is invertible: its Dirichlet face gives the Poincare inequality, and the weak solution obtained from its coercive quadratic form is smooth by boundary regularity. At any solution, the linearization is a Laplacian with a smooth drift of norm at most b and with the homogeneous mixed boundary conditions. The strong maximum principle and the Hopf boundary principle make its kernel zero. It is a Fredholm perturbation of the mixed Laplacian of index zero, and is therefore invertible. The boundary estimates used here are the ordinary elliptic estimates for Dirichlet and normal Neumann conditions: positive definiteness of g verifies their principal complementing condition, and the disjoint boundary components can be covered separately [1]. The maximum principles and local elliptic estimates below are used in their usual uniformly elliptic forms [21].

To close continuation at t=1, first work on a fixed truncation. The elliptic W^{2,p} estimate and gradient interpolation give

The constants include the fixed boundary data and are uniform in t. If the supremum norms of solutions on this truncation were unbounded, divide them by their supremum norms. The preceding estimate with p>3, compactness, and the equation give a nonzero limit v with homogeneous mixed data satisfying

This can be written \Delta_gv+A\cdot dv=0 with a bounded measurable drift: take A=t_*b\nabla v/|dv| off the critical set and zero on it. The strong and Hopf maximum principles exclude a nonzero solution with the homogeneous mixed conditions. Thus the supremum norms are bounded. The W^{2,p} estimate, Schauder estimates, and the smooth equation now give the bounds needed for closedness of continuation. Existence on each truncation follows.

Every such solution is nonnegative. A negative minimum cannot be interior because the right side of Equation (7.19) is positive. At an inner-boundary minimum, the outward-domain derivative would be negative by the Hopf principle, whereas the prescribed derivative in that direction is 1. The outer value is zero.

It remains to make these bounds independent of R. For fixed sufficiently large A and then sufficiently large r_0, the positive radial function

satisfies, on r\ge r_0,

The constants are fixed independently of the truncation. Since b\zeta=o(w), a sufficiently large multiple of v_0 is a supersolution of the regularized equation. The comparison principle gives

The notation for the compact core includes all points outside the end chart. Were M_R unbounded along expanding truncations, the normalized solutions \phi/M_R would have locally uniform W^{2,p} bounds, including at B, by the preceding local estimates and Equation (7.20). A subsequence would converge locally in C^1 to a nonnegative function with maximum one on the core, homogeneous inner Neumann data, and a bounded-drift homogeneous equation. Equation (7.20) makes its value tend to zero at infinity. It consequently attains a positive maximum, contradicting the strong or Hopf principle for that homogeneous equation. This proves the uniform core bound. The contradiction uses the homogeneous bounded-drift equation.

Local compactness and diagonal extraction give a smooth solution of Equation (7.19), with \phi=O(r^{-1}). The claimed derivative count can be checked directly on annuli. Put

On smaller fixed annuli the equation is

Its coefficients have uniform C^{1,1} bounds from the two derivatives in Equation (7.9); the scaled drift is O(\rho^{-\beta}) and the scaled source is O(\rho^{-\delta}). The bounded zeroth-order norm, interior W^{2,p} estimates, and gradient interpolation first give uniform C^{1,\alpha} bounds for some \alpha>0. The right side is then uniformly C^\alpha. Schauder estimates give uniform C^{2,\alpha} bounds on smaller annuli. This proves \phi=O_2(r^{-1}) without using a third derivative of g or a second derivative of K.

The right side of Equation (7.19) is w+O(r^{-3-\beta}). It is integrable and proves Equation (7.15). The divergence theorem, with outward-domain normal -\nu on B, gives

The Euclidean flux differs by O(r^{-q}) and has the same finite limit.

For every fixed finite s\ge0, the factor u_s in Equation (7.16) is bounded and at least one, so the transformed data are complete up to B. The u-form of Lemma 7.4 gives

This proves the asserted signs for all finite s, without a smallness restriction.

Since u_s-1=O_2(r^{-1}), the transformed decay exponent is \min(q,1)>1/2. The exact energy transformation is

Here \Delta_g\phi=O(r^{-3-\delta}), and the additional current term is bounded by a fixed-s constant times |K|_g|d\phi|_g=O(r^{-3-q}). Both are integrable in dimension three. The bounded positive conformal factor also preserves integrability in the transformed volume measure.

The leading metric change is 4s\phi\delta, whose ADM energy flux is -8s\int_{S_r}\partial_r\phi\,\dd A_\delta before division by 16\pi. All remaining metric flux terms are O_s(r^{-q})+O_s(r^{-1}). Writing \pi=K-(\tr_gK)g, one has \pi_s=u_s^2\pi, so the change in its Euclidean momentum flux is O_s(r^2r^{-1}r^{-1-q})=O_s(r^{-q}). This proves Equation (7.17); Equation (7.21) gives A_\phi>0. No 1/r coefficient expansion for \phi is needed. Finally, the cut class is unchanged and every cut has area \int_\Gamma u_s^4\,\dd A_g. Bounding u_s between 1 and 1+s\|\phi\|_\infty and taking infima proves Equation (7.18), including coincident and disconnected cuts.

For the final assertion, take b nonnegative, compactly supported, and at least |K|; the proof is unchanged. The end equation is then simply -\Delta_g\phi=w. Once the two-derivative estimate has been established, successive annular elliptic estimates use the assumed symbol bounds for g and w to give \phi=O_k(r^{-1}) for every k.□

Proof. For purposes of perimeter minimization only, extend the metrics through a collar of B and attach a fixed compact filling. The filling need not satisfy any constraint or curvature condition. Extend g_j so that the extensions converge uniformly on this fixed collar and filling. A filled enclosure is then a bounded set containing the prescribed inner obstacle.

For each fixed metric, sufficiently large spheres in its asymptotically flat chart have strictly positive outward mean curvature. Clipping a bounded enclosure at any such sphere cannot increase perimeter. To see this, integrate the divergence of the outward unit normal field of the sphere foliation over the part of the enclosure outside the sphere. Its divergence is positive. Its flux through the exterior boundary is at most the area of that boundary, whereas its flux on the cutting sphere is the negative of the new cutting area. Thus the discarded boundary area is at least the inserted area. Approximation gives the same statement for finite-perimeter sets. The direct method on the resulting bounded domain now gives a minimizer. The clipping observation makes this a minimizer against every bounded competitor, not merely those inside the chosen truncation.

