Proof. The exact isotropic metric is \[-\left(\frac{1-M/(2|y|^p)}{1+M/(2|y|^p)}\right)^2db^2 +\left(1+\frac{M}{2|y|^p}\right)^{4/p}|dy|^2.\] Its leading perturbation of Minkowski space is \(2M|y|^{-p}(db^2+p^{-1}|dy|^2)\). On \(T=0\) put \[\rho^2=\gamma^2(x^1)^2+\sum_{a=2}^n(x^a)^2,\qquad f=\rho^{-p}, \qquad A=\gamma^2(v^2+p^{-1}),\quad B=p^{-1},\quad C=kp^{-1}\gamma^2v.\] 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 \[\partial_1f=-p\gamma^2x^1\rho^{-n},\qquad \partial_af=-px^a\rho^{-n},\qquad \partial_Tf=-p\gamma^2vx^1\rho^{-n}.\] Thus the energy flux vector \(Q_i=\partial_j\mathsf H_{ij} -\partial_i\mathsf H_{jj}\) satisfies \[Q_1=-2MkB\partial_1f=2Mk\gamma^2x^1\rho^{-n},\qquad Q_a=-2M(A+pB)\partial_af=2Mp(A+1)x^a\rho^{-n}.\] Since \(p(A+1)=k\gamma^2\), these expressions give the pointwise identity \[ Q_i=2Mk\gamma^2x_i\rho^{-n}. \tag{14}\]

The chosen sign of the second fundamental form gives, to first order, \[K_{ij}=\tfrac12\bigl(\partial_T\mathsf H_{ij} -\partial_i\mathsf H_{0j} -\partial_j\mathsf H_{0i}\bigr).\] Direct substitution gives \[\begin{align*} K_{11}&=M\gamma^4v(2p+1-pv^2)x^1\rho^{-n},& K_{aa}&=-M\gamma^2vx^1\rho^{-n},\\ K_{1a}&=Mk\gamma^2vx^a\rho^{-n},& K_{ab}&=0\quad(a\ne b). \end{align*}\] Their Euclidean trace is \(M\gamma^4v(p+v^2)x^1\rho^{-n}\). Consequently, for the leading Euclidean trace reversal \(\pi_{ij}=K_{ij}-(\mathop{\mathrm{tr}}_\delta K)\delta_{ij}\), \[ \pi_{1j}=Mk\gamma^2v x_j\rho^{-n},\qquad \pi_{a1}=Mk\gamma^2vx^a\rho^{-n},\qquad \pi_{ab}=-Mk\gamma^4vx^1\rho^{-n}\delta_{ab}. \tag{15}\] 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 \[ \int_{S^{n-1}}|\operatorname{diag}(\gamma,1,\ldots,1)\theta|^{-n} \,d\theta=\frac{\omega}{\gamma}. \tag{16}\] Equations (14) and (15), 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 (Costabel and McIntosh 2010, sec. 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 \[(\mathcal Bf)(x)=\int f(y)(x-y) \int_1^\infty\eta(y+t(x-y))t^{n-1}\,dt\,dy.\] Differentiation in distributions gives \(\mathop{\mathrm{div}}\mathcal Bf=f-\eta\int f=f\); the segments occurring in the integral remain in the ball. For compact input support and compact \(\mathop{\mathrm{supp}}\eta\), 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 \(h=Q-k^{-1}(\mathop{\mathrm{tr}}_\delta Q)\delta\) 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 \[0=\int(z_iF_j-z_jF_i)=\int(B_{ij}-B_{ji}).\] Put \(A_{ij}=(B_{ij}-B_{ji})/2\) and solve \(\partial_lD_{ijl}=-A_{ij}\), choosing \(D_{ijl}=-D_{jil}\). Set \[E_{ij}=\partial_l(D_{ijl}-D_{ilj}-D_{jli}),\qquad C_{ij}=B_{ij}+E_{ij}.\] 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 \(b=C-k^{-1}(\mathop{\mathrm{tr}}_\delta C)\delta\) 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 \[ \mathsf S(h)=2\mu+O(r^{-2q-2}),\qquad \mathsf D(K)=J+O(r^{-2q-2}). \tag{17}\] 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, \[ \int_{R_0<r<4L}r|f|\,dx=o(L). \tag{18}\] 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 \(Q=H-(\mathop{\mathrm{tr}}_\delta H)\delta\). For an affine function \(a(x)\) define its Green flux by \[\mathcal B_a(R)=\int_{S_R} \bigl(a\,\partial_jQ_{ij}\nu^i_\delta -(\partial_i a)Q_{ij}\nu^j_\delta\bigr)\,dA_\delta.\] 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 (18) gives \(\mathcal B_{x^a}(R)=o(R)\). For the scalar commutator \(C_L\), supported where the cutoff varies, one consequently has \[ \int_{L<r<4L}a C_L =-\mathcal B_a(L)-\int_{L<r<4L}\chi_La\,\mathsf S(H). \tag{19}\] This is \(o(1)\) for \(a=1\) and \(o(L)\) for \(a=x^a\).

For momentum put \(Q_{ij}=(K-K^{\mathrm b})_{ij} -\mathop{\mathrm{tr}}_\delta(K-K^{\mathrm b})\delta_{ij}\). 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 (18) give \(o(R)\). Equation (19) 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 \[-\Delta_g u-tB\frac{\langle d\phi,du\rangle_g} {\sqrt{|d\phi|^2+a^2}}, \qquad \partial_\nu u=0\text{ on }S,\quad u=0\text{ on }S_R.\] 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 (Gilbarg and Trudinger 2001, chap. 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 \[W(r)=r^{-p}\bigl(1-r^{-\alpha}\bigr),\qquad r\ge r_0,\] is positive and satisfies \[ -\Delta_gW-B|dW|_g\ge c r^{-n-\alpha}. \tag{30}\] 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 \[ 0\le\phi(x)\le C(1+A)r^{-p}\quad(r_0\le r\le R), \tag{31}\] 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 (31) 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 \[-\Delta_gu=t_*B|du|,\qquad \partial_\nu u=0.\] 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 (27); 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 (31) bounds \(\Phi\). The rescaled equation has uniformly controlled metric coefficients and the form \[-\Delta_{g(Lz)}\Phi =O_{5}(L^{-q})\sqrt{|d\Phi|_{g(Lz)}^2+|z|^{-2k}} +L^{-\beta}|z|^{-n-\beta},\] 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 (28).

The source is integrable. Integrating the equation on \(\Omega_R\), remembering that its inner outward normal is \(-\nu\), gives \[\int_{S_R}\partial_{\nu_g}\phi\,dA_g =-\mathop{\mathrm{Area}}_g(S)-\int_{\Omega_R} \bigl(B\sqrt{|d\phi|^2+a^2}+b\bigr)\,dV_g.\] 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 (27) implies, for every unit-ball vector \(v\), \(-\Delta\phi+K(\nabla\phi,v)\ge b\). Use Equation (10) 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 (29) 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. 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 (10) gives, as scalar and covector identities, \[a^2\mu_U=\mu-\frac{2k}{pU}\Delta_gU,\qquad aJ_U=J+\frac{2k}{pU}K(\nabla U,\cdot).\] 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 \[ a^2(\mu_U-|J_U|_{g_U}) \ge \mu-|J|_g+\frac{2k}{pU} \bigl(-\Delta_gU-|K|_g|dU|_g\bigr). \tag{33}\] The boundary identity is \[a\theta_{+,U}=\theta_++\frac{2k}{pU}\partial_\nu U.\] 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 \[T(r)=r^{-p}-r^{-p-\xi}.\] 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 \[-\Delta_gT-|K|_g|dT|_g =\xi(p+\xi)r^{-n-\xi}+O(r^{-n-q})>0\] after increasing the radius. Choose a nondecreasing smooth switch \(\beta\) from zero to one in this region, constant near its endpoints, and put \[G(r)=-\int_r^\infty\beta(t)T'(t)\,dt.\] Extend \(G\) constantly through the interior. Then \(G\ge0\), it is constant near \(S\), and \(G=T\) sufficiently far out. Because \(dG=\beta\,dT\), \[-\Delta_gG-|K|_g|dG|_g =\beta(-\Delta_gT-|K|_g|dT|_g)-\beta'T'|dr|_g^2\ge0.\] Use \(U_\lambda=1+\lambda G\). Equation (33) 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 \[g_\lambda-\delta=O_6(r^{-q}),\qquad K_\lambda=O_5(r^{-1-q}).\] Moreover, \[\Delta_gU_\lambda=O(r^{-n-\xi})+O(r^{-n-q}),\qquad |K|_g|dU_\lambda|_g=O(r^{-n-q}).\] 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 \[f_\lambda=1+\frac{4\lambda}{p}G+O_1(r^{-2p}).\] In the metric flux of Equation (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 \[\lim_{R\to\infty}\int_{S_R}\partial_rG\,dA_\delta =-p\omega_{n-1},\] the current normalization gives \[E_\lambda-E =-\frac{2\lambda}{p\omega_{n-1}} \lim_{R\to\infty}\int_{S_R}\partial_rG\,dA_\delta =2\lambda.\] If \(\pi=K-(\mathop{\mathrm{tr}}_gK)g\), its transformed tensor is exactly \(\pi_\lambda=U_\lambda^{2/p}\pi\). The additional momentum flux in Equation (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 2 are unchanged. For their entire intrinsic boundaries, \[\mathop{\mathrm{Area}}_{g_\lambda}(\partial D) =\int_{\partial D}U_\lambda^{2k/p}\,dA_g \ge \mathop{\mathrm{Area}}_g(\partial D).\] 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 \(\mathbb R_z\times\mathcal M_N\) consider the stationary Lorentz metric \[\mathbf G=-l^2(dz-df)^2+\bar g =-l^2dz^2+2l^2dz\,df+g.\] 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 \[ b=-U^{-1}\mathop{\mathrm{sym}}\nabla(Uw) =-H^f-p(dt\otimes w+w\otimes dt). \tag{40}\] This sign follows directly from \(\partial_zg=2Ub+\mathcal L_{Uw}g=0\) and the prescribed future-normal convention. Write \(B_0=\mathop{\mathrm{tr}}b\). Contracted Gauss and the normal lapse equation give \[\mathbf R+2\mathbf\mathop{\mathrm{Ric}}(\mathbf n,\mathbf n)=R_g-\mathsf B(b,b), \qquad \mathbf\mathop{\mathrm{Ric}}(\mathbf n,\mathbf n) =-\mathbf n(B_0)-|b|^2+U^{-1}\Delta_gU.\] Since \(B_0\) is stationary, \(\mathbf n(B_0)=-w(B_0)\), and taking the trace in Equation (40) gives \(\mathop{\mathrm{div}}(Uw)=-UB_0\). Substitution proves \[ \mathbf R=R_g+\mathsf B(b,b) -2U^{-1}\mathop{\mathrm{div}}\bigl(\nabla U+B_0Uw\bigr). \tag{41}\]

