Proof. Choose \[ \frac32<\beta<\min\{\tau,3\},\qquad 0<\delta<\min\{2\beta-3,1\},\qquad \eta_R=R^{\beta-\tau}. \tag{19}\] Take \(R\) beyond the support of \(K\) and a fixed cutoff \(\chi_R=\chi(r/R)\), equal to one for \(r\leq R\) and zero for \(r\geq2R\). Set \(q_R=b+\chi_R(g-b)\) on the end and \(q_R=g\) elsewhere. For large \(R\) this is positive definite. The curvature cutoff error \[E_R=\frac18\{\chi_R(R_g+6)-(R_{q_R}+6)\}\] is supported in \(R\leq r\leq2R\) and satisfies \[ \|E_R\|_{C^{0,\alpha'}_\beta}\leq C\eta_R. \tag{20}\] Indeed, the scalar-curvature formula uses two derivatives of the metric, physical derivatives of \(\chi(r/R)\) are uniformly bounded on this annulus, and every term contains \(g-b\) or one of its first two physical derivatives. The quadratic terms decay faster. The metrics \(q_R\) have uniform bounded-geometry coordinate estimates of the orders used here.

Choose a nonnegative smooth compactly supported function \(D\geq|K|_g\), with support in the region where \(q_R=g\), and a smooth positive \(w_0\) equal to \(r^{-3-\delta}\) far out. We solve \[ \begin{cases} (-\Delta_{q_R}+3)v =D\sqrt{\eta_R^2+|dv|_{q_R}^2}+E_R+\eta_Rw_0,\\ \partial_\nu v=0\quad\hbox{on }S,\qquad v\longrightarrow0 \quad\hbox{at infinity}. \end{cases} \tag{21}\] Here the sign of the normal in the homogeneous Neumann condition is immaterial.

We give the existence and uniform estimates. On a finite truncation put \(v=0\) on its outer sphere, and multiply the entire right side of Equation (21) by \(\lambda\in[0,1]\). Write its gradient term as \(W\cdot dv+F_D\), where \(|W|\leq D\) and \(0\leq F_D\leq D\eta_R\); for example this follows from \[\sqrt{\eta_R^2+|dv|^2} =\frac{|dv|^2}{\sqrt{\eta_R^2+|dv|^2}} +\frac{\eta_R^2}{\sqrt{\eta_R^2+|dv|^2}}.\] The maximum principle for \(-\Delta-W\cdot d+3\), with the indicated mixed boundary conditions, bounds \(|v|\) by \(C\eta_R\). Outside a fixed sphere containing \(\mathop{\mathrm{supp}}D\), the radial function \(r^{-\beta}\) satisfies \[(-\Delta_{q_R}+3)r^{-\beta} =\{(3-\beta)(\beta+1)+o(1)\}r^{-\beta} \geq c_\beta r^{-\beta}\] uniformly in \(R\). Comparison using Equation (20) therefore sharpens the bound to \[ |v|\leq C\eta_R(1+r)^{-\beta}. \tag{22}\] The boundary constants in these comparisons are independent of the outer truncation.