Off the obstacle the boundary is locally perimeter minimizing. If it contains a point of radius r sufficiently large, the coordinate ball of radius c_1r about that point avoids the obstacle. After scaling this ball to unit size, Equation (7.22) gives uniform ellipticity and uniform first-derivative bounds for its area integrand. The local perimeter density estimate gives

Here c_1,c_2>0 are independent of j and r. To justify the uniform lower bound, work in a rescaled coordinate ball avoiding the obstacle, and let v(s)=|F_j\cap B_s|_\delta. Removing F_j\cap B_s is an admissible local comparison. For almost every s, minimality and uniform metric comparability give

The Euclidean isoperimetric inequality therefore yields v(s)^{2/3}\le C''v'(s). At a point in the support of the minimizing frontier, both phases meet every ball. Integrating this differential inequality gives v(s)\ge c s^3. Filling the complement in B_s gives the same estimate for |B_s\setminus F_j|_\delta. The relative isoperimetric inequality now gives P_\delta(F_j;B_s)\ge c's^2 on a smaller fixed ball. Returning to the original scale proves the displayed area bound. The argument also applies at all support points by taking limits of reduced-boundary points. No uniform curvature or higher metric derivative bound is used. The fixed competitor bounds the total area independently of j. Consequently the minimizing boundaries lie in a fixed compact set.

The local obstacle regularity theorem for pure area minimization with a C^2 obstacle gives C^{1,1} regularity, and smoothness off contact, in dimension three; its local estimates require only the appropriate bounded geometry of the smooth metric [24, Regularity Theorem 1.3]. This applies to the fixed smooth obstacle and each of the smooth metrics g_j and g after the collar extension. Push a minimizing boundary slightly away from its contact set by a smooth vector field agreeing there with the outward obstacle normal. In a small obstacle collar its flow increases the signed distance to the obstacle. The resulting C^1 embedded boundary is a positive distance from the obstacle. Smooth local graph approximation, within that distance, preserves enclosure and changes area by an arbitrarily small amount. This proves equality of the smooth and perimeter infima for each metric.

Enlarge the fixed compact region so that its complement is a connected coordinate tail. Each minimizing inner set is either empty or the whole tail there, since its frontier is absent. Boundedness excludes the latter, so the sets themselves also lie in a common compact set. Choose a fixed smooth auxiliary metric on this compact set. Uniform metric comparability gives uniformly bounded auxiliary perimeters, and BV compactness gives a limiting filled enclosure. The obstacle is still contained in the limit. Fill any newly bounded complementary pockets; this only removes frontier. Uniform metric convergence makes the area integrands converge uniformly relative to the auxiliary metric, so lower semicontinuity implies

Conversely, using a fixed enclosure with g-area arbitrarily close to a_g(B) gives the reverse upper-limit inequality.□

Proof. Apply Lemma 3.2 with n=3, k=2, p=1, and M=m. In static isotropic coordinates (T,y) its spacetime metric is

For P\ne0, put v=P/E and rotate the boost axis into its direction; for P=0, take v=0. Then \gamma=(1-|v|^2)^{-1/2}=E/m, and the reference plane has charges (m\gamma,m\gamma v)=(E,P). The common lemma uses the same future-normal convention for K, and its ADM prefactors become

Thus its charges have precisely the present signs and normalization; the use of physical constraint densities here does not rescale them.

For the later bend, write the plane as T=v\cdot y and set y=Ax, where A=(I-v\otimes v)^{-1/2}. The induced Minkowski metric in the x coordinates is \delta. The perturbation of the metric above from Minkowski space has symbol decay of order r^{-1} on this fixed spacelike plane, and Lemma 3.2 gives the induced metric and tensor orders r^{-1} and r^{-2} through every fixed derivative order. These estimates concern the exact reference end and impose no additional derivative assumption on the original data.□

Proof. Apply Lemma 3.3 in dimension three, with the prescribed source support Q and the same connected annulus. The scalar operator there is exactly \mathcal L here, and its affine moments are the four conditions in (i). The vector operator is exactly \mathcal D; its translational and rotational moments are (ii). The inverse of trace reversal in that proof is S\mapsto S-(\operatorname{tr}S)\delta/(n-1), which becomes S\mapsto S-(\operatorname{tr}S)\delta/2. In particular its symmetric correction is obtained from a skew tensor D_{ijk}=-D_{jik} by

The identities C_{ijk}=-C_{ikj} and (C_{ijk}-C_{jik})/2=D_{ijk} cancel the skew part without changing divergence. The two scalar inversions gain two derivatives, and the row-divergence inversion followed by that correction gains one. All supports and bounds are those of the fixed annulus. Thus the full conclusions, including linear dependence and every stated Sobolev order, are precisely the specializations of that lemma. This is an operator statement about smooth sources; the estimates available from the original weakly decaying data are verified separately in the annular replacement proof below.□

Proof. The scalar-curvature expansion has remainder bounded by

and the replacement of the momentum constraint by Euclidean divergence has remainder bounded by C(|h||\partial K|+|\partial h||K|). These are O(r^{-2-2q}), using exactly the derivatives in Equation (7.9). They are integrable because 2+2q>3.

For any integrable function F on the end and fixed c>1,

Indeed the contribution inside any fixed radius M tends to zero after division by R; the remaining contribution is at most c\int_{|x|>M}|F|. Let R\to\infty and then M\to\infty.

For an affine function f, the vector field

satisfies \partial_i\mathcal Q_f(h)^i=f\mathcal Lh. For a Euclidean Killing field Y,

because \pi is symmetric. Integrating from a fixed inner sphere and using Equation (7.27) proves the asserted o(R) estimates. This argument does not assert the existence of a center-of-mass or angular-momentum limit and requires no parity condition.□

Proof. Fix \tfrac12<\beta<\min(q,1) and work on the annulus \mathcal A=\{1<|y|<4\} with x=Ry. The scaled fields are

and likewise for the reference data. Their constraint densities are the original densities multiplied by R^2. If h_R=g_R-\delta, then

Choose \chi\in C^\infty(\mathcal A) with 0\le\chi\le1, equal to one near |y|=1 and zero near |y|=4, with all its derivatives supported in a fixed compact subannulus. Blend both fields by this cutoff.

Write the constraints in the unnormalized form

Their Euclidean linear part is (\mathcal Lh,\mathcal Dk). On fields satisfying Equation (7.29), the nonlinear remainder is bounded in supremum norm by CR^{-2\beta}. Thus the blended constraints equal the blended original constraint sources, plus a quadratic error and the compact commutators

They have C^{0,1} norms at most CR^{-\beta}. In detail the scalar commutator applied to a symmetric tensor h is

one derivative of this expression uses at most two derivatives of h. The vector commutator is (\partial_j\chi)(k_{ij}-(\tr k)\delta_{ij}); one derivative uses at most one derivative of k. No derivative of the complete matter sources is needed.