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 \[l^{-1}\Delta_{\bar g}l =U^{-1}\mathop{\mathrm{div}}\bigl((U/l)A_d\nabla l\bigr).\] Differentiating \(U=l\sqrt D\) and using Equation (37) proves the vector identity \[\nabla U-(U/l)A_d\nabla l=-Ub(w,\cdot).\] Consequently Equation (41) becomes \[ \frac12R_{\bar g} =\frac12R_g+\frac12\mathsf B(b,b) +U^{-1}\mathop{\mathrm{div}}(UP_bw). \tag{42}\]

Set \(Q=K-b=L+p(dt\otimes w+w\otimes dt)\). For any symmetric \(C\), the product rule and Equation (40) give the useful exact contraction \[U^{-1}\mathop{\mathrm{div}}(UP_Cw)=(\mathop{\mathrm{div}}P_C)(w)-\mathsf B(C,b).\] Expanding \(\mathsf B(K-b,K-b)\) and applying this formula to \(C=K\) transforms Equation (42) into \[ \frac12R_{\bar g}=\mu+J(w)+\frac12\mathsf B(Q,Q) -U^{-1}\mathop{\mathrm{div}}(UP_Qw). \tag{43}\] For example, the terms involving \(K\) on the right cancel to \(R_g/2-\mathsf B(b,b)/2+(\mathop{\mathrm{div}}P_b)(w)\), as they must.

The conformal scalar law in dimension \(k+1\) is \[\frac12e^{2t}R_{\widehat g} =\frac12R_{\bar g}-k\Delta_{\bar g}t -\frac{k(k-1)}2|dt|_{A_d}^2.\] The volume formula gives \[\Delta_{\bar g}t =U^{-1}\mathop{\mathrm{div}}(UA_d\nabla t)-p|dt|_{A_d}^2.\] Changing the divergence weight from \(U\) to \(u=e^{(k-1)t}U\) now yields \[ \begin{aligned} \frac12e^{2t}R_{\widehat g} ={}&\mu+J(w)+\frac12\mathsf B(Q,Q) +(k-1)P_Q(w,dt)+ks_p|dt|_{A_d}^2\\ &-u^{-1}\mathop{\mathrm{div}}\bigl(u(P_Qw+kA_d\nabla t)\bigr). \end{aligned} \tag{44}\] 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: \[ \begin{aligned} u(P_Qw+kA_d\nabla t)&=V-uFw,\\ u^{-1}\mathop{\mathrm{div}}(uw)&=\mathop{\mathrm{tr}}_gL-\tau+2s_pw(t) =F+(1-\chi)j-\tau+2s_pa_wx. \end{aligned} \tag{45}\] For the first, \(P_Qw+kA_ddt\) has transverse and axial components \[a_wM+(k+pv)y,\qquad -a_w\mathop{\mathrm{tr}}T+kd x,\] whereas \(V/u\) has components \(a_wM+(k+pv)y\) and \(\chi a_wj+kd x\) by Equation (37). Their difference is \(Fw\). For the second, use \(\mathop{\mathrm{div}}(Uw)=-U\mathop{\mathrm{tr}}b\) and change the weight. Thus the smooth invariant definition of the claimed quadratic form is \[ \mathcal T=\frac12\mathsf B(Q,Q)+(k-1)P_Q(w,dt) +ks_p|dt|_{A_d}^2+F\bigl(\mathop{\mathrm{tr}}_gL+2s_pw(t)\bigr). \tag{46}\] This proves smoothness and Equation (38). To obtain its displayed block form, the blocks of \(Q\) are \(T\), \(M+pa_wy\), and \(j+2pa_wx\). The transverse terms are \[|M+pa_wy|^2+(k-1)a_w(M+pa_wy)\cdot y =|M+s_pa_wy|^2-\frac{(k-1)^2}{4}v|y|^2.\] 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 (39).

For completeness this derivation retains derivatives of \(p\). In the flux of Equation (44) 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 (38) 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 \(u^{-1}\mathop{\mathrm{div}}V\) are respectively \[p'\chi_p a_wxj, \qquad p'\bigl(v|y|^2+\chi_p a_wxj\bigr).\] Their difference is exactly \(-p'v|y|^2\). No derivative of the cutoff has been discarded. ◻

Proof. The entire axial part of Equation (39) is exactly \[ \begin{aligned} &ks_pd\left(x+\frac{a_wj}{k+pv}\right)^2 +\frac{k+1}{2k}\left(F-\frac{\chi j}{k+1}\right)^2 +\frac{kd B_k(v)}{(k+pv)^2}j^2,\\ &B_k(v)=d\frac{k(k+2)}{2(k+1)}+v\frac{k+1}{2} =\frac{k(k+2)+v}{2(k+1)}. \end{aligned} \tag{49}\] Indeed completing first the \(x\) square and then the \(F\) square leaves \[\chi-\frac{k}{2(k+1)}\chi^2 -s_p\frac{\chi^2v}{kd} =\frac{kd B_k(v)}{(k+pv)^2}.\] The last coefficient is at least \(c_kd/(1+N)^2\), and \[ks_p-\frac{(k-1)^2}{4}v\geq\frac{k^2-1}{4}>0.\] 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 \(\mathop{\mathrm{tr}}T=F-\chi j\) then controls \(|T|^2\). These observations give Equation (47) with the stated single quadratic polynomial; they do not remove the factors \(d\).

When \(dt=0\) the axial expression instead completes as \[\frac{k+1}{2k}\left(F-\frac{\chi j}{k+1}\right)^2 +\chi\left(1-\frac{k\chi}{2(k+1)}\right)j^2.\] Its second parenthesis is bounded above and below by positive constants, and \(\chi^2\leq\chi\). This proves Equation (48) independently of \(N\). ◻

Proof. At \(dt=0\), direct block contraction gives \[\mathcal S(L)=\frac{k-1}{2k}(F-\chi j)^2 -\frac12|T^0|^2+dj(F-\chi j)-d|M|^2.\] Subtracting from Equation (39) gives the full difference, including its mixed term: \[ \begin{aligned} \mathcal T-\mathcal S(L)={}&|T^0|^2+(1+d)|M|^2+\frac{F^2}{k}\\ &+\left(\frac{k-2}{k}\chi-d\right)Fj +\chi\left(1+d-\frac{k-1}{k}\chi\right)j^2. \end{aligned} \tag{53}\] 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 \[|T^0|^2+2|M|^2+\frac{(F-j)^2}{k}+j^2=|L|^2.\] Perturbation of this finite-dimensional positive form, and Equation (48), therefore bound Equation (53) 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 (53) 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 \[|T^0|^2+|M|^2+\tfrac12\chi j^2 +\left(\frac1k-\frac{d^2}{2\chi}\right)F^2.\] Here \[\frac{d^2}{\chi}=\frac{d(k+pv)}k\leq1+\frac Nk.\] It follows, again using Equation (48), 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 \[|L|_{A_d\otimes A_d}^2 =|T|^2+2d|M|^2+d^2j^2\leq C_k(1+N)\mathcal T.\] Moreover \(|K|_{A_d\otimes A_d}\leq|K|\). Therefore \[|\mathcal S(L)-\mathcal S(L-K)| \leq \zeta\mathcal T+C_{k,\zeta}(1+N)|K|^2\] for every fixed \(\zeta>0\). Taking \(\zeta\) below half the positive constant already obtained proves Equation (52).

At an interior minimum of \(t\), \(dt=0\) and \(\mathop{\mathrm{Hess}}_g t\geq0\). 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 \(l^{-1}\mathop{\mathrm{Hess}}l=p\mathop{\mathrm{Hess}}t\). Its ambient scalar curvature is \(R_g-2p\Delta t\), its horizontal Ricci tensor is \(\mathop{\mathrm{Ric}}_g-p\mathop{\mathrm{Hess}}t\), 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 \[ \begin{aligned} \frac12e^{2t}R_{\widehat g} ={}&\frac12R_g-v\mathop{\mathrm{Ric}}_g(e,e)+\mathcal S(H^f)\\ &-pv\mathop{\mathrm{tr}}_{e^\perp}\mathop{\mathrm{Hess}}t-k\mathop{\mathrm{tr}}_{A_d}\mathop{\mathrm{Hess}}t\\ \leq{}&\frac12R_g-v\mathop{\mathrm{Ric}}_g(e,e)+\mathcal S(H^f). \end{aligned} \tag{54}\] When \(v=0\), terms using \(e\) vanish. In particular all \(p'(dt)^2\) terms vanish at this point.