For completeness, the required derivative bounds follow from scalar elliptic estimates without any coupled-system assumption. Divide the equation by \(\eta_R\) and write \(u=v/\eta_R\). Its gradient dependence is \(D\sqrt{1+|du|^2}\), with uniformly bounded first derivative in \(du\). Local \(W^{2,p}\) estimates on physical balls, the corresponding Neumann or Dirichlet half-ball estimates, and first-derivative interpolation give a bound for \(u\) in \(W^{2,p}\) on smaller patches in terms of its already bounded height and the source \(E_R/\eta_R+w_0\). One can see the absorption directly by applying the estimate to nested cutoffs and using \[\|du\|_{L^p}\leq\varepsilon\|u\|_{W^{2,p}} +C_\varepsilon\|u\|_{L^p}.\] Choose \(p>3\) sufficiently large. Sobolev embedding makes \(du\) Hölder, and the scalar Schauder estimate then gives \(C^{2,\alpha'}\) control. On the end the same argument is applied after multiplication by the local weight, which is comparable on every physical unit patch. Thus \[ \|v\|_{C^{2,\alpha'}_\beta}\leq C\eta_R. \tag{23}\] Lemma 119 supplies these scalar estimates and the finite-domain inverse. Its local reflection argument permits bounded drift, and its Schauder step is applied to the original one-sided equation after the gradient becomes Hölder. The ellipticity, boundary smoothness, source regularity and gradient-growth hypotheses have just been specified. The linearization of the finite-domain equation is \[-\Delta_{q_R}+3- \lambda D\frac{\langle dv,d(\,\cdot\,)\rangle} {\sqrt{\eta_R^2+|dv|^2}}.\] Its kernel vanishes by the same maximum principle. Its index is zero by continuation of the bounded drift from zero. It is therefore an isomorphism with these boundary conditions. The a priori bounds close the continuation from \(\lambda=0\) to \(1\). Exhaustion, the uniform estimates, and a diagonal subsequence give a smooth solution of Equation (21).

On its exact hyperbolic tail, this solution satisfies \((-\Delta_b+3)v=\eta_Rr^{-3-\delta}\). Rotations commute with the operator and annihilate this radial source. Applying difference quotients, comparison by \(r^{-\beta'}\) for any fixed \(\beta'<3\), and local elliptic estimates gives the same decay for all fixed angular derivative orders. The hyperbolic Laplacian is \[\Delta_b=(1+r^2)\partial_r^2+(2/r+3r)\partial_r +r^{-2}\Delta_\sigma.\] Consequently, for \(\beta'\) sufficiently close to \(3\), \[(-r^2\partial_r^2-3r\partial_r+3)v =\eta_Rr^{-3-\delta}+O_\infty(r^{-2-\beta'}).\] Variation of constants for the two Euler solutions \(r\) and \(r^{-3}\), with the growing solution excluded by decay, yields \[v=r^{-3}v_3(\omega) -\frac{\eta_R}{\delta(4+\delta)}r^{-3-\delta} +O_\infty(r^{-4}).\] Angular difference quotients give a smooth \(v_3\), and differentiating the radial equation gives the stated symbol estimates. In particular \(d_0=\eta_R/[\delta(4+\delta)]>0\).

Define \[\widetilde g=e^{4v}q_R,\qquad \widetilde K=e^{2v}K.\] The stress transforms as \(\widetilde K-(\mathop{\mathrm{tr}}_{\widetilde g}\widetilde K)\widetilde g =e^{2v}(K-(\mathop{\mathrm{tr}}_{q_R}K)q_R)\), so its sign is retained. The conformal boundary formula is \[H_{\widetilde g}+\mathop{\mathrm{tr}}_T\widetilde K =e^{-2v}(H_{q_R}+\mathop{\mathrm{tr}}_TK+4\partial_\nu v),\] which preserves the original expansion sign. The scalar and divergence formulas are \[\begin{align*} e^{4v}\mathcal H_{\widetilde g}(\widetilde K) &=\mathcal H_{q_R}(K)+6(e^{4v}-1) -8\Delta_{q_R}v-8|dv|_{q_R}^2, \tag{24}\\ \mathcal J_{\widetilde g}(\widetilde K) &=e^{-2v}\{\mathcal J_{q_R}(K)+4K(\nabla v,\cdot)\}. \tag{25}\end{align*}\] The momentum norm in the second formula gains a further \(e^{-2v}\). Substituting Equation (21), using \(D\sqrt{\eta_R^2+|dv|^2}\geq |K||dv|\), and using the original DEC on the support of \(K\) and its scalar version outside that support, gives \[e^{4v}\{\mathcal H_{\widetilde g}(\widetilde K) -2|\mathcal J_{\widetilde g}(\widetilde K)|\} \geq8\eta_Rw_0-8|dv|^2+6(e^{4v}-1)-24v.\] The last two terms are nonnegative. Equation (23) and \(2\beta>3+\delta\) give \(|dv|^2\leq C\eta_R^2(1+r)^{-3-\delta}\). For large \(R\) the required strict margin follows.