Every affine moment of f_R and every Killing-field moment of V_R is o(R^{-1}). For the constant moments, integration by parts on \mathcal A gives R^{-1} times the difference between the physical translation fluxes on its two boundary spheres, less the integrals of the blended linear sources. The flux difference tends to zero because the reference and original ADM vectors agree. Replacing the physical momentum integrand by its Euclidean trace reversal costs O(R^{1-2q}), which tends to zero. By Lemma 7.9, the physical linear-source integrals on this receding annulus tend to zero as well. For a linear scalar test or a rotational vector test the corresponding factor is R^{-2}. Its physical boundary fluxes are o(R) by the same lemma, while the weighted source integrals are o(R) by Equation (7.27). This proves the claim.

Remove these small moments by fixed smooth bumps. Explicitly, for a basis p_1,\ldots,p_4 of affine functions and a fixed nonnegative bump \omega positive on a ball inside the annulus, the Gram matrix

is positive definite. Subtract \omega\sum_{a,b}p_a(G^{-1})_{ab}\int p_bf_R from f_R. For the vector moments use in exactly the same way the Gram matrix \int\omega Y_a\cdot Y_b of a basis of the six Euclidean Killing fields. It too is positive definite, since a nonzero Killing field cannot vanish on an open ball. Both bump corrections have every fixed smooth norm o(R^{-1}).

Apply Lemma 7.8 with the negative of the moment-free commutators. Since their W^{1,p} norms are O(R^{-\beta}), choosing p>3 gives metric and tensor corrections of size O(R^{-\beta}) in W^{3,p} and W^{2,p}, respectively. Sobolev embedding gives the needed C^{2,\alpha} and C^{1,\alpha} bounds. Their supports stay in a fixed compact subset of \mathcal A. Extend them by zero and undo the scaling. The metric remains positive definite for large R.

To check DEC, denote the original scaled sources by (A_R,B_R), so A_R\ge2|B_R|_{g_R}. The reference sources vanish. The corrected sources therefore have the form

in supremum norm. The norm of B_R changes by at most CR^{-\beta}|B_R|\le CR^{-2\beta} when the metric is changed, since the pointwise constraint formula gives |B_R|\le CR^{-\beta}. All other nonlinear correction terms have the same quadratic bound. Because 2\beta>1, the scaled DEC deficit is o(R^{-1}). Multiplication by R^{-2} gives Equation (7.28). Outside the correction annulus the data are either the original DEC data or the exact vacuum reference data. Smoothness follows from smoothness of the original fields and of the compact inverse constructions, irrespective of the sizes of their unneeded higher derivatives.□

Proof. Use the description T=v\cdot y, y=Ax, from Lemma 7.7. Choose R_b=c_bR so large, with c_b fixed, that the region |y|\ge R_b is entirely inside the exact reference part. Let \psi be a smooth function equal to one on (-\infty,0], zero on [1,\infty), and taking values in [0,1]. Replace the plane by the graph

Its Euclidean gradient satisfies

Choose a fixed L making the right side strictly smaller than one. The lapse and spatial coefficients in Equation (7.23) converge to their Minkowski values, so for all sufficiently large R this graph is uniformly spacelike. In fact for positive Schwarzschild mass the lapse is smaller than one and the spatial conformal factor exceeds one, which only improves this estimate on the static exterior.

The graph agrees with the original plane before R_b and with a constant-time slice after e^LR_b. Its first derivative is bounded, and its second and third derivatives are O(r^{-1}) and O(r^{-2}). The induced metric is uniformly comparable with \delta; differentiating its pullback formula gives the first bound in Equation (7.30). The second fundamental form uses the graph’s second derivatives and ambient first derivatives, divided by a uniformly positive spacelikeness factor, and gives the other bound. These estimates remain true in the x coordinates because A is a fixed invertible linear map. The constraints vanish on the entire bent part by the Gauss–Codazzi equations in exact vacuum Schwarzschild spacetime.

It remains to make the radius precise. On the bent region keep \rho_R=|x|=|A^{-1}y|. Beyond the bend, where the slice is already static, write y=r\omega and put a(\omega)=|A^{-1}\omega|. Choose R_c a sufficiently large fixed multiple of R_b e^L, and a cutoff \chi_0 equal to one before zero and zero after one. In that static region set

For fixed sufficiently large L_0 its radial derivative is bounded below by a positive constant: differentiating in r gives the positive factor \rho_R/r times 1+L_0^{-1}\chi_0'\log a, which is at least 1/2. The function and its first two Euclidean derivatives have the bounds \rho_R\asymp r, |d\rho_R|\le C, and |\partial^2\rho_R|\le C/r. Before this interpolation, the same bounds hold for |A^{-1}y|, and uniform metric comparability gives the gradient lower bound. Combining these facts with Equation (7.30) proves Equation (7.31). After the interpolation, \rho_R=r is precisely the static isotropic radius. All radius ratios used in the construction are fixed, independently of R.□

Proof. All geometry estimates below are uniform in R. Enlarge the fixed transition region slightly so that, in terms of t=\rho_R/R, it lies inside an interval (a,b) with 0<a<b<\infty. Arrange that t\ge b is in the static spherical region, and that the support of the possible deficit lies in a fixed smaller interval I\Subset(a,b). Increasing b if needed makes these assertions follow from Lemma 7.11. We take a=1: the compact supports in the annular correction leave a fixed neighborhood of t=1 free of deficit. The factor constructed below will consequently be constant throughout the original region r\le R, and is extended by that constant over the compact core of \Omega.

Let \eta_R\to0 be as in Equation (7.28), replacing it by a positive majorant if necessary. We will choose \varepsilon_R=C_0\eta_R and construct

where F_R and the relevant derivatives are bounded independently of R. Write z_R=-F_R'. On [a,b+1], initially define z_R by

where \omega_1 is smooth, nonnegative, zero near a, and at least one on I. Take C_1 sufficiently large, depending only on the constants in Equations (7.30) and (7.31). The solution is nonnegative and has uniform bounds on this fixed interval. For a function with F_R'=-z_R the chain rule gives

Choose C_1 so that this is nonnegative and is at least c_1\varepsilon_RR^{-3} on I.

The preceding differential inequality need not be imposed on the spherical tail. There K_R=0 and the Schwarzschild radial Laplace flux coefficient is (r+m/2)^2. Put c_R=m/(2R) and, on the still spherical interval near b, define

It has positive derivative wherever z_R>0. Continue it to a positive constant on [b+1,\infty) while keeping A_R'\ge0. For an explicit smooth continuation, continue the solution of Equation (7.34) to b+1, multiply the derivative of (t+c_R)^2z_R(t) by a nonnegative cutoff equal to one near b and zero near b+1, and integrate with the initial value A_R(b). Define z_R=A_R(t)/(t+c_R)^2 on this tail, and set it to zero for t\le a. All joins are smooth, the functions are nonnegative, and their bounds on finite intervals are uniform in R. The constant tail value, denoted A_R^\infty, is bounded independently of R.