Combining Equations (38), (52), and (54), and using \(F=\eta f\), yields \[ u^{-1}\mathop{\mathrm{div}}V\geq c_1\mathcal T+\eta a_w|df| -\mathcal P_N(\rho_0+v\rho_1). \tag{55}\] 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, \(v|\mathop{\mathrm{Ric}}_g|\leq Cv\varrho^{-n}\leq Cv\rho_1\) 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 (51) from the lower bound in Equation (55) gives \[(c_1-\delta_0)\mathcal T+(1-\delta_0)\eta a_w|df| +(C_N-\mathcal P_N)(\rho_0+v\rho_1)-\rho.\] 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 \(\mathop{\mathrm{tr}}_{A_\chi}H^f=\eta f\). 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 \(\mathop{\mathrm{div}}(u\nabla Z)=0\) 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 (37) give \[D-1=O_j(e^{-c_N'r}),\quad Z-t=O_j(e^{-c_N'r}),\quad V=k\nabla t+O_N(r^{3-2n})+O(e^{-c_N'r}).\] 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 \[(e^{2t}-1)g=2t\delta+O_1(r^{4-2n}).\] 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 \[\widehat E-E=-\omega^{-1}\lim_{r\to\infty} \int_{S_r}\partial_{\nu_g}t\,dA_g =-\frac{\mathfrak F_N}{k\omega}.\]

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, \[ u\leq C(\ell+\ell^2\sigma),\qquad uv\leq C\ell^2\sigma,\qquad ua_w\sigma=e^{(k-1)t}l^2\sigma^2\geq c\ell^2\sigma^2. \tag{60}\] 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 \[C C_N\ell^2\sigma(\rho_0+\rho_1) \leq\tfrac12\delta_0c\eta\ell^2\sigma^2 +C' C_N^2\frac{\ell^2}{\eta}(\rho_0^2+\rho_1^2).\] Consequently \[\mathfrak F_N\geq -C C_N\ell\int\rho_0 -C' C_N^2\frac{\ell^2}{\eta}\int(\rho_0^2+\rho_1^2).\] 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 \[\eta f=\mathop{\mathrm{tr}}K_a+l\Delta_g f\le \mathop{\mathrm{tr}}K_a.\] 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 \[A_\chi:\mathop{\mathrm{Hess}}H =Cb_1r^{-b_1-2}\big((b_1+1)\chi-k\big).\] 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 \[A_\chi:\mathop{\mathrm{Hess}}h =\frac{\eta\sqrt D}{l}\,h.\] 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 \[ |df|_{S_R}\le C\eta^{-1}R^{-b_1-1}. \tag{66}\] 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 \(S_R\), \[0\ge t\ge-\frac1{2k} \log\bigl(1+e^2C^2\eta^{-2}R^{-2b_1-2}\bigr).\] Choosing \(R_{\min}(N)\) sufficiently large proves the assertion. ◻

Proof. Write \(\sigma=|df|\), \(B=K_a(w,\cdot)\), \(A=A_\chi\), and \[d\mu_D=\sqrt D\,dV_g,\qquad W=e^{(k-1)t}l,\qquad u=W\sqrt D=l e^{kZ-t}.\] The floor already implies \[ \ell\le l\le e,\quad t\le Z,\quad D=e^{2k(Z-t)},\quad \sigma\le\ell^{-1}e^{k(Z+\epsilon)}. \tag{67}\] 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, \[\eta a_w\sigma=\frac\eta l(\sqrt D-D^{-1/2}).\] 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 \[ \Xi_a\ge\delta_0\mathcal T+\rho +\frac{\delta_0}2\eta a_w\sigma \quad\hbox{when }Z>Z_0. \tag{68}\] 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 \[\int \zeta(Z)u\Xi_a =-\int\zeta'(Z)u\{k|dZ|_A^2+\langle B,dZ\rangle_A\} \le C\int_{\{Z_0<Z<Z_0+1\}\cap\mathop{\mathrm{supp}}K}u|K|^2.\] 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 \[ \int_{\{Z>Z_0+1\}}u (\delta_0\mathcal T+\rho+\tfrac12\delta_0\eta a_w\sigma) \,dV_g\le\exp(P(N)). \tag{69}\] 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 \(Q\), \[ \int_Q e^{\alpha Z}\,d\mu_D\le\exp(P(N)). \tag{70}\] Below \(Z_0+1\), Equation (67) proves this directly. Above that level, when \(D\le2\) use \[\frac{e^{\alpha Z}\sqrt D}{u} =l^{-1}e^{(\alpha-k+1)t}D^{\alpha/(2k)} \le\exp(P(N)).\] When \(D\ge2\), use instead \[u\eta a_w\sigma=\eta e^{(k-1)t}(D-1),\qquad e^{\alpha Z}\sqrt D=e^{\alpha t}D^{1/2+\alpha/(2k)}.\] 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 (69) proves the claim.

Sobolev inequality on the product graph. We use the ordinary graph of \(f\) in \((Q\times\mathbb R,g+dz^2)\) 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 \[\sqrt{1+\sigma^2}\,dV_g, \qquad A_* = I-\frac{df\otimes df}{1+\sigma^2}.\] 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 \((1+\sigma^2)^{-1/2}\mathop{\mathrm{tr}}_{A_*}\mathop{\mathrm{Hess}}f\). 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 16 therefore gives \[|H_{\mathrm{graph}}|\le\exp(P(N))(1+\sqrt{\mathcal T}).\] Extend the metric of a slightly larger fixed base patch to a closed manifold and use Nash’s smooth isometric embedding theorem (Nash 1956); 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 (Michael and Simon 1973), in the formulation of (Simon 2018, chap. 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, \[ \left(\int |\varphi|^{2\kappa_n}\,d\mu_D\right)^{1/\kappa_n} \le\exp(P(N))\int \bigl(|d\varphi|_A^2+(1+\mathcal T)\varphi^2\bigr)\,d\mu_D, \qquad \kappa_n=\frac n{n-2}. \tag{71}\] 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 \[\begin{align*} \int \psi^2e^{2qZ}\Xi_a\,d\mu_D =-\int e^{2qZ} \bigl(2q\psi^2dZ-\psi^2\beta_W+2\psi d\psi\bigr) \cdot A(kdZ+B)\,d\mu_D. \end{align*}\] 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 \(q\ge q_*(N)\), \[ \int e^{2qZ}\psi^2 (\mathcal T+q|dZ|_A^2)\,d\mu_D \le P(N)(1+q)\int e^{2qZ} (\psi^2+|d\psi|_g^2)\,d\mu_D. \tag{72}\] Only \(dt\) occurs in this test; it does not differentiate \(p\).

Apply Equation (71) with \(\varphi=\psi e^{qZ}\) and then use Equation (72). For nested base patches \(Q'\Subset Q\) separated by a coordinate distance \(b\le1\), this gives \[\|e^Z\|_{2\kappa_nq,Q'} \le\bigl[e^{P(N)}(1+q)^2b^{-2}\bigr]^{1/(2q)} \|e^Z\|_{2q,Q}.\] 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 \[ \sup_{Q'}e^Z \le e^{P(N)}b^{-C}\|e^Z\|_{2q_*,Q}, \tag{73}\] where \(C\) is independent of \(N\).

Finally interpolate the norm on the right with Equation (70): \[\|e^Z\|_{2q_*,Q} \le \bigl(\sup_Qe^Z\bigr)^\theta e^{P(N)}, \qquad \theta=1-\frac\alpha{2q_*}<1.\] 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 \[M_0\le\theta^mM_m+ \sum_{j=0}^{m-1}\theta^j(P(N)+Cj)\le\theta^mM_m+P(N).\] 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 (68), \(\Xi_a>0\); at the same point \(u^{-1}\mathop{\mathrm{div}}(kuA\nabla Z)=kA:\mathop{\mathrm{Hess}}Z\le0\), 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 (67) and the height bound prove the remaining bounds in Equation (62). In particular, \[d\ge e^{-2k(P(N)+\epsilon)},\qquad \chi\ge\frac{k}{k+N}e^{-2k(P(N)+\epsilon)}.\] 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 (Gilbarg and Trudinger 2001). 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 \[a^{ij}=g^{ij}-(1-\chi)e^i e^j.\] If \(g=I\), then \(|I-a|=1-\chi\leq1-c\). Consequently, on a sufficiently small chart, depending only on \(c\) and the metric modulus, \[ |I-a|\leq1-\frac c2. \tag{76}\] 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 \(\mathbb R^n\), 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 \[\|\mathcal R\|_{L^s\to L^s}(1-c/2)<1, \qquad 2\leq s\leq s_0.\] For a compactly supported \(v\) the identity \(\Delta v=a:D^2v+(I-a):D^2v\) now gives, by absorption, \[\|D^2v\|_{L^s}\leq C\|a:D^2v\|_{L^s}, \qquad s=2,s_0.\] Cutoffs, first-derivative interpolation and absorption on nested balls give the local version \[ \|D^2v\|_{L^s(B_{r/2})} \leq C\bigl(\|a:D^2v\|_{L^s(B_r)} +r^{-2}\|v\|_{L^s(B_r)}\bigr), \qquad s=2,s_0. \tag{77}\] 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 (77). 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 \(P=\nabla_g u\), \(H=\mathop{\mathrm{Hess}}_g u\), 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 \[ A\nabla_g q-GP =\bigl(H^\sharp-(\Delta_g u)\mathop{\mathrm{id}}\bigr)P=:\mathcal F. \tag{78}\] Both sides vanish when \(P=0\), so no choice of axis at those points affects this identity. For smooth \(u\), commuting third derivatives gives \[ \mathop{\mathrm{div}}_g\mathcal F =|H|_g^2-(\Delta_g u)^2+\mathop{\mathrm{Ric}}_g(P,P). \tag{79}\] 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 (78) 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 \[((1-\chi)H(e,e)+G)^2 \leq(1-\gamma/2)|H|_g^2+C_cG^2.\] Thus, when \(|P|_g\leq1\) and \(s=1-|P|_g^2\geq0\), \[ \mathop{\mathrm{div}}_g(A\nabla_g s+2GP) \leq-\gamma|H|_g^2+C_cG^2+2|\mathop{\mathrm{Ric}}_g|_g. \tag{80}\]