Uniform bilinear convergence of \(\widetilde g\) to \(g\) gives convergence of every full-cut infimum: if \((1-\epsilon)g\leq\widetilde g\leq(1+\epsilon)g\), the same factors bound all two-dimensional areas, hence their infima. To verify mass convergence, observe outside \(\mathop{\mathrm{supp}}D\) that \[e^{4v}(R_{\widetilde g}+6) =\chi_R(R_g+6)+8\eta_Rw_0+O(v^2+|dv|^2).\] The weighted integrals of the right side on distant tails are uniformly small. The scalar-linearization divergence formula in Lemma 6 has quadratic remainder bounded by \(C V_0r^{-2\beta}\), whose integral is finite because \(\beta>3/2\). Compare the fluxes first on a fixed sphere, where local convergence applies, and then bound both tails by these integrals. Since \(|V_a|\leq V_0\), this proves convergence for all four fluxes. Finally, for \(K=0\), \(D=0\) and the correction equation is linear on each fixed compact subset for all sufficiently large \(R\). Differentiated local estimates prove the asserted smooth convergence. Relabeling a sequence \(R\to\infty\) proves the lemma. ◻

Proof. Use disjoint Gaussian collars \([0,L]\times S_j\) with metric \(dx^2+q_j(x)\), where \(L>0\) is fixed. Extend \(q_j\) smoothly to \([-\varepsilon,0]\). Since \(q_j(x)=q_j(0)+O(\varepsilon)\) there, a cutoff to the constant \(q_j(-\varepsilon)\) near \(-\varepsilon\) changes only the metric values by \(O(\varepsilon)\); derivatives may depend on \(\varepsilon\). Choose a smooth map \(F_\varepsilon:[-\varepsilon,L]\to[0,L]\) equal to the identity near \(L\), mapping endpoints to endpoints, and satisfying \(F_\varepsilon-x=O(\varepsilon)\) and \(F_\varepsilon'=1+O(\varepsilon/L)\). Its product with the identity on \(S_j\), extended by the identity elsewhere, is a diffeomorphism \(\Phi_\varepsilon:\Omega_\varepsilon\to\Omega\) with \[(1-o(1))g_\varepsilon\leq\Phi_\varepsilon^*g \leq(1+o(1))g_\varepsilon.\] This diffeomorphism maps a connected exterior submanifold and its entire intrinsic boundary to another of the same type. Applying the area comparison to every such cut proves Equation (26). The new product face has mean curvature zero. ◻

Proof. Below \(h/2\) the normalization gives \(t_h(s)=c_hs\) with \(c_h\asymp h\). On the transition interval \(s\asymp h\), \(t_h\asymp h^2\), and \(b_h\in[1,2]\). Above it the inequalities follow from \(t/s^2>0\) and \(b>0\) on a bounded interval. The radial and tangential eigenvalues of the covector derivative are \(t_h'\) and \(t_h/s\). Metric differentiation costs at most a constant times \(|p|_g\) in differentiating \(|p|_g\), and \(s t_h'=b_h t_h\leq2t_h\). These facts prove (34). For example, outside a slightly larger bounded-gradient interval one may interpolate the logarithmic derivative to \(2\) and then continue by a quadratic flux law. Positive lower and upper bounds for that logarithmic derivative give the required global extension. This extension is used only after the gradient bound; it is not an existence argument for the saturating law.