Now define

It is nonnegative, uniformly bounded, and constant before a. The preceding construction gives a superharmonic u_R on the spherical tail, because its signed radial flux is -\varepsilon_RA_R(t) and is nonincreasing. It also gives Equation (7.33). For large R, 1\le u_R\le2. Increasing C_0 ensures

On the original inner region the conformal factor is constant and the original data satisfy DEC. Elsewhere this inequality and Lemma 7.4 prove DEC for the repaired data. Using u_R, rather than estimating \log u_R separately, incorporates the gradient-square term exactly.

On the final end,

Writing a_R=\varepsilon_RA_R^\infty=o(1), one obtains the exact identity

Thus the final mass is m+2a_R=m+o(1) and the momentum is zero. Multiplication of K_R by u_R^2 preserves its compact support. On the inner boundary, where u_R is constant, the expansion is simply multiplied by u_R^{-2}, so its weak sign is preserved.□

Proof. Apply Proposition 7.10, Lemma 7.11, and Lemma 7.12 along any sequence R_j\to\infty. The resulting DEC exteriors have compactly supported second fundamental form and an exact rest Schwarzschild end with energy tending to m. They agree with the original data on expanding compact subsets except for a constant conformal factor tending to one. Hence they converge smoothly on each fixed compact subset. Their boundary expansion is weakly negative.

Apply the last part of Lemma 7.5 separately to each of these fixed exteriors, using the same chosen 0<\delta<1 and a compactly supported drift coefficient. Choose its positive parameter s in Equation (7.16) small enough that the energy change, \|\log(1+s\phi)\|_\infty, and the supremum of its first derivative multiplied by the fixed original coordinate radius are at most 1/j. Also require the change in the first j smooth norms on the first j members of a compact exhaustion to be at most 1/j. All these quantities are finite for the fixed j-th exterior, so these choices are possible. No estimate uniform in the receding bending radius is required for this final strictification.

The resulting data have pointwise strict DEC and strict trapping. The quantitative end gap is at least c_j r^{-3-\delta}. On its compact complement the continuous positive function (\mu_j-|J_j|_{g_j})\rho^{3+\delta} has a positive minimum. Replacing c_j by the smaller of this minimum and the end constant proves the global weighted assertion in (i). The strict conformal factor has symbol decay of all orders and does not enlarge the support of K. On the far Schwarzschild end the drift vanishes, so -\Delta\phi=w=r^{-3-\delta}. Since the rest Schwarzschild metric has zero scalar curvature and K=0 there, the exact u-formula gives

This proves the scalar-curvature bound in (ii). The Ricci bound follows from the two-derivative metric decay. The energy change tends to zero and the momentum change is zero. The prescribed smallness gives smooth local convergence, proving (ii) and the local convergence and charge assertions in (iii).

The metrics before strictification satisfy the uniform comparisons and scaled first-derivative bounds of Lemma 7.6, in the fixed original end chart: the reference plane, the bend, and the radius interpolation involve only fixed linear maps and fixed radius ratios. The radial repair preserves these bounds. The final strictification parameters were chosen to preserve them as well. That lemma therefore gives the convergence of enclosing infima.

This completes the preparation; no numerical inequality was used in constructing these data.□

Proof of Theorem 7.2. Let m=\sqrt{E^2-|P|^2}>0, and take the sequence in Proposition 7.13. Each datum is a smooth complete one-ended exterior on \Omega with the same nonempty compact boundary B. The proposition gives a compactly supported tensor, O_d(r^{-1}) metric decay for every fixed derivative order d, scalar curvature O(r^{-3-\delta}) with 0<\delta<1, integrable constraints, strict future trapping, and a global weighted DEC margin. These are the geometric and asymptotic conditions in Definition 2.1.

The enclosing class also agrees with its full-cut class: both retain a connected end-containing outer domain and count its entire intrinsic frontier, including coincidence with the obstacle. Filling pockets removes only surplus frontier and leaves either infimum unchanged. Thus the prepared area is A_j=a_{g_j}(B)>0, where positivity follows from Lemma 7.3.

There is one normalization conversion. The densities used in this section are physical densities, while those of the prepared theorem satisfy 2\mu_{\mathrm A}=R+(\operatorname{tr}K)^2-|K|^2 and J_{\mathrm A}=\operatorname{div}(K-(\operatorname{tr}K)g). Hence

The global weighted margin becomes \mu_{\mathrm A,j}-|J_{\mathrm A,j}|_{g_j} \ge8\pi c_j\rho^{-3-\delta}, and all sign and integrability conditions are preserved. At n=3, k=n-1=2 and \omega_2=4\pi, so the energy and momentum prefactors in that theorem are respectively

Thus neither ADM energy nor momentum nor invariant mass is rescaled.

Theorem 2.3 therefore applies to each fixed prepared datum and gives

Its elliptic exhaustion and its numerical limits in N and \epsilon have already been completed for that datum; no estimate uniform in j is required. Now use E_j\to m and the full convergence a_{g_j}(B)\to a_g(B) to obtain

as asserted.□

Proof. Lemma 7.5 supplies a positive function \phi, a constant A_\phi>0, and, for every finite s\ge0, the data

These data retain completeness, integrable constraint densities, DEC, weak future trapping of B, and the required differentiated decay. The lemma makes no sign assumption on the original ADM vector. Its charge and area identities, Equations (7.17) and (7.18), give

where the last strict inequality is Lemma 7.3.

Suppose E\le|P|_\delta, and set s_0=(|P|_\delta-E)/A_\phi. This is a finite nonnegative number. For every s>s_0 the deformed vector is future timelike, so Theorem 7.2, applied to these fixed data, implies

As s\downarrow s_0 the left side tends to zero, while the final lower bound is fixed and positive. This contradiction proves E>|P|_\delta. The timelike exterior inequality now applies directly to (\Omega,g,K) and gives the stated bound.□

Proof. Let D_0 be the connected component of D^+ containing the distant chosen end. Its boundary is a union of components of S. It is nonempty: otherwise D_0 would be both open and closed in the connected manifold M, and hence equal to M, contradicting S\ne\varnothing. The closure X_0=\overline{D_0} is therefore a smooth manifold with nonempty compact boundary. Since S is compact, each sufficiently distant end of M lies either entirely in D_0 or entirely outside it. Thus X_0 has finitely many ends, precisely those original end tails contained in D_0.