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 \[\kappa(y)=\frac12 h^{ab}(y,0)\partial_t h_{ab}(y,0+), \qquad p(y)=u_t(y,0).\] 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 \[j(y)=4J(y,0)\kappa(y)p(y)^2.\] Adding the coordinate vector \[ B_{\rm face}(y,t)=-j(y)\mathbf1_{\{t>0\}}\partial_t \tag{81}\] 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 \[ \mathop{\mathrm{div}}(a_1Ds+B_1) \leq-\gamma_1|H|_g^2+C\varepsilon, \qquad a_1^{ij}=Ja^{ij},\qquad \|B_1\|_\infty\leq C\varepsilon, \tag{82}\] 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 \[ \sup_{B_{r_*}}|\nabla_g u|_g\leq k_*, \tag{83}\] or there is an affine \(L\) such that \[ \frac12\leq|DL|\leq2,\qquad \sup_{B_{r_*}}|u-L|\leq b_*r_*. \tag{84}\]

Here is the compactness argument with its energy step. Otherwise choose a sequence with errors tending to zero, no approximation in Equation (84), and gradient supremum tending to one on \(B_{r_*}\). Its functions are uniformly Lipschitz and vanish at zero. Equation (77), applied to \[a:D^2u=G+a^{ij}\Gamma^\ell_{ij}u_\ell,\] 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 (82) after discarding the negative Hessian term. It gives, for some \(p_0>0\), \[\left(\frac1{|B_{1/3}|}\int_{B_{1/3}}s^{p_0}\right)^{1/p_0} \leq C\left(\inf_{B_{1/3}}s+\|B_1\|_\infty+\varepsilon\right) \longrightarrow0.\] 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 \[\int(a_1Ds+B_1)\cdot D\varphi \geq\gamma_1\int|H|_g^2\varphi-C\varepsilon\int\varphi, \qquad\varphi\geq0.\] Dropping the nonnegative Hessian term, expanding the test derivative and using Young’s inequality proves \[ \int_{\{s<b\}}\psi^2|Ds|^2 \leq C_\psi b^2+C_\psi\|B_1\|_\infty^2 +C_\psi b\|B_1\|_\infty+C_\psi\varepsilon b. \tag{85}\] 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 (82) 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 (84) 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 \[ \|Y_0\|_{C^{1,\alpha_1}(B_{1/2})} \leq C\|Y_0\|_{W^{2,2}(B_1)} \quad\text{if}\quad \Delta_{e_0^\perp}Y_0+\chi\partial_{e_0}^2Y_0=0 \ \text{a.e. in }B_1, \tag{86}\] 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 \[ \sum_{a<n}\partial_a^2w+\partial_n(\chi\partial_nw)=0, \quad\text{or}\quad \mathop{\mathrm{div}}\bigl((I-e_0\otimes e_0+\chi e_0\otimes e_0)Dw\bigr)=0. \tag{87}\] 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, \[ \int_{B_r(x)}|Dw|^2\leq C r^{n-2+2\alpha_1}. \tag{88}\] The ordinary Laplacian of \(Y_0\) is \(f=(1-\chi)\partial_nw\). By Cauchy–Schwarz, \[ \int_{B_r(x)}|f|\leq C r^{n-1+\alpha_1}. \tag{89}\]

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 \[C\sum_{j\geq0}(2^{-j}r)^{1-n}(2^{-j}r)^{n-1+\alpha_1} \leq Cr^{\alpha_1}.\] For distances \(R\geq2r\), the difference of the two gradient kernels is at most \(CrR^{-n}\). Summing over dyadic annuli and using Equation (89) gives \[Cr\sum_{r\leq R\leq1}R^{\alpha_1-1} \leq Cr^{\alpha_1}.\] 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 (86). 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 \[ \frac14\leq|DL|\leq4,\qquad \|u-L\|_{L^\infty(B_1)}\leq b,\quad 0<b\leq b_0, \qquad \|g-I\|_\infty+\|\Gamma\|_\infty+\|G\|_\infty\leq b\delta, \tag{90}\] then there is an affine \(L'\) such that \[ \sup_{B_\lambda}|u-L'|\leq b\lambda^{1+\alpha_2}, \qquad |DL'-DL|\leq Cb. \tag{91}\] The metrics may again be smooth or reflected.

Set \(Y=(u-L)/b\). Its coordinate equation is \[a:D^2Y=b^{-1}\bigl(G+a^{ij}\Gamma^\ell_{ij}u_\ell\bigr),\] whose right side has absolute value at most \(C\delta\). Equation (77) and interpolation give \[ \|Y\|_{W^{2,s_0}(B_{7/8})}\leq C. \tag{92}\] 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 \[\|a-a_0\|_{L^{s_0}(B_{7/8})}\leq Cb, \qquad\|a-a_0\|_\infty\leq C.\] 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 \[ R:=a_0:D^2Y,\qquad \|R\|_{L^2(B_{3/4})}\leq C(\delta+b^\sigma), \quad \sigma=\min\{1,(s_0-2)/2\}>0. \tag{93}\]

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 \[\int_{\mathcal B}|D^2v|^2 \leq\int_{\mathcal B}|\Delta v|^2.\] 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 \[v\longmapsto\Delta_D^{-1}\bigl(R+(1-\chi)\partial_{e_0}^2v\bigr)\] 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 \[a_0:D^2v=R,\qquad \|v\|_{W^{2,2}(\mathcal B)}\leq C\|R\|_2.\] Consequently \(Y_0=Y-v\) satisfies the homogeneous frozen equation and has bounded \(W^{2,2}(\mathcal B)\) norm. By Equation (86), \(\|Y_0\|_{C^{1,\alpha_1}(B_{3/8})}\leq C\).

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 \[ \|v\|_{L^\infty(B_{1/3})} \leq C\|v\|_2^{2\theta/(n+2\theta)} \leq C(\delta+b^\sigma)^{2\theta/(n+2\theta)}. \tag{94}\] 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 (94) is at most \(\tfrac12\lambda^{1+\alpha_2}\). The affine function \[L'=L+b\bigl(Y_0(0)+DY_0(0)\cdot x\bigr)\] then satisfies Equation (91).

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 (91). 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 \[u_j(x)=\frac{u_0(r_*^j x)}{r_*^j k_*^j}.\] 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 \[\sup_{B_{r_*^j}}|u_0-u_0(0)| \leq C r_*^j k_*^j =C(r_*^j)^{1+\alpha_d}, \qquad\alpha_d=\frac{\log k_*}{\log r_*}\in(0,1).\]

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 \[U_i(x)=\frac{U(\lambda^i x)}{\lambda^i}, \qquad b_i=b_0\lambda^{i\alpha_2}.\] 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 \[\lambda^i b_0\delta\leq b_i\delta \quad\text{because}\quad\alpha_2<1.\] 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 (90).

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 \[C a_e r_e\lambda^{i(1+\alpha_2)} =C r^{1+\alpha}\, \bigl(a_e r_e^{-\alpha}\bigr) \lambda^{i(\alpha_2-\alpha)} \leq C r^{1+\alpha},\] 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 (75), 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 \[a:D^2(u-L_x)=G+a^{ij}\Gamma^\ell_{ij}u_\ell,\] Equation (77) with \(s=2\) yields \[\|D^2u\|_{L^2(B_r(x))} \leq C\bigl(r^{n/2}+r^{n/2-1+\alpha}\bigr).\] The bounded connection term converting \(D^2u\) to \(\mathop{\mathrm{Hess}}_g u\) has \(L^2\) norm \(O(r^{n/2})\). Squaring the estimate and using \(\alpha<1\) proves the Hessian estimate in Equation (75). 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 \[ Z^e(x)=-Z(Jx),\quad A^e(x)=JA(Jx)J,\quad B^e(x)=-JB(Jx),\quad S^e(x)=-S(Jx). \tag{97}\] 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 (95) 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 (95). 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 \[\begin{align*} \mathcal L Q_\varepsilon &=\partial_i\bigl(\varepsilon LQ_\varepsilon B^i\bigr) -L^2Q_\varepsilon B\cdot DZ +\varepsilon LQ_\varepsilon S -L^2Q_\varepsilon A DZ\cdot DZ. \tag{98}\end{align*}\] 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, \[ \mathcal L Q_\varepsilon \le \mathop{\mathrm{div}}B_\varepsilon+C\nu, \qquad B_\varepsilon=\varepsilon LQ_\varepsilon B, \qquad \|B_\varepsilon\|_\infty\le C. \tag{99}\] 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 \[ \mathcal L v_\varepsilon=\mathop{\mathrm{div}}B_\varepsilon+C\nu, \qquad v_\varepsilon|_{\partial B_r}=0. \tag{100}\] 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 (96) enters the estimate. The part with right side \(\mathop{\mathrm{div}}B_\varepsilon\) 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 (Littman et al. 1963, sec. 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 \[\int_{B_r}G_r(x,y)\,d\nu(y) \le C\sum_{j\ge0}(2^{-j}2r)^{2-n} \nu\bigl(B_{2^{-j}2r}(x)\bigr) \le C\sum_{j\ge0}(2^{-j}2r)^a\le Cr^a.\] We work sufficiently far inside the coordinate patch that the mass bound applies to the balls in this sum. Consequently \[ \|v_\varepsilon\|_{L^\infty(B_r)}\le E_r, \qquad E_r=C(r+r^a). \tag{101}\]