We specify the hypotheses of the regularity theorem. For \(-\partial_i A^i(x,u,Du)=B(x,u,Du)\) on a \(C^{1,\alpha}\) domain, Appendix Theorem A.1 of (Porretta and Véron 2009) permits \[\lambda(\mu^2+|p|^2)^{(p_0-2)/2}I \leq D_pA\leq \Lambda(\mu^2+|p|^2)^{(p_0-2)/2}I,\] an analogous bound for \(|D_pA|\), and the spatial and scalar continuity condition \[|A(x,s,p)-A(y,t,p)| \leq C(1+|p|^{p_0-2}+|p|^{p_0-1}) (|x-y|^\alpha+|s-t|^\alpha), \qquad |B|\leq C(1+|p|^{p_0}).\] For bounded weak solutions it yields a Hölder continuous gradient up to a \(C^{1,\alpha}\) Dirichlet boundary; the same local argument gives the interior estimate. We apply it with \(p_0=3\), \(\mu=h\), no scalar dependence, and \(B=0\). The metric bounds and (34) verify every displayed condition; constants on a fixed height and gradient range are uniform in \(h\). At a finite Dirichlet face the datum is constant; subtracting that constant reduces a boundary patch to the homogeneous theorem. The finitely many smooth boundary components are disjoint, so these patches can be chosen separately. At conformal infinity the datum is already zero. The interior gradient estimates originate in (Tolksdorf 1984), and the frozen boundary estimate used in the proof is the boundary regularity estimate of (Lieberman 1988). We use the stated Appendix theorem as the precise boundary input.

For completeness, normalization removes any additive constant in the interior gradient bound. Extend the unregularized density with \(D_pA\asymp|p|\) and \(|\partial_xA|\leq C|p|^2\). If \(m=\operatorname{osc}_{B_1}f>0\), subtract the infimum and put \(u=f/m\). The equation for \(u\) has density \(A_m(x,p)=m^{-2}A(x,mp)\), whose structural constants do not depend on \(m\). Since \(0\leq u\leq1\), the local gradient estimate for \(u\) is uniform. Multiplication by \(m\) proves (35). The case \(m=0\) is immediate. ◻

Proof. At a noncritical point put \(s=|df|_g\), \(e=\nabla f/s\), and \[Lf=A^{ij}\nabla_i\nabla_jf=0, \qquad A=g^{-1}+(b_h-1)e\otimes e.\] The maximum principle gives the height bound. Let \(d\) be the inward distance from the finite boundary. Its level sets have mean curvature \(H(d)=\Delta_g d\leq C d\) in a fixed collar, since \(H(0)\leq0\). Consider \[\underline f=D'-\gamma(d),\qquad \gamma'(d)=\frac{c}{\sqrt{d^2+d_*^2}}.\] The radial operator on this function is \[ L\underline f =\gamma'\left(\frac{b_h(\gamma')d}{d^2+d_*^2}-H(d)\right). \tag{37}\] Uniformly in \(h\), \(b_h(s)(1+s^2)\geq c_0>0\), and hence \[\frac{b_h(\gamma')}{d^2+d_*^2} \geq\frac{c_0}{d^2+d_*^2+c^2}.\] Choose the collar length \(\ell\) and \(c>0\) small first; then choose \(d_*>0\) sufficiently small. The last bound makes (37) nonnegative on \([0,\ell]\), while \(\gamma(\ell)=c\operatorname{arsinh}(\ell/d_*)\geq D\). Comparison with \(\underline f\) and the constant upper barrier gives a uniform finite-boundary gradient bound \(c/d_*\).

On the end, for \(F(r)=r^{-\xi}\), the leading radial operator is \(\xi(b_h\xi-2)r^{-\xi}\), with relative error tending to zero uniformly for \(0<b_h\leq2\). Thus \(F\) is a strict supersolution beyond a fixed sphere, uniformly in any positive multiple of \(F\). Comparison with \(K_D(r^{-\xi}-R^{-\xi})\) gives both the last estimate in (36) and the outer-boundary gradient bound. Choose \(K_D\) at the fixed inner sphere before sending \(R\) to infinity; for \(R\) at least twice its radius the required \(K_D\) is uniform.