For every end of X_0 other than the chosen one, remove a sufficiently distant open coordinate tail. The tails can be chosen disjoint and disjoint from S. Denote the remainder by \Omega. It is connected: a path segment entering one of the removed tails can be replaced by a path on the connected coordinate sphere bounding that tail. Consequently \Omega is a one-ended smooth exterior with compact boundary

This boundary is nonempty because \partial D_0 is nonempty.

The set \Omega is closed in the complete manifold (M,g), and it is complete for its intrinsic distance with B included. Indeed, an intrinsic Cauchy sequence is ambient Cauchy and has its ambient limit in \Omega. At an interior limit point the two distances are locally comparable. At a boundary limit point, a smooth boundary chart identifies \Omega locally with a half-ball and gives the same comparison, so the sequence converges intrinsically as well. Restricting the data preserves DEC and integrability. The chosen end is unchanged, so its decay and ADM vector remain the original ones.

On \partial D_0 the normal into \Omega is the normal into D^+, and the future expansion is nonpositive. On an added sphere S_{R_i}, the normal into \Omega points toward decreasing radius in the removed end. The differentiated asymptotic assumptions give

Choose each R_i large enough that their sum is negative. Every component of B is then weakly future outer trapped for the normal into \Omega. Theorem 7.14 applies and yields

It remains to compare the enclosing classes. We claim

Take first a smooth enclosing cut \Gamma contained in the interior of \Omega. The connected component on its distant chosen-end side is a smooth open region D'^+\subset D_0\subset D^+. Its boundary consists of components of \Gamma, so it is compact and smooth and has area at most \operatorname{Area}_g(\Gamma). It is therefore an admissible competitor in (7.36). For a cut that meets or coincides with B, a smooth collar push into the interior, with the obstacle side filled, gives enclosing cuts whose areas converge to its full area. This applies also to contact with the added coordinate spheres. Hence every full enclosing cut of \Omega has area at least A_{\min}(S), proving (7.39) after taking the infimum. The discarded components of D^+ do not obstruct this comparison: the ambient definition permits the chosen-end component alone as a competitor. Combining the area comparison with (7.38) proves the theorem.□

Proof. A sufficiently distant sphere in the chosen end bounds an admissible outer domain, so the infimum is finite. To obtain a positive lower bound, choose an embedded arc from an open disk in S to the chosen end, and continue it along a straight coordinate ray. The compact part of the arc can be chosen embedded and disjoint from the boundary except at its initial point. A boundary collar and the tubular neighborhood theorem give a proper tube with coordinates

Its initial section lies in S; its far part is a Euclidean tube of fixed transverse coordinate size. Choose a smooth nonnegative function \eta compactly supported in B^3, with integral one, and define in this tube

Extend \alpha by zero across the lateral tube boundary. It is a smooth closed three-form on M. Its comass, meaning the supremum of its absolute value on unit simple three-vectors, has a finite bound C_\alpha: the initial part is compact, and the far metric is uniformly comparable with the Euclidean metric. A sufficiently distant selected-end sphere has \alpha-integral one, after fixing orientation.

Let D_e be any admissible outer domain with cut \Gamma. Cut off its selected-end tail at a sphere beyond \Gamma. The resulting domain is compact, since D_e contains no distant part of another end. Its boundary is the distant sphere together with the entire intrinsic boundary \Gamma, with the boundary orientations. Stokes’ theorem gives

This proof applies to disconnected cuts and to frontier on S. In particular, it proves positivity before any end is truncated.

On an unwanted end, the normal of the truncation sphere that points into the retained manifold is the inward radial normal. The given decay therefore gives

The new boundary has strictly negative future expansion for large T. The other boundary components are the original ones. Removing the open tails leaves a connected manifold: a path entering a removed tail can have that portion replaced by a path along its connected spherical cross-section. The retained manifold is closed in M. It is complete for its intrinsic metric as well. Indeed, an intrinsic Cauchy sequence is ambient Cauchy and has a limit in the retained set; near a smooth boundary or an interior point, the intrinsic and ambient local distances are comparable in a half-ball or ball chart.

Every admissible outer domain in M_T, viewed as a submanifold of M, is an admissible outer domain for e. A part of its frontier on a truncation sphere becomes a hypersurface in the interior of M, and remains part of its intrinsic boundary. This proves the first inequality in (8.5). Conversely, a fixed original admissible domain has compact frontier and contains no tail of any unwanted end. It is consequently contained in M_T for all sufficiently large truncation radii and is then an admissible domain there. Applying this observation to cuts with area within any prescribed positive error of A_e(S) proves the limit. The selected-end data have not been changed, so neither selected-end charge changes.□

Proof. Use Equation (3.2) with n=4, k=3, p=2, and s=\log u. The constraint normalization is unchanged. At zero vector argument its first identity gives u^2\mu_u=\mu-3\Delta\log u-3|d\log u|^2 =\mu-3u^{-1}\Delta u. The covector formula proved with that identity gives uJ_u=J+3K(\nabla\log u,\cdot); taking its g_u norm adds another factor u^{-1}. The triangle inequality then gives Equation (8.8), equivalently the U=u specialization of Equation (3.3). The boundary formula reads \Theta_{K_u}=u^{-1}(\Theta_K+3\partial_\nu\log u), with the same chosen normal. The identities require smooth fields and u>0 only, so this application does not strengthen the original O_2/O_1 decay hypotheses.□

Proof. Choose 0<\xi<\min(q,1). For T(r)=r^{-2}-r^{-2-\xi}, at sufficiently large radius,

Take a nondecreasing smooth switch \beta from zero to one in this region, constant on neighborhoods of its endpoints, and set

Extend G constantly throughout the interior. Then G\ge0, it equals T sufficiently far out, and

Use u=1+\lambda G in Lemma 8.4. The dominant energy condition is preserved, as is the boundary expansion sign. The new densities are integrable: outside a compact set the additional terms in (8.6)–(8.7) are bounded by constants, depending on \lambda, times r^{-4-\xi}+r^{-4-q}. The new falloff exponent can be taken to be \min(q,2)>1.