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 (99) and (100), the functions \[W_+=e^{LM}-Q_++v_++E_r, \qquad W_-=e^{-Lm}-Q_-+v_-+E_r\] are nonnegative weak supersolutions of \(\mathcal L W\ge0\) on \(B_r\). At least one of the sets \[\{Z\le(m+M)/2\}\cap B_{r/2},\qquad \{Z\ge(m+M)/2\}\cap B_{r/2}\] 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 \[c\omega\le C\left(\frac1{|B_{r/2}|}\int_{B_{r/2}}W_\varepsilon^q\right)^{1/q} \le C\inf_{B_{r/4}}W_\varepsilon.\] 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 \(\theta\in(0,1)\), \[ \operatorname{osc}_{B_{r/4}}Z \le (1-\theta)\operatorname{osc}_{B_r}Z+C(r+r^a). \tag{102}\] Iteration gives a uniform \(C^b\) bound for any fixed \[0<b<\min\left\{a,\frac{-\log(1-\theta)}{\log4}\right\}.\] 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 21 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 22, applied to the trace equation, gives some \(a\in(0,1)\) such that \[ [df]_{C^a}\le C, \qquad \int_{B_r\cap\mathcal M_{N,R}}|D^2f|^2\,dx \le C r^{n-2+2a},\qquad 0<r\le r_0. \tag{103}\] 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, \[ |Dt|\le C(1+|DZ|+|D^2f|). \tag{104}\] 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 (104) and the bounded zeroth- and first-order quantities imply the following coordinate form of the second equation: \[ -\partial_i(A^{ij}\partial_j Z)=\partial_i B^i+S, \qquad \lambda I\le A\le\Lambda I, \qquad |B|\le C, \qquad |S|\le C\bigl(1+|DZ|^2+|D^2f|^2\bigr). \tag{105}\] 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 23 with \(F_0=|D^2f|^2\). Its coefficient and source hypotheses are precisely Equation (105). On patches of size at most one, Equation (103) 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 (103) 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 \[ \|f\|_{C^{2,c}}\le C \tag{106}\] 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 (106) and the formula for \(Dt\) show that the expanded equation has the form \[ A^{ij}D_{ij}Z=H, \qquad |H|\le C(1+|DZ|^2), \qquad \|A\|_{C^c}\le C. \tag{107}\] 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 \[ \|D^2Z\|_{s,Q_r} \le C_s\|DZ\|_{2s,Q_{2r}}^2+C_{s,r}. \tag{108}\] The last constant includes the bounded height, the constant term in Equation (107), and the scaled cutoff costs. For a ball or a fixed-shape half-ball the scaled interpolation inequality is \[ \|DZ\|_{2s,Q}^2 \le C_s\operatorname{osc}_Q Z\,\|D^2Z\|_{s,Q} +C_s r^{n/s-2}(\operatorname{osc}_Q Z)^2, \tag{109}\] 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 \[\mathcal M_r=\sup_x\|D^2Z\|_{s,Q_r(x)}.\] 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 (108) and (109) imply \[\mathcal M_r \le C_s J_0^{1/s} \sup_x\operatorname{osc}_{Q_{2r}(x)}Z\,\mathcal M_r +C_{s,r}.\] 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 \(\sigma=|df|\), \[ b\le C(1+\sigma),\qquad \frac{c}{1+\sigma}\le\chi\le1. \tag{110}\] 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 \[e^{2kz}=e^{2kt}+\sigma^2e^{2(k+N)t}.\] Uniformly on a bounded \(z\) range, as \(\sigma\to\infty\), \[t=\frac{kz-\log\sigma}{k+N}+o(1),\qquad d\asymp\sigma^{-2k/(k+N)},\qquad \chi\asymp\frac{k}{k+N}\sigma^{-2k/(k+N)}.\] 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 (110).

2. A global gradient bound. Continue from \(\Delta_g f=f\) by solving, for \(0\le q\le1\), \[ A_q:\mathop{\mathrm{Hess}}f-c_q f=-q b\mathop{\mathrm{tr}}_{A_\chi}K_a, \qquad A_q=(1-q)I+qA_\chi, \qquad c_q=1-q+q\eta b. \tag{111}\] The scalar \(c_q\) is positive. Extrema give a common height bound \(\|f\|_\infty\le H\), since \[\frac{q b|\mathop{\mathrm{tr}}_{A_\chi}K_a|}{c_q}\le\frac C\eta.\] On this height range the right side of the equation written as \(A_q:\mathop{\mathrm{Hess}}f=G(x,f,df)\) satisfies, for a covector \(\xi\), \[ |G(x,f,\xi)|\le C(1+|\xi|), \tag{112}\] 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 \[ \psi(d)=K_0^{-1}\log(1+K_0Md),\qquad \psi''=-K_0(\psi')^2. \tag{113}\] 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 \[\chi_q\psi''\le-\frac{cK_0\sigma^2}{1+\sigma} \le-\frac{cK_0}2\sigma.\] Choose \(K_0\) large enough to dominate the collar curvature and Equation (112). 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 \[ |f(x)|\le\psi(d(x,\partial\mathcal D)) \quad\hbox{when }d(x,\partial\mathcal D)\le d_0. \tag{114}\]

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 (114) 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 \[\mathop{\mathrm{Hess}}f_x(e,e)\le\psi'',\qquad \mathop{\mathrm{Hess}}f_y(e,e)\ge-\psi''.\] Varying both endpoints in parallel transverse directions and summing the second-variation inequalities gives \[\mathop{\mathrm{tr}}_{e^\perp}\mathop{\mathrm{Hess}}f_x- \mathop{\mathrm{tr}}_{e^\perp}\mathop{\mathrm{Hess}}f_y\le C d(x,y)\sigma.\] 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 \[\begin{align*} A_{q,x}:\mathop{\mathrm{Hess}}f_x-A_{q,y}:\mathop{\mathrm{Hess}}f_y &\le Cd(x,y)\sigma+(\chi_{q,x}+\chi_{q,y})\psi''\\ &\le Cd_0\sigma-cK_0\sigma. \end{align*}\] The right sides of Equation (111) 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 (110) now gives uniform ellipticity throughout the continuation. Its normalized forcing is bounded. Lemma 22 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 (Gilbarg and Trudinger 2001) 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 \[[0,1]\times C^{1,\gamma}\times C^{3,\gamma}_0 \longrightarrow C^{1,\gamma}.\] 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 19. Keep \(N\) fixed and let \(R\to\infty\). The estimates of Proposition 24 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 17.

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 \[ A_\chi:\mathop{\mathrm{Hess}}f-\frac{\eta\sqrt D}{l}f=0, \qquad \frac{\eta\sqrt D}{l}\ge\frac\eta e>0. \tag{117}\] For small \(a_N>0\) and a sufficiently large fixed \(r_2(N)\), \(v_N(r)=e^{-a_N(r-r_2)}\) obeys \[A_\chi:\mathop{\mathrm{Hess}}v_N-\frac{\eta\sqrt D}{l}v_N<0 \quad(r\ge r_2).\] In fact \(0<A_\chi\le I\) and the end metric bounds give \(A_\chi:\mathop{\mathrm{Hess}}v_N\le C(a_N^2+a_N/r)v_N\); 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 (117), 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 \[ \begin{gathered} k\Delta_g Z+k B_N(Z)|dZ|_g^2 =\rho-\vartheta(N(Z+\epsilon))C_N\rho_0+O_j(e^{-a_Nr}),\\ B_N(z)=(k-1)+p(z)-\delta_0s_{p(z)}. \end{gathered} \tag{118}\] 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 \[Q_N(0)=0,\qquad Q_N'(z) =\exp\left(\int_0^z B_N(\tau)\,d\tau\right).\] Its derivative has positive upper and lower bounds at fixed \(N\). Equation (118) and the chain rule give \[ |\Delta_g Q_N(Z)|\le C_N' r^{-n-\beta}, \qquad Q_N(Z)|_{S_R}=0. \tag{119}\] Here \(C_N'\) denotes a constant at fixed \(N\), not necessarily the penalty polynomial. The function \[b_*(r)=r^{2-n}(1-r^{-\beta/2})\] is positive for \(r>1\) and satisfies \[\Delta_g b_* =-\frac\beta2\left(n-2+\frac\beta2\right)r^{-n-\beta/2} +O(r^{2-2n})<0\] 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 (119) 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 (118) 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 \[ H_s+\mathop{\mathrm{tr}}_{TS_s}K\le-4\delta \tag{125}\] 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 \[ \mathcal Q(v_*)= \mathop{\mathrm{tr}}_{A_\chi}K+ \frac{\mathop{\mathrm{tr}}_{A_\chi}\mathop{\mathrm{Hess}}_g v_*} {\sqrt{a_0^2+|dv_*|_g^2}}. \tag{126}\] The equation is \(\mathcal Q(h)=h\). The coefficients in Equation (126) are evaluated at the solution; we do not replace its value of \(t\) by one computed from the barrier. Nevertheless, \[ 0<\chi\le d=\frac{a_0^2}{a_0^2+|dh|_g^2}, \qquad \frac\eta e\le a_0\le\frac\eta\ell=\ell^{1/2}. \tag{127}\] 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 \[L_N=\log(q_*/\eta)+O(1)=O(1+N).\] The cutoff can be flat at its initial endpoint. This construction gives \[ |q_N'|\le C(q_N+\eta),\qquad \int q_N\,ds\le Cq_*+C\eta, \tag{128}\] 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 (127)–(128) give, even where the slope vanishes, \[\frac{\chi|v_*''|}{\sqrt{a_0^2+|v_*'|^2}} \le C\frac{q_N+\eta}{\sqrt{(\eta/e)^2+q_N^2}}\le C'.\] 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 (128), 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, \[\mathcal Q(v_*)= \mathop{\mathrm{tr}}_{TS_s}K+\chi K(\partial_s,\partial_s) +\frac{v_*'}{\sqrt{a_0^2+(v_*')^2}}H_s +\frac{\chi v_*''}{\sqrt{a_0^2+(v_*')^2}}.\] Equation (127) shows that this converges uniformly to the left side of Equation (125). 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 \(\mathop{\mathrm{Hess}}h\le\mathop{\mathrm{Hess}}v_*\). Positive definiteness of \(A_\chi\) would give \[h=\mathcal Q(h)\le\mathcal Q(v_*)<v_*<h,\] 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 19 has \(t>-\epsilon\) and the parameters of the scalar construction. Thus Equation (56) 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 (129); the remaining terms are nonnegative and retain the positive contribution \((1-\delta_0)\mathcal T\).