It remains to exclude a large interior gradient. The following calculation is for \(s\geq s_0\), where \(b_h=b\). Writing \(x=t^{3/2}\) gives \[b+s b'=b\left(1-\frac{18x}{(2+x)^2}\right).\] Thus, for fixed sufficiently large \(s_0\), \(b<1\) and \(b+s b'<0\). At a point with first frame vector \(e\), differentiating \(Lf=0\) and commuting covariant derivatives yields the exact identity \[ \begin{aligned} L\log s &=\frac{1}{s^2}\Bigl( |\mathop{\mathrm{Hess}}f|_{e^\perp\times e^\perp}^2 +(1-b)|\mathop{\mathrm{Hess}}f(e,\cdot)|_{e^\perp}^2\Bigr)\\ &\quad-\frac{b+s b'}{s^2}\mathop{\mathrm{Hess}}f(e,e)^2+\mathop{\mathrm{Ric}}_g(e,e). \end{aligned} \tag{38}\] To see the coefficients, differentiate \(A\) in the \(e\) direction: its contraction with \(\mathop{\mathrm{Hess}}f\) is \(b'\mathop{\mathrm{Hess}}f(e,e)^2+ 2(b-1)s^{-1}|\mathop{\mathrm{Hess}}f(e,\cdot)|_{e^\perp}^2\). The two derivatives of \(\log s\) then leave exactly the three quadratic terms displayed. In the curvature commutator the additional \((b-1)e\otimes e\) term vanishes by antisymmetry, leaving \(\mathop{\mathrm{Ric}}(e,e)\). The Ricci curvature is bounded below on the whole exterior, so \(L\log s\geq-C\). Moreover \(L(f^2)=2b s^2\) and \(\inf_{s\geq s_0}b s^2>0\). Choose \(c_1\) so large that \[L(\log s+c_1f^2)>0\qquad(s\geq s_0).\] An interior maximum in this range is impossible. The two boundary gradient bounds and \(0\leq f\leq D\) therefore bound the maximum of \(\log s+c_1f^2\), and then \(s\), everywhere. At a critical point \(\log s=-\infty\), so such points cause no difficulty in this maximum argument. ◻

Existence, uniqueness, and finite-boundary regularity in Theorem [str-pde-fixed-height]. Fix \(h>0\) and a finite smooth truncation. Use the homotopy of boundary values \(\lambda D\), \(0\leq\lambda\leq1\), for the original regularized saturating equation. At \(\lambda=0\) the zero solution exists. For fixed \(h\) and on the gradient range of Lemma 15, the operator is uniformly elliptic and smooth in the covector. Its linearization has no zeroth-order term, is invertible with homogeneous Dirichlet data by the maximum principle and linear elliptic solvability, and gives openness. For closedness, first apply Lemma 14 using the already established gradient bound. Its extension agrees with the original flux on every solution. This gives a common \(C^{1,\theta}\) bound; the ordinary uniformly elliptic Schauder estimates for this fixed \(h\) then give \(C^{2,\theta'}\) bounds and the required compactness. The continuity method therefore reaches \(\lambda=1\); the linear Dirichlet solvability and Schauder estimates used here follow from Lemma 119, including its general-principal all-Dirichlet conclusion. No uniform ellipticity as \(h\downarrow0\) is asserted at this step.

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

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

Proof. There is a useful radial identity that also applies to the families considered below. For \(g=z^{-2}\bar g\), put \(u=|dz|_{\bar g}^2\) and \(\bar e=\nabla_{\bar g}z/\sqrt u\). If \(F'>0\), the operator on \(F(z)\), divided by \(z^2u\), is \[ bF''+\left(\frac{b-2}{z}+E\right)F',\qquad E=\frac{\Delta_{\bar g}z+(b-1)\mathop{\mathrm{Hess}}_{\bar g}z(\bar e,\bar e)}u, \qquad b=b(z\sqrt u F'). \tag{40}\] Indeed \(\mathop{\mathrm{Hess}}_g z=\mathop{\mathrm{Hess}}_{\bar g}z+ 2z^{-1}dz\otimes dz-z^{-1}u\bar g\) and \(\Delta_g z=z^2\Delta_{\bar g}z-zu\). For (29), \[E=\frac{b z}{1+z^2}-2(b-2)v_z=O(z)\] uniformly for \(0<b\leq2\). More generally, if \(\bar g-h_0=O_2(z^\beta)\) with \(h_0=(1+z^2)^{-1}dz^2+\sigma\) and \(\beta>1\), then \(E=O(z)+O(z^{\beta-1})\), again uniformly in \(b\).