For completeness, the leading change in the metric flux numerator is -6\partial_i(\lambda G). Products involving g-\delta, u-1, or their first derivatives have vanishing limiting flux. Hence (8.1) gives

The transformed momentum tensor is exactly u(K-\tau g). Its additional flux is O(r^{-q}), so P_\lambda=P by (8.2). Since u\ge1, every cut has at least its original area, proving the cut comparison.□

Proof. Use Lemma 3.2 with n=4, k=3, p=2, and M=m. Its exact isotropic spacetime metric is

Rotate the first axis into the direction of P and put v=|P|/E, \gamma=(1-v^2)^{-1/2}. The Lorentz plane in that lemma has E_*=m\gamma=E and P_{*1}=m\gamma v=|P|, with all transverse momenta zero. These are the present factors 1/(6\omega_3) and 1/(3\omega_3): in particular no mass is rescaled. The leading energy flux is 6m\gamma^2 x_i\rho^{-4}, the leading momentum tensor is the k=3 instance of (3.7), and the angular integral in (3.8) is \omega_3/\gamma. Thus the sign and normalization are the same ones used here. The second form uses the same future-normal convention. Finally the metric and tensor remainders in that lemma are O_d(r^{-4}) and O_d(r^{-5}), with leading orders r^{-2} and r^{-3}, through every fixed derivative order. Gauss–Codazzi in the vacuum metric (8.10) proves that the reference data are vacuum.□

Proof. Lemma 3.3 applies with n=4 on the given connected annulus. Its scalar operator \mathsf S is \mathcal L, and its vector operator \mathsf D is \mathcal D. Its affine and Euclidean Killing moments are exactly (8.12). The inverse trace reversal specializes to

so b-(\operatorname{tr}_\delta b)\delta=S. The scalar construction gains two Sobolev derivatives. The symmetric divergence construction first gains one derivative, uses a further divergence inverse to correct the skew part, and differentiates that correction once; its net gain is one. This proves (8.13), with support in a fixed larger compact subset and linear dependence on the sources. No quantitative derivative bound for the initial data is being inferred here; the commutator calculation below establishes the needed source norms from O_2/O_1.

For necessity, integration by parts pairs \mathcal Lb with \operatorname{Hess}\phi-(\Delta\phi)\delta, which vanishes for affine \phi, and pairs \mathcal Dk with the symmetric gradient of a Killing field after trace reversal. The latter also vanishes. Thus all and only the stated moments obstruct these compact inverses.□

Proof. Use Lemma 8.6 for a reference end with exactly the original charges. We first join the data to this plane on R<r<4R, controlling the full pointwise constraint deficit.

The scaled commutators. Fix 1<p<\min(2,q), write x=Rz, and on 1<|z|<4 use the scaled component fields

and their reference counterparts. This corresponds to rescaling lengths by R^{-1}, so both constraint densities scale by R^2. The metric deviations and tensors have size O(R^{-p}) in C^2 and C^1, respectively. Expanding the constraint map about (\delta,0) gives its linear part (\mathcal L,\mathcal D) from (8.11), with remainder bounded in C^0 by CR^{-2p}. This follows directly from the scalar curvature formula: the nonlinear metric terms are bounded by C(|b||D^2b|+|Db|^2); the other scalar terms are quadratic in k. The momentum remainder is bounded by C(|b||Dk|+|Db||k|).

Choose a fixed smooth radial \psi, equal to one near |z|=1 and zero near |z|=4, with 0\le\psi\le1. Blend the metric components and tensors by \psi. If b is the difference of the two scaled metric perturbations and k the difference of the two scaled tensors, the errors of the linear constraints relative to the blended sources are f_R=[\mathcal L,\psi]b and F_R=[\mathcal D,\psi]k. For T_{ij}=b_{ij}-(\tr_\delta b)\delta_{ij},

These expressions are supported in a fixed compact subannulus, and their C^1 norms are O(R^{-p}). In particular, the first bound uses only two metric derivatives, and the second only one tensor derivative.

All compatibility moments are small. We prove the sharper bound o(R^{-2}) for every moment in (8.12). Work momentarily in the physical coordinates, with

Both f and F belong to L^1 on the end. Indeed, the difference between the physical constraints and their Euclidean linear parts is O(r^{-2-2\min(q,2)}), which is integrable in dimension four. The physical densities are integrable by hypothesis and by reference vacuum, and Euclidean and physical measures and norms are uniformly comparable there.

An elementary consequence of integrability is

To see this, split at a fixed radius L. The integral up to L, divided by R, tends to zero. The remaining integral divided by R is at most four times the L^1 tail beyond L, which can be made arbitrarily small. A finite first moment is not required.

For an affine function \phi, define the flux primitive

Its divergence is \phi f. For a Killing field Y, symmetry of T gives \partial_j(Y^iT_{ij})=Y^iF_i. Let \mathfrak F_\phi(r) and \mathfrak G_Y(r) be the corresponding outward Euclidean sphere fluxes. For constant tests, matching the ADM charges gives

In the momentum flux the replacement of \tr_gK\,g by (\tr_\delta K)\delta costs at most O(r^{2-2\min(q,2)})=o(1). For homogeneous linear scalar tests and rotational fields, integration from a fixed sphere and (8.17) instead give

Put \psi_R(x)=\psi(x/R), and let A_R=\{R<r<4R\}. Integration by parts, with \psi_R=1 near the inner sphere and zero near the outer sphere, gives the exact formulas

In scaled coordinates, constant moments have the factor R^{-2}: the source scales by R^2 and volume by R^{-4}. A linear test in z contributes the additional factor R^{-1}. Thus (8.18)–(8.19) and the preceding estimates show

for each element of a fixed basis of the affine functions or Killing fields.

Correction and the physical deficit. Choose a nonnegative smooth bump \zeta, supported in a fixed ball within the annulus and positive on a smaller ball. For a basis of the affine tests, its Gram matrix \int\zeta\phi_a\phi_b is positive definite. For a basis of Killing fields, the matrix \int\zeta\,Y_a\cdot Y_b is likewise positive definite: a nonzero affine Killing field cannot vanish on an open ball. Subtract the corresponding bump combinations from f_R,F_R to erase their moments. By (8.20), the coefficients, and every fixed smooth norm of these bump corrections, are o(R^{-2}).

Apply Lemma 8.7 to the negatives of the moment-free commutators and add the resulting tensors to the blend. With j=1 and p_1>4, Sobolev embedding and (8.13) bound the metric correction in C^2 and the tensor correction in C^1 by CR^{-p}. Their supports stay in a fixed enlarged compact subannulus. The new metric is positive for sufficiently large R and agrees exactly with the original data inside R and the plane outside 4R.

Let (\widetilde g_R,\widetilde K_R) denote these physical data. Their scaled constraints equal \psi times the original scaled constraints, plus an error o(R^{-2})+O(R^{-2p}); the reference constraints vanish. The change in the momentum norm costs another O(R^{-2p}), because the metric change is O(R^{-p}) and the scaled momentum density is O(R^{-p}). Since \psi\ge0, the original dominant energy condition implies, after undoing the scaling,

Indeed the remaining physical error is R^{-2}[o(R^{-2})+O(R^{-2p})]=o(R^{-4}), since p>1. The possible deficit is confined to a fixed closed subannulus on the scale R. Replace \eta_R by a positive majorant tending to zero if necessary.