For the new conformal factor we use Equation (58), 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 \[\begin{align*} \tfrac12e^{2(t+\psi_N)}R_{\widetilde g_N} =\tfrac12e^{2t}R_{\widehat g} -k e^{2t}\Delta_{\widehat g}\psi_N -\tfrac{k(k-1)}2e^{2t}|d\psi_N|_{\widehat g}^2. \tag{132}\end{align*}\] The chain rule, Equation (58), and the fixed derivative bounds of \(\zeta\) imply \[\begin{align*} |e^{2t}\Delta_{\widehat g}\psi_N| &\le P_N\ell^{1/4}(1+\sqrt{\mathcal T})+C\ell^{3/4},\\ e^{2t}|d\psi_N|_{\widehat g}^2&\le C\ell^{1/2}. \end{align*}\] On the support of these errors \(b_3/4\le|h|\le b_3/2\). This region is disjoint from the filling by Lemma 27, 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 (56) therefore proves nonnegative scalar curvature there for large \(N\). Where \(\zeta\) is constant, Equation (132) 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 (131) is exactly Proposition 18. 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 (123). 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 \(\mathop{\mathrm{div}}Y=H_r>0\). For a bounded finite-perimeter competitor \(F\) and almost every large \(R\), Gauss–Green on \(F\cap\{r>R\}\) gives \[\int_{F\cap\{r>R\}}H_r\,dV =\int_{\partial^*F\cap\{r>R\}}\langle Y,\nu_F\rangle\,dA -\mathcal H^k(F^{(1)}\cap S_R).\] 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 \[ P_{\widetilde g_N}(F\cap\{r<R\}) \le P_{\widetilde g_N}(F) -\int_{F\cap\{r>R\}}H_r\,dV_{\widetilde g_N}. \tag{136}\] The Gauss–Green and truncation formulas hold at almost every radius by the trace and slicing theorems for BV functions (Ambrosio et al. 2000). 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 (136) 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 (122) implies \(\widehat g\le e^{P_N}g\). Its filled side is a competitor; increasing \(P_N\) gives Equation (135). 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 (136) 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 \[P(F\cup G)+P(F\cap G)\le P(F)+P(G).\] 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 \(\mathop{\mathrm{dist}}_g(p,C_N^h)=s_N\). 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 \[ \mathop{\mathrm{Vol}}_g(F_N\cap B^g_{s_N}(p))\ge c s_N^n. \tag{138}\] 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 (122) gives, on this ball and every \(k\)-plane, \[ e^{-P_N}g\le\gamma\le e^{P_N}g, \qquad e^{-P_N}dA_g\le dA_\gamma\le e^{P_N}dA_g, \tag{139}\] after changing the polynomial. No derivative of \(\gamma\) appears in this comparison.

Put \(U=\mathcal M_N\setminus F_N\) and \(V(r)=\mathop{\mathrm{Vol}}_g(U\cap B^g_r(p))\). 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 \[P_\gamma(U;B^g_r(p)) \le\mathcal H^k_\gamma(U^{(1)}\cap\partial B^g_r(p)).\] 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 (139) then yield \[ P_g(U;B^g_r(p))\le e^{P_N}V'(r) \quad\text{for almost every }r\in(0,s_N). \tag{140}\]

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: \[V(r)^{(n-1)/n} \le C\bigl(P_g(U;B^g_r(p))+V'(r)\bigr) \le e^{P_N}V'(r).\] 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 \(n\)th root satisfies \[(V^{1/n})'(r)\ge n^{-1}e^{-P_N} \quad\text{almost everywhere on }(0,s_N).\] Integrate from \(a>0\) to \(r\) and let \(a\downarrow0\). The result is \[ V(r)\ge e^{-P_N}r^n\qquad(0<r\le s_N), \tag{141}\] 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 (138) and (141), now gives \[P_g(F_N;B^g_{s_N}(p))\ge e^{-P_N}s_N^{n-1}.\] Absorb \(s_N^{n-1}=e^{-(n-1)Q_N}\) into the polynomial exponential, use Equation (139), and restore the constant multiplier. We obtain \[ P_{\widetilde g_N}(F_N) \ge\exp(kA_N-P_N) =\exp\bigl(k e^{N\epsilon/4}-P_N\bigr). \tag{142}\] For fixed \(\epsilon>0\), \(e^{N\epsilon/4}\) dominates every polynomial in \(N\). Equation (142) contradicts Equation (135) for large \(N\). ◻

Proof of Theorem 7. The connected exterior and its curvature region.

Choose \(F_N\) as in Lemma 29. Its frontier is compact and, by Lemma 30, 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 (Simon 2018, chap. 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 (Simon 2018, chap. 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 (Ambrosio et al. 2000). 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 28 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 \[ \int_{\partial^*F_N}\mathop{\mathrm{div}}_{T_x\partial^*F_N}Y\, d\mathcal H^k_{\widetilde g_N}=0. \tag{143}\] 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 27 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 \[ \mathbf1_{F_j}\longrightarrow\mathbf1_{F_N}\text{ in }L^1, \qquad P_g(F_j)\longrightarrow P_g(F_N), \tag{144}\] 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 \[P_g(\{u_j>t_j\}) \le\frac{\int|du_j|_g\,dV_g}{1-2\delta_j}+o(1) =P_g(F_N)+o(1).\] 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 (144). This is the strict approximation/coarea construction of (Ambrosio et al. 2000), 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 2. Thus \[a\le\mathop{\mathrm{Area}}_g(\Gamma_j)\le P_g(F_j).\] 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 \[ \mathcal H^k_{\widetilde g_N}(\Sigma_N) =P_{\widetilde g_N}(F_N) \ge e^{-k\epsilon}P_g(F_N)\ge e^{-k\epsilon}a. \tag{145}\] 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 (123) shows \[R_{\widetilde g_N}=O(r^{-n-\beta})+O(r^{-2n+2}).\] 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 28. 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 \[C_\omega=\sup_{x\in \Omega}\sup_{ v,w\ \text{orthonormal in }(T_x\Omega,g)}|\omega_x(v,w)|<\infty.\] 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 \[\int_\Gamma\omega=\int_{S_R}\omega=1, \qquad 1\le C_\omega\operatorname{Area}_g(\Gamma).\] 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 (10) 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 (155). 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 (157) with precisely its factor \(8\). The boundary formula has \(k\partial_\nu s=4\partial_\nu f\) and proves Equation (158). Finally Equation (11) 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 \[ -\Delta_g\phi=b\sqrt{|d\phi|_g^2+\zeta^2}+w, \qquad \partial_\nu\phi=-1, \qquad \phi\longrightarrow0. \tag{164}\] 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 (164) 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 (Agmon et al. 1959). The maximum principles and local elliptic estimates below are used in their usual uniformly elliptic forms (Gilbarg and Trudinger 2001).