Choose \(F_+(z)=A(z-c_2z^2)\) with a fixed \(c_2>0\). The term \((b-2)F_+'/z\) is nonpositive. Make \(z_0\) small and take \(A\) sufficiently large for \(F_+(z_0)\geq f(z_0,\cdot)\). The order of these choices can be made consistent with small physical slope: the coarse bound allows \(A=2C_0z_0^{\xi-1}\), so \(z\sqrt u F_+'\leq C C_0z_0^\xi\). Thus \(b\geq1\) on the whole collar after reducing \(z_0\). The negative term \(-2Ac_2b\) then dominates \(E F_+'\). Hence \(F_+\) is a supersolution, and comparison gives \(f\leq F_+\). For the lower barrier take \(F_-(z)=a_0(z+c_2z^2)\), with \(a_0>0\) small enough to lie below the positive minimum of \(f\) on \(z=z_0\). Here \(b=2-O(a_0^3z^3)\), so the unfavorable term \((b-2)F_-'/z\) is \(O(a_0^4z^2)\), while \(bF_-''\) has a positive constant multiple of \(a_0\). Reducing the collar if needed makes this a subsolution. For the upper comparison test by \((f-F_+-\varepsilon)_+\); for the lower comparison use \((F_--f-\varepsilon)_+\). The limiting zero boundary values make both tests compactly supported for every \(\varepsilon>0\), and their traces vanish on the inner sphere. Monotonicity and then \(\varepsilon\downarrow0\) justify both comparisons at infinity. They prove the height bounds in (39).

Physical balls of a fixed small radius in this collar have uniform metric bounds; \(z\) is comparable to its central value throughout each ball. The height estimate gives oscillation \(O(z)\) there. Apply (35) using the bounded-gradient extension furnished by the global bound (or the asserted uniform local bound for a family). It gives \(|df|_g=O(z)\). All the choices above use only the quantities listed in the statement, proving the uniform version. ◻

Proof. Put \(q=|df|_{\bar g}\). In dimension three, change of the volume density and inverse metric gives the exact equation \[ \mathop{\mathrm{div}}_{\bar g}\big(a(zq)q\nabla_{\bar g}f\big)=0. \tag{41}\] For reference, the conformal factor in (29) can also be cancelled exactly: with \(Q=|df|_{h_0}\) the equation is \(\mathop{\mathrm{div}}_{h_0}[a(ze^{-2v}Q)Q\nabla_{h_0}f]=0\). The bounds (39) give \(q\leq Q_0\), so \(f\) is Lipschitz up to \(z=0\) in the compact metric and has zero trace there. In a \(\bar g\)-orthonormal frame, the covector derivative of the compact flux has eigenvalues \[a(zq)q,\qquad a(zq)q\,b(zq).\] They are uniformly comparable to \(q\), and the first spatial derivatives of the coordinate-density flux are bounded by \(Cq^2\) on this bounded covector range. A continuation beyond that range gives the global three-Laplace structure used in Lemma 14.