Bending the plane to a static slice. Only the exact reference region is used for this step. In its spatial coordinates y, write the plane as b=v\cdot y. Choose a smooth switch \psi_1 from one to zero on [0,1], constant outside that interval, and replace the plane farther out by the graph

Here R_b is a sufficiently large fixed multiple of R, so the graph initially lies wholly beyond the gluing annulus. For fixed |v|<1, choose the fixed L so large that

The Minkowski graph is uniformly spacelike. Since the perturbation (8.10) is O(r^{-2}), it remains uniformly spacelike in that metric for large R. It agrees with the plane on an open inner region and with b=0 on an open outer region. These induced data are vacuum everywhere in the bend, so it adds no deficit.

In the fixed x chart, use y=Ax where A=(I-v\otimes v)^{-1/2}, with the chosen rotation. The derivatives of (8.22) give D^2b=O(r^{-1}), and the induced data, now denoted (g_R^0,K_R^0), satisfy on all transition regions

Here c,C>0 may depend on the fixed boost and switches, but not on large R. Outside a fixed multiple of R, the data are static in the y chart and have tensor zero.

For the repair choose a smooth radius \mathfrak r, equal to |x| through the gluing annulus and through the bend, and equal to |y| farther out in the static region. It can be chosen with

Here is an explicit way to make the interpolation. In the static region put a(n)=|A^{-1}n|, n=y/|y|, and interpolate

Take R_c beyond the bend and make \chi change from zero to one over a sufficiently long fixed logarithmic interval. Its slope can be made so small that the derivative of \log\mathfrak r with respect to \log|y| lies between 1/2 and 3/2. Angular derivatives are bounded, second spatial derivatives are O(r^{-1}), and (8.23) proves (8.24). All transition intervals are contained between fixed multiples of R.

A conformal repair with vanishing mass cost. Write s=\mathfrak r/R. We construct a nonnegative function F_R(s), constant on the inside, and use

Choose a fixed interval [a,b_1] in the s variable that starts before the possible deficit, contains all transition regions, and ends where the radius is |y| and the data are static. This interval lies in the end for all large R. Put z_R=-F_R' and prescribe

where \omega_1\ge0 is smooth, vanishes on a neighborhood of a, and is at least one on a neighborhood of the possible deficit interval. Choose it to vanish near b_1. The solution is nonnegative and its norms on this fixed interval are bounded independently of large R.

Differentiating u_R and using (8.23) and (8.24) gives fixed constants c',C'>0 such that

In fact the first positive term is \varepsilon_RR^{-4}z_R'|\dd\mathfrak r|^2; the other terms are \varepsilon_RR^{-3}z_R\Delta\mathfrak r and the negative tensor term, both bounded below by -C\varepsilon_RR^{-4}z_R here. Choose C_1>C'/c'. Equation (8.25) then makes (8.26) nonnegative everywhere on this interval and at least c'\varepsilon_RR^{-4} on the deficit region.

On the static region put c_m=m/2 and continue z_R so that

is nondecreasing and becomes a positive constant q_{R,\infty}. This can be done smoothly with a uniform bound on q_{R,\infty}. Indeed, immediately past b_1, continue (8.25); both its derivative and that of the positive prefactor make q_R' nonnegative for large R. Multiply this nonnegative derivative by a smooth switch that equals one initially and zero after one more fixed interval, and integrate. This preserves all matching derivatives and keeps q_R nondecreasing. The radial Laplacian of the static metric then shows -\Delta_{g_R^0}u_R\ge0 there, since its radial divergence is proportional to -q_R'; also K_R^0=0.

Set F_R(s)=\int_s^\infty z_R(t)\,\dd t, extending it constantly before a. Its norm is bounded uniformly: the finite intervals have bounded length and data, while on the tail z_R=O(s^{-3}). Thus 1\le u_R\le2 for large R. Choose the fixed C_0 large enough that 3u_R^{-1}c'\varepsilon_R\ge\eta_R. Lemma 8.4, (8.21), and (8.26) now prove the dominant energy condition everywhere for

The repair is constant in a neighborhood of the original boundary, and therefore preserves its expansion sign.

On the ultimate tail the integral for F_R is explicit:

Consequently

Its energy is m+2a_R=m+o(1), with zero momentum, by the direct static instance of Lemma 8.6. The densities vanish on this tail and are smooth on the remaining compact part, so they are integrable. The data are complete with boundary: a fixed compact core is smooth, and the end has a uniform positive metric lower bound for each R.

Comparison of every enclosing cut. The final metric dominates the original g for r\le R, because the intermediate data there are the original ones and u_R\ge1. For r\ge R, (8.23) and the static tail give g_R\ge c\delta in the original x chart with a fixed c>0. Choose 0<\kappa<\min(1,\sqrt c/2). Let L_R=R^{1/2}, and choose a smooth strictly increasing radial map \rho_R equal to r for r\le L_R, with derivative decreasing from one to \kappa on [L_R,2L_R], and derivative \kappa afterward. It defines a diffeomorphism

on the end, extended by the identity on the core. It is proper because \rho_R(r)\to\infty, preserves the selected end, and fixes the boundary. Its radial and tangential Euclidean dilations are \rho_R' and \rho_R/r. They are at most one everywhere, and for r\ge R are at most \kappa+O(R^{-1/2}).

On the nonidentity region with r\le R, both r and \rho_R(r) tend uniformly to infinity; the original metric is therefore (1+o(1))\delta at both points. The dilation bound and g_R\ge g show \Phi_R^*g\le(1+o(1))g_R there. For r\ge R, the choice of \kappa, the same original-end decay, and g_R\ge c\delta give this inequality as well. It is immediate on the identity core. We have thus proved the global tensor comparison

Let \Gamma be the full boundary of any admissible outer domain for g_R. Its image under this proper diffeomorphism is the full boundary of an admissible original outer domain, with all coincident frontier retained. Equation (8.28) gives

Taking infima gives

which proves the last assertion without a compactness assumption on minimizing sequences.□

Proof. The generic strictification lemma, Lemma 3.5, applies to this fixed rest exterior. Indeed the exact static tail has g-\delta=O_d(r^{-2}) for every d, and K is compactly supported. Choose any 1<q_*<2 to express its O_6/O_5 bounds, set n=4, k=3, and take \beta=\delta_1. The condition 0<\delta_1<\min(1,q_*) is exactly the one required by that lemma. Its smooth nonnegative majorant can be chosen compactly supported: write it as b_0\ge|K|_g. The complete mixed-boundary continuation, uniform core and tail estimates, exhaustion, and differentiated-decay proof of that lemma therefore give a positive smooth solution of

The exponent -6 is -2k. The all-order clause of the same lemma gives

This application occurs after rest replacement, so it uses no higher derivatives of the original weakly asymptotically flat end.