To close continuation at \(t=1\), first work on a fixed truncation. The elliptic \(W^{2,p}\) estimate and gradient interpolation give \[\|\phi\|_{W^{2,p}} \le C_p\bigl(1+\|\phi\|_{L^p}+\|d\phi\|_{L^p}\bigr) \le \tfrac12\|\phi\|_{W^{2,p}} +C'_p(1+\|\phi\|_{L^p}).\] 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 \[-\Delta_gv=t_*b|dv|_g.\] 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 (164) 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 \[v_0(r)=r^{-1}(1-Ar^{-\delta})\] satisfies, on \(r\ge r_0\), \[\begin{align*} -\Delta_gv_0-b|dv_0|_g &=A\delta(1+\delta)r^{-3-\delta} +O(r^{-3-q})+O(r^{-3-\beta})\ge c_0w. \end{align*}\] 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 \[ 0\le\phi(x)\le C(1+M_R)v_0(r),\qquad M_R=\sup_{\Omega\cap\{r\le r_0\}}\phi, \quad r_0\le r\le R. \tag{165}\] 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 (165). 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 (165) 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 (164), with \(\phi=O(r^{-1})\). The claimed derivative count can be checked directly on annuli. Put \[\phi_{\rho}(y)=\rho\phi(\rho y),\qquad g_{\rho}(y)=g(\rho y),\qquad 1<|y|<4.\] On smaller fixed annuli the equation is \[-\Delta_{g_{\rho}}\phi_{\rho} =\rho b(\rho y) \sqrt{|d\phi_{\rho}|_{g_{\rho}}^2+ (\rho^2\zeta(\rho y))^2} +\rho^3w(\rho y).\] Its coefficients have uniform \(C^{1,1}\) bounds from the two derivatives in Equation (154); 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 (164) is \(w+O(r^{-3-\beta})\). It is integrable and proves Equation (160). The divergence theorem, with outward-domain normal \(-\nu\) on \(B\), gives \[ \lim_{r\to\infty}\int_{S_r}\partial_{n_g}\phi\,\,\mathrm dA_g =-|B|_g-\int_{\Omega} \bigl(b\sqrt{|d\phi|_g^2+\zeta^2}+w\bigr)\,\,\mathrm dV_g. \tag{166}\] 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 (161) is bounded and at least one, so the transformed data are complete up to \(B\). The \(u\)-form of Lemma 35 gives \[\begin{align*} 16\pi u_s^4(\mu_s-|J_s|_{g_s}) &\ge16\pi(\mu-|J|_g) +\frac{8s}{u_s}(-\Delta_g\phi-|K|_g|d\phi|_g) \ge \frac{8s}{u_s}w,\\ \theta_{+,s} &=u_s^{-2}\left(\theta_+-\frac{4s}{u_s}\right). \end{align*}\] 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 \[16\pi u_s^4\mu_s=16\pi\mu-8s u_s^{-1}\Delta_g\phi.\] 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 \(-8s\int_{S_r}\partial_r\phi\,\,\mathrm dA_\delta\) before division by \(16\pi\). All remaining metric flux terms are \(O_s(r^{-q})+O_s(r^{-1})\). Writing \(\pi=K-(\mathop{\mathrm{tr}}_gK)g\), 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 (162); Equation (166) gives \(A_\phi>0\). No \(1/r\) coefficient expansion for \(\phi\) is needed. Finally, the cut class is unchanged and every cut has area \(\int_\Gamma u_s^4\,\,\mathrm dA_g\). Bounding \(u_s\) between \(1\) and \(1+s\|\phi\|_\infty\) and taking infima proves Equation (163), 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 (167) gives uniform ellipticity and uniform first-derivative bounds for its area integrand. The local perimeter density estimate gives \[\operatorname{Area}_{g_j} (\partial F_j\cap B_{c_1r})\ge c_2r^2.\] 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 \[P_\delta(F_j;B_s)\le C v'(s),\qquad P_\delta(F_j\cap B_s)\le C'v'(s).\] 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 (Huisken and Ilmanen 2001, 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 \[a_g(B)\le\liminf_j a_{g_j}(B).\] Conversely, using a fixed enclosure with \(g\)-area arbitrarily close to \(a_g(B)\) gives the reverse upper-limit inequality. ◻

Proof. Apply Lemma 10 with \(n=3\), \(k=2\), \(p=1\), and \(M=m\). In static isotropic coordinates \((T,y)\) its spacetime metric is \[ -\left(\frac{1-m/(2r)}{1+m/(2r)}\right)^2\,\mathrm dT^2 +\left(1+\frac{m}{2r}\right)^4\delta_{ij}\,\mathrm dy^i\,\mathrm dy^j, \qquad r=|y|. \tag{168}\] 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 \[\frac1{2k\omega_2}=\frac1{16\pi},\qquad \frac1{k\omega_2}=\frac1{8\pi}.\] 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 10 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 11 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 \[ C_{ijk}=D_{ijk}-D_{ikj}-D_{jki}. \tag{170}\] 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 \[C\bigl(|h||\partial^2h|+|\partial h|^2+|K|^2\bigr),\] 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 (154). They are integrable because \(2+2q>3\).

For any integrable function \(F\) on the end and fixed \(c>1\), \[ \frac1R\int_{r_0<|x|<cR}|x|\,|F(x)|\,\,\mathrm dx\longrightarrow0. \tag{172}\] 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 \[\mathcal Q_f(h)^i =f(\partial_jh_{ij}-\partial_i\mathop{\mathrm{tr}}h) -(\partial_jf)h_{ij}+(\partial_if)\mathop{\mathrm{tr}}h\] satisfies \(\partial_i\mathcal Q_f(h)^i=f\mathcal Lh\). For a Euclidean Killing field \(Y\), \[\partial_j(\pi_{ij}Y^i)=Y^i\partial_j\pi_{ij},\] because \(\pi\) is symmetric. Integrating from a fixed inner sphere and using Equation (172) 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 \[g_R(y)=g(Ry),\qquad k_R(y)=RK(Ry),\] and likewise for the reference data. Their constraint densities are the original densities multiplied by \(R^2\). If \(h_R=g_R-\delta\), then \[ \|h_R\|_{C^2(\mathcal A)}+\|k_R\|_{C^1(\mathcal A)} +\|h_{*,R}\|_{C^2(\mathcal A)}+\|k_{*,R}\|_{C^1(\mathcal A)} \le CR^{-\beta}. \tag{174}\] 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 \[\mathcal C(g,k) =\bigl(\mathop{\mathrm{Scal}}_g+(\mathop{\mathrm{tr}}_gk)^2-|k|_g^2, \mathop{\mathrm{div}}_g(k-(\mathop{\mathrm{tr}}_gk)g)\bigr).\] Their Euclidean linear part is \((\mathcal Lh,\mathcal Dk)\). On fields satisfying Equation (174), 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 \[\begin{align*} f_R&=\mathcal L(\chi(h_R-h_{*,R})) -\chi\mathcal L(h_R-h_{*,R}),\\ V_R&=\mathcal D(\chi(k_R-k_{*,R})) -\chi\mathcal D(k_R-k_{*,R}). \end{align*}\] They have \(C^{0,1}\) norms at most \(CR^{-\beta}\). In detail the scalar commutator applied to a symmetric tensor \(h\) is \[\begin{align*} &2\partial_i\chi(\partial_jh_{ij}-\partial_i\mathop{\mathrm{tr}}h) +(\partial_i\partial_j\chi)h_{ij}-(\Delta\chi)\mathop{\mathrm{tr}}h; \end{align*}\] one derivative of this expression uses at most two derivatives of \(h\). The vector commutator is \((\partial_j\chi)(k_{ij}-(\mathop{\mathrm{tr}}k)\delta_{ij})\); 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 40, 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 (172). 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 \[G_{ab}=\int\omega p_ap_b\] 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 39 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 \[\bigl(\chi A_R,\chi B_R\bigr) +o(R^{-1})+O(R^{-2\beta})\] 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 (173). 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 38. 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 \[ T_R(y)=(v\cdot y) \psi\left(\frac{\log(|y|/R_b)}{L}\right). \tag{177}\] Its Euclidean gradient satisfies \[|dT_R|_\delta\le |v|\bigl(1+\|\psi'\|_\infty/L\bigr).\] Choose a fixed \(L\) making the right side strictly smaller than one. The lapse and spatial coefficients in Equation (168) 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 (175). 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 \[\rho_R(r,\omega) =r\exp\left[ \chi_0\left(\frac{\log(r/R_c)}{L_0}\right)\log a(\omega)\right].\] 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 (175) proves Equation (176). 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 42. 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 (173), replacing it by a positive majorant if necessary. We will choose \(\varepsilon_R=C_0\eta_R\) and construct \[u_R=1+\frac{\varepsilon_R}{R}F_R(\rho_R/R),\] 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 \[ z_R'=C_1z_R+\omega_1, \qquad z_R(a)=0, \tag{179}\] 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 (175) and (176). The solution is nonnegative and has uniform bounds on this fixed interval. For a function with \(F_R'=-z_R\) the chain rule gives \[\begin{align*} -\Delta_{g_R}u_R-|K_R|_{g_R}|du_R|_{g_R} &=\frac{\varepsilon_R}{R^3} \left[z_R'|d\rho_R|_{g_R}^2 +Rz_R\Delta_{g_R}\rho_R -Rz_R|K_R|_{g_R}|d\rho_R|_{g_R}\right]\\ &\ge\frac{\varepsilon_R}{R^3} (c_1z_R'-C_2z_R). \end{align*}\] 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 \[A_R(t)=(t+c_R)^2z_R(t).\] 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 (179) 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 \[F_R(t)=\int_t^\infty z_R(s)\,\,\mathrm ds.\] 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 (178). For large \(R\), \(1\le u_R\le2\). Increasing \(C_0\) ensures \[-\Delta_{g_R}u_R-|K_R||du_R| \ge \frac{u_R}{8}\, \bigl[16\pi(|J_R|_{g_R}-\mu_R)\bigr]_+.\] On the original inner region the conformal factor is constant and the original data satisfy DEC. Elsewhere this inequality and Lemma 35 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, \[u_R=1+\frac{\varepsilon_RA_R^\infty}{r+m/2}.\] Writing \(a_R=\varepsilon_RA_R^\infty=o(1)\), one obtains the exact identity \[ u_R^4\left(1+\frac{m}{2r}\right)^4\delta =\left(1+\frac{m/2+a_R}{r}\right)^4\delta. \tag{180}\] 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 41, Lemma 42, and Lemma 43 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 36 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 (161) 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 \[R_{g_j}=8s(1+s\phi)^{-5}r^{-3-\delta}.\] 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 37, 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 33. Let \(m=\sqrt{E^2-|P|^2}>0\), and take the sequence in Proposition 44. 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 6.

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 34.

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 \[\mu_{\mathrm A}=8\pi\mu,\qquad J_{\mathrm A}=8\pi J.\] 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 \[\frac1{2k\omega_2}=\frac1{16\pi},\qquad \frac1{k\omega_2}=\frac1{8\pi}.\] Thus neither ADM energy nor momentum nor invariant mass is rescaled.