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

The compact metric is generally \(C^{3,\delta}\), rather than smooth in its normal variable. Every one of its angular derivatives has the same finite normal regularity. Choose \(0<\gamma<\min(\theta,\delta)\). The first boundary upgrade can be obtained without assuming a second derivative in advance. Take angular difference quotients of the divergence equation. Their principal coefficient is the average of \(A_p\) between two gradients; it has a common Hölder modulus and is uniformly elliptic. Their divergence source is the difference quotient of the explicit spatial dependence of the flux, with the same Hölder control. The boundary trace remains zero. The linear divergence boundary estimate (Agmon et al. 1959, Theorem 9.1, p. 675, with \(l=1\)) therefore bounds these difference quotients in \(C^{1,\gamma}\) on smaller half-balls, using their already bounded \(C^0\) norm. Passing to the limit gives tangential second derivatives and mixed derivatives. The equation then gives the normal derivative of the normal flux from its tangential derivatives. Its derivative with respect to \(f_z\) is bounded away from zero, so the implicit function theorem recovers the remaining normal second derivative with its Hölder bound. Thus \(f\in C^{2,\gamma}\), and subsequent ordinary Schauder estimates use Lemma 119. To record explicitly why arbitrary angular derivatives are allowed, write (41) in density form \(\partial_j A^j(x,Df)=0\) and put \(B^{jk}=A^j_{p_k}(x,Df)\). For an angular multi-index \(\nu\ne0\), \(w_\nu=\partial_\omega^\nu f\) satisfies \[ \partial_j(B^{jk}\partial_k w_\nu)=-\partial_j F_\nu^j, \qquad w_\nu|_{z=0}=0. \tag{42}\] After the top derivative \(B Dw_\nu\) is removed, the chain-rule remainder \(F_\nu\) contains only explicit angular derivatives of the flux and \(D\partial_\omega^\mu f\) with \(|\mu|<|\nu|\). Inductively \(B,F_\nu\in C^{1,\gamma}\), and the preceding stage gives \(w_\nu\in C^{1,\gamma}\). Expansion of (42) gives a uniformly elliptic nondivergence equation whose right side is \(C^{0,\gamma}\). Linear boundary Schauder estimates give \(w_\nu\in C^{2,\gamma}\). Finite covers and nested boundary patches complete this induction for each fixed \(\nu\), with constants allowed to depend on \(\nu\).

Set \(C_f=f_z|_{z=0}\). It is positive and smooth angularly. Taylor’s formula applied to each \(w_\nu\) gives \(\partial_\omega^\nu(f-zC_f)=O(z^2)\). The identities \[z\partial_z(w_\nu-z(w_\nu)_z|_0) =z((w_\nu)_z-(w_\nu)_z|_0),\] \[(z\partial_z)^2(w_\nu-z(w_\nu)_z|_0) =z((w_\nu)_z-(w_\nu)_z|_0)+z^2(w_\nu)_{zz}\] prove the two symbol derivative estimates. Finally \(q=C_f+O(z)\), and the outward normal to the physical end points toward decreasing \(z\). Substituting in the physical flux, with \(dA_g=z^{-2}e^{4v}dA_\sigma\), gives (32). This proves the remaining assertions of Theorem [str-pde-fixed-height]. ◻

Proof. Increase \(s_0\) in (38) if necessary. Since \(b\to0\) and \(-(b+s b')/b\to1\), that identity implies \[ L\log s\geq\tfrac12|d\log s|_A^2-C \qquad(s\geq s_0). \tag{43}\] Choose a smooth positional cutoff \(\psi\) that is negative near the ball boundary and positive on the smaller ball, with controlled first two derivatives. Choose \(\varepsilon>0\) depending only on \(H\) so that \(\phi=\psi+\varepsilon f\) is still negative near the outer boundary and uniformly positive on the smaller ball. Because \(Lf=0\), \(L\phi=L\psi\) is uniformly bounded. Moreover, when \(s\) is sufficiently large, \[ |d\phi|_A^2\geq b(\varepsilon s+e\psi)^2 \geq c\varepsilon^2, \tag{44}\] using \(bs^2\to4/3\). On the relatively compact region \(\phi>0\), put \(q(\phi)=e^{k\phi}-1\) and consider \(W=\log s+4\log q(\phi)\). At an interior maximum with large gradient, \(d\log s=-4(q'/q)d\phi\). Equations (43)–(44) then give \[ LW\geq 4\left((q'/q)^2+q''/q\right)|d\phi|_A^2 +4(q'/q)L\phi-C. \tag{45}\] Here \(q'/q\geq k\) and \(q''/q=kq'/q\). A fixed sufficiently large \(k\) makes the right side positive, a contradiction. Since \(W\to-\infty\) at \(\phi=0\), and \(q\) is bounded below on the smaller ball, a bound at the remaining, bounded-gradient maxima gives the desired interior gradient bound. Every calculation takes place where the gradient is large and the solution is smooth, so critical points introduce no additional regularity requirement. ◻