We now verify the geometric conclusions. Put

The inner outward normal is -\nu, and the finite-flux conclusion of the lemma is

The geometric and Euclidean fluxes differ by O(T^{-2}) on the static tail, by (8.33).

Apply Lemma 8.4 with u=e^{s\phi}. It gives

The number B=\sup_M r^{4+\delta_1}|\mathrm d\phi|^2 is finite: smoothness controls the compact core and the tail is O(r^{-2+\delta_1}). Choose sB\le1/2. This proves the global margin with c_s=(3s/2)e^{-2s\|\phi\|_\infty}>0. On each boundary component,

Furthermore

Thus completeness with the boundary included is preserved, and the tensor remains compactly supported. On the tail, R_g=0, K_s=0 and

This proves the scalar decay and integrability there; the rest is smooth and compact. All differentiated metric bounds follow from (8.33), and momentum is zero because the tensor vanishes on a distant tail.

The leading metric change is 2s\phi\delta+O_1(r^{-4}), whose energy flux numerator is -6s\partial_i\phi+O(r^{-5}). The latter remainder has vanishing sphere flux. Division by 6\omega_3 therefore gives

Every full cut has its three-dimensional area element multiplied by e^{3s\phi}; the cut class itself is unchanged. Taking infima yields

The infimum is finite, since a distant coordinate sphere is admissible. Its convergence as s\downarrow0 follows without an attained minimizer. Together these estimates prove every asserted output at this fixed rest datum. If it is (g_R,K_R), every constant above may depend on R; strictification is removed with R fixed.□

Proof. First let the original exterior have one end and E>|P|. Set m=(E^2-|P|^2)^{1/2}>0. Proposition 8.8 gives data (g_R,K_R) with zero momentum and energy m_R\to m. Its full-cut comparison is

Fix one sufficiently large replacement radius R. For any fixed 0<\delta_1<1, Lemma 8.9 makes (g_{R,s},K_{R,s}) a prepared exterior for all sufficiently small s>0. Indeed it is the same smooth connected orientable one-ended exterior, complete with its compact boundary included and with compact complement of its coordinate end. It has finite ADM energy and a compactly supported tensor, so its ADM momentum is zero. On the end, g_{R,s}-\delta=O_d(r^{-2}) for every fixed derivative order d, its scalar curvature is O(r^{-4-\delta_1}), and its constraint densities are integrable. The boundary is strictly future trapped, and the global margin is

where \rho is a positive smooth extension of the rest-end radius. These are the geometric and decay conditions of Definition 2.1, with n=4 and \beta=\delta_1.

The area condition follows along the same comparisons. In one end, Definition 8.1 is exactly the prepared full-cut class: the outer domain is connected and closed, its interior lies in the open exterior, and its entire compact intrinsic boundary is counted, including every component and all coincidence with the original boundary. Lemma 8.3 gives A_*(g)>0; Equation (8.38) and strictification give A_*(g_{R,s})\ge A_*(g_R)>0. The constraint and ADM normalizations also agree: 2\mu=R+(\operatorname{tr}K)^2-|K|^2, J=\operatorname{div}(K-(\operatorname{tr}K)g), and the energy and momentum factors are 1/(6\omega_3) and 1/(3\omega_3). No rescaling is made.

Theorem 2.3 therefore yields, at every such fixed pair (R,s),

This is the completed numerical theorem: its elliptic exhaustion and its limits N\to\infty and \epsilon\downarrow0 have already been taken for the fixed prepared datum. None of those estimates is required to be uniform in R or s.

Now let s\downarrow0 with R fixed. Strictification gives both E_{R,s}\to m_R and A_*(g_{R,s})\to A_*(g_R), so

Only at this point let R\to\infty. The result is

Full convergence of A_*(g_R) has not been used or asserted.

It remains to remove the timelike assumption. If instead E\le|P|, take any sequence a_j>0 tending to zero and set

Lemma 8.5 gives, separately for each j, data of energy |P|+a_j, momentum P, and enclosing infimum at least A_*(g). Their charges are future timelike, so (8.40) gives

The right side is fixed and positive by Lemma 8.3, while the left side tends to zero. This contradiction includes zero, negative, and null energy cases. No preparation geometry is passed to a limit as the boost degenerates: the timelike numerical result was applied afresh to each shell datum. Thus E>|P| and (8.40) hold for every one-ended original datum in the theorem.

Finally let the original manifold have finitely many ends and fix one end e. Truncate every other end at a sufficiently large sphere as in Lemma 8.3. The retained manifold is a complete connected one-ended exterior. Its added boundary components are strictly future trapped, its selected-end charges are still (E_e,P_e), and its full-cut infimum is at least A_e(S). Applying the one-ended result gives

Positivity of A_e(S) was proved before any truncation. This gives both the asserted inequality and strict future timelikeness at e.□

Proof. In the given coordinates set

The differentiated decay assumptions give

To verify the second bound, expand the covariant divergence as

Replacing g^{jk} by \delta^{jk} costs O(r^{-q})O(r^{-2-q}); replacing \pi^g by \pi^\delta costs the derivative of the first bound in (8.41). Each connection term has size O(r^{-1-q})O(r^{-1-q}). These are exactly the asserted errors.

Their Euclidean radial volume bound is Cr^{1-2q}, integrable because q>1. The asymptotic metric and Euclidean volume and covector norms are uniformly comparable, so \partial_j\pi^\delta_{ij}\in L^1. The Euclidean divergence theorem between two large spheres makes the flux \int_{S_R}\pi^\delta_{ij}n_\delta^j\,\dd A_\delta a Cauchy function of R, with finite limit. By the first bound in (8.41), its difference from the flux of \pi^g is O(R^{2-2q})\to0. This is the tensor appearing in (8.2), which proves the lemma.□

Proof. Maximality identifies the momentum density with J_X=\divg_{g_X}K_X. Lemma 8.11 supplies the finite ADM momentum. All hypotheses of Theorem 8.2 therefore hold for this one-ended exterior. Its invariant-mass inequality in particular gives the displayed energy bound. The proof of Theorem 8.2 uses Theorem 2.3, independently of this corollary. This numerical consequence retains its broad maximal-DEC scope; the separate direct maximal-vacuum construction has its own hypotheses.□