Theorem 8 therefore applies to each fixed prepared datum and gives \[E_j\ge\frac12\left(\frac{a_{g_j}(B)}{4\pi}\right)^{1/2} =\sqrt{\frac{a_{g_j}(B)}{16\pi}}.\] 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 \[\sqrt{E^2-|P|^2}\ge\sqrt{\frac{a_g(B)}{16\pi}},\] as asserted. ◻

Proof. Lemma 36 supplies a positive function \(\phi\), a constant \(A_\phi>0\), and, for every finite \(s\ge0\), the data \[g_s=(1+s\phi)^4g,\qquad K_s=(1+s\phi)^2K.\] 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 (162) and (163), give \[E_s=E+sA_\phi,\qquad P_s=P,\qquad a_{g_s}(B)\ge a_g(B)>0,\] where the last strict inequality is Lemma 34.

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 33, applied to these fixed data, implies \[\sqrt{(E+sA_\phi)^2-|P|_\delta^2} \ge\sqrt{\frac{a_{g_s}(B)}{16\pi}} \ge\sqrt{\frac{a_g(B)}{16\pi}}>0.\] 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 \[B=\partial D_0\ \cup\ \bigcup_{\text{other ends of }X_0} S_{R_i}.\] 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 \[H_{S_{R_i}}=-\frac{2}{R_i}+O(R_i^{-1-q_i}),\qquad \operatorname{tr}_{S_{R_i}}K=O(R_i^{-1-q_i}).\] 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 45 applies and yields \[ E>|P|_\delta,\qquad \sqrt{E^2-|P|_\delta^2}\ge\sqrt{\frac{a_g(B)}{16\pi}}. \tag{183}\]

It remains to compare the enclosing classes. We claim \[ a_g(B)\ge A_{\min}(S). \tag{184}\] 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 (181). 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 (184) 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 (183) 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 \[[0,\infty)\times B^3\longrightarrow M.\] 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 \[\alpha=\eta(z)\,\,\mathrm dz^1\wedge\,\mathrm dz^2\wedge\,\mathrm dz^3.\] 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 \[\left|\int_\Gamma\alpha\right|=1, \qquad |\Gamma|_g\ge C_\alpha^{-1}.\] 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 \[H=-3T^{-1}+O(T^{-1-q}),\qquad \mathop{\mathrm{tr}}_{S_T}K=O(T^{-1-q}).\] 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 (189). 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 (10) 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 (192), equivalently the \(U=u\) specialization of Equation (11). 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, \[T'<0,\qquad -\Delta_gT-|K|_g|\,\mathrm dT|_g =\xi(2+\xi)r^{-4-\xi}+O(r^{-4-q})>0.\] Take a nondecreasing smooth switch \(\beta\) from zero to one in this region, constant on neighborhoods of its endpoints, and set \[G(r)=-\int_r^\infty\beta(s)T'(s)\,\,\mathrm ds.\] Extend \(G\) constantly throughout the interior. Then \(G\ge0\), it equals \(T\) sufficiently far out, and \[-\Delta_gG-|K|_g|\,\mathrm dG|_g =\beta(-\Delta_gT-|K|_g|\,\mathrm dT|_g) -\beta'T'|\,\mathrm dr|_g^2\ge0.\] Use \(u=1+\lambda G\) in Lemma 51. 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 (190)–(191) 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 (185) gives \[E_\lambda-E=-\frac{\lambda}{\omega_3} \lim_{r\to\infty}\int_{S_r}\partial_rG\,\,\mathrm dA_\delta =2\lambda.\] The transformed momentum tensor is exactly \(u(K-\tau g)\). Its additional flux is \(O(r^{-q})\), so \(P_\lambda=P\) by (186). Since \(u\ge1\), every cut has at least its original area, proving the cut comparison. ◻

Proof. Use Lemma 53 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.

Proof. The generic strictification lemma, Lemma 13, 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 \[ -\Delta_g\phi=b_0\sqrt{|\mathrm d\phi|_g^2+r^{-6}}+r^{-4-\delta_1}, \qquad \partial_\nu\phi=-1,\qquad \phi\longrightarrow0. \tag{216}\] The exponent \(-6\) is \(-2k\). The all-order clause of the same lemma gives \[ |\partial^j\phi|\le C_jr^{-2-j}\qquad(j\ge0). \tag{217}\] 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 \[F=b_0\sqrt{|\mathrm d\phi|^2+r^{-6}}+r^{-4-\delta_1}.\] The inner outward normal is \(-\nu\), and the finite-flux conclusion of the lemma is \[ L_\phi=\lim_{T\to\infty}\int_{S_T}\partial_r\phi\,\mathrm dA_\delta =-|\partial M|_g-\int_MF\,\mathrm dV_g<0. \tag{218}\] The geometric and Euclidean fluxes differ by \(O(T^{-2})\) on the static tail, by (217).

Apply Lemma 51 with \(u=e^{s\phi}\). It gives \[e^{2s\phi}(\mu_s-|J_s|_{g_s}) \ge3s\bigl(r^{-4-\delta_1}-s|\mathrm d\phi|^2\bigr).\] 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, \[\Theta_{K_s}=e^{-s\phi}(\Theta_K+3s\partial_\nu\phi) =e^{-s\phi}(\Theta_K-3s)<0.\] Furthermore \[ g\le g_s\le e^{2s\|\phi\|_\infty}g. \tag{219}\] Thus completeness with the boundary included is preserved, and the tensor remains compactly supported. On the tail, \(R_g=0\), \(K_s=0\) and \[R_{g_s}=6e^{-2s\phi} \bigl(sr^{-4-\delta_1}-s^2|\mathrm d\phi|^2\bigr) =O(r^{-4-\delta_1}).\] This proves the scalar decay and integrability there; the rest is smooth and compact. All differentiated metric bounds follow from (217), 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 \[ E_{g_s}-E_g=-\frac{s}{\omega_3}L_\phi\longrightarrow0. \tag{220}\] Every full cut has its three-dimensional area element multiplied by \(e^{3s\phi}\); the cut class itself is unchanged. Taking infima yields \[ A_*(g)\le A_*(g_s)\le e^{3s\|\phi\|_\infty}A_*(g). \tag{221}\] 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 55 gives data \((g_R,K_R)\) with zero momentum and energy \(m_R\to m\). Its full-cut comparison is \[ A_*(g_R)\ge(1+\epsilon_R)^{-3/2}A_*(g), \qquad \epsilon_R\longrightarrow0. \tag{222}\] Fix one sufficiently large replacement radius \(R\). For any fixed \(0<\delta_1<1\), Lemma 56 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 \[\mu_{R,s}-|J_{R,s}|_{g_{R,s}} \ge c_{R,s}\rho^{-4-\delta_1},\qquad c_{R,s}>0,\] where \(\rho\) is a positive smooth extension of the rest-end radius. These are the geometric and decay conditions of Definition 6, with \(n=4\) and \(\beta=\delta_1\).

The area condition follows along the same comparisons. In one end, Definition 48 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 50 gives \(A_*(g)>0\); Equation (222) 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 8 therefore yields, at every such fixed pair \((R,s)\), \[ E_{R,s}\ge\frac12 \left(\frac{A_*(g_{R,s})}{\omega_3}\right)^{2/3}. \tag{223}\] 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 \[m_R\ge\frac12 \left(\frac{A_*(g_R)}{\omega_3}\right)^{2/3} \ge\frac12(1+\epsilon_R)^{-1} \left(\frac{A_*(g)}{\omega_3}\right)^{2/3}.\] Only at this point let \(R\to\infty\). The result is \[ \sqrt{E^2-|P|^2}\ge\frac12 \left(\frac{A_*(g)}{\omega_3}\right)^{2/3} \qquad (E>|P|). \tag{224}\] 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 \[\lambda_j=\frac{|P|+a_j-E}{2}>0.\] Lemma 52 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 (224) gives \[\sqrt{(|P|+a_j)^2-|P|^2} \ge\frac12\left(\frac{A_*(g)}{\omega_3}\right)^{2/3}>0.\] The right side is fixed and positive by Lemma 50, 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 (224) 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 50. 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 \[E_e\ge\sqrt{|P_e|^2+ \frac14\left(\frac{A_e(S)}{\omega_3}\right)^{4/3}}.\] 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 \[\pi^g_{ij}=K_{ij}-(\mathop{\mathrm{tr}}_gK)g_{ij},\qquad \pi^\delta_{ij}=K_{ij}-(\mathop{\mathrm{tr}}_\delta K)\delta_{ij}.\] The differentiated decay assumptions give \[ \pi^g-\pi^\delta=O_1(r^{-1-2q}),\qquad \partial_j\pi^\delta_{ij}-J_i=O(r^{-2-2q}). \tag{225}\] To verify the second bound, expand the covariant divergence as \[J_i=g^{jk}\bigl(\partial_k\pi^g_{ij} -\Gamma^a_{ki}\pi^g_{aj}-\Gamma^a_{kj}\pi^g_{ia}\bigr).\] 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 (225). 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 \(\int_{S_R}\pi^\delta_{ij}n_\delta^j\,\,\mathrm dA_\delta\) a Cauchy function of \(R\), with finite limit. By the first bound in (225), its difference from the flux of \(\pi^g\) is \(O(R^{2-2q})\to0\). This is the tensor appearing in (186), which proves the lemma. ◻

Proof. Maximality identifies the momentum density with \(J_X=\mathop{\mathrm{div}}_{g_X}K_X\). Lemma 58 supplies the finite ADM momentum. All hypotheses of Theorem 49 therefore hold for this one-ended exterior. Its invariant-mass inequality in particular gives the displayed energy bound. The proof of Theorem 49 uses Theorem 8, independently of this corollary. This numerical consequence retains its broad maximal-DEC scope; the separate direct maximal-vacuum construction has its own hypotheses. ◻