Proof. First obtain an upper height bound without using \(D_i\). Write \(\eta=\operatorname{arsinh}r\) and choose a large fixed \(\eta_0\). On \(x=\eta-\eta_0>0\) take \[ w(\eta)=\frac{A e^{-\xi x}}x,\qquad 0<\xi<1. \tag{48}\] Uniformly in \(i\), \(|d\eta|_{g_i}^2=1+o(1)\) and \[\Delta_{g_i}\eta+(b-1)\mathop{\mathrm{Hess}}_{g_i}\eta(e,e)=2+o(1), \qquad e=\frac{\nabla_{g_i}\eta}{|d\eta|_{g_i}},\] for \(0<b\leq2\). The radial operator is therefore \(b w''|d\eta|^2+w'(2+o(1))\). A direct differentiation gives \[\frac{w''}{-w'}=\xi+\frac1x+\frac1{x(1+\xi x)}.\] Far from the pole, \(b\leq2\) and \(\xi<1\) give a strict supersolution after fixing \(\eta_0\) sufficiently large. Near the pole, \(|w'|\asymp A/x^2\), \(w''\asymp A/x^3\), and \(b(|w'||d\eta|)=O(x^4/A^2)\): the positive term is \(O(x/A)\), whereas the negative term has order \(-A/x^2\). On the remaining compact interval, increasing \(A\) makes \(b\) uniformly small. Thus one fixed \(A\) makes \(w\) a supersolution on the entire end, for every \(i\).

For each finite \(D_i\) compare first on a truncated domain starting sufficiently near the pole that \(w>D_i\). At its outer boundary the Dirichlet solution is zero. Passage to the exhaustion limit gives \(f_i\leq w\). In particular \(f_i\leq C_0z^\xi\) beyond a fixed sphere, independently of \(i,D_i\). Applying Lemma 18 on uniform physical balls there gives a common local physical gradient bound. This is the bound needed before extending the flux to a three-Laplace structure law on these patches.

For a uniform positive lower value on a fixed sphere, let \(u_i\) be the height-one solution for \(g_i\). Comparison gives \(f_i\geq u_i\). The fixed-height estimates have uniform constants in this family: local smooth convergence controls the finite-boundary collars and compact geometry, while (46) controls the end. Thus every subsequence has a further subsequence converging locally in \(C^1\) to the height-one solution for \(g\), with zero limiting boundary value at infinity. This limit is positive in the interior by the local annular comparison used above. Uniqueness gives the same limit for every subsequence, so the minima of \(u_i\) on a fixed interior sphere have a positive common lower bound. Reducing it to handle finitely many initial terms if necessary supplies the desired lower bound. Lemma 16 now gives (47), using its general radial formula (40).

We spell out the regularity conversion for the metrics. If \(E_i=z^2(g_i-b)\), tensor-to-coordinate conversion in (46) gives \[ |\partial^j E_i|\leq C M z^{\beta-j},\qquad 0\leq j\leq2. \tag{49}\] Indeed, a tensor of rank \(2+j\) with \(b\)-norm \(O(z^\beta)\) has coordinate components \(O(z^{\beta-2-j})\), and \(\Gamma_b=O(z^{-1})\), \(\partial\Gamma_b=O(z^{-2})\). The compact metrics are consequently uniformly \(C^{1,\alpha}\) for any \(0<\alpha<\min(\beta-1,1)\). To verify the only nontrivial case \(1<\beta<2\), let two points be separated by coordinate distance \(d\). If \(d\) is smaller than half the larger \(z\) value, integrate the second derivative bound along a path on which \(z\) is comparable to that value: \[|DE_i(x)-DE_i(y)|\leq C d z^{\beta-2} \leq C d^{\beta-1}.\] For the remaining pairs use the first derivative amplitude \(Cz^{\beta-1}\) at the two endpoints. Pairs on the boundary follow by continuity. Smooth interior convergence and this uniform estimate also give compact \(C^{1,\alpha'}\) convergence for \(\alpha'<\alpha\). In particular, a cutoff annulus moving toward \(z=0\) does not require any extra ordinary-derivative bound.

The compact flux in (41) has uniform three-Laplace structural constants, and (47) gives a common compact gradient bound. The weak zero-trace formulation and Lemma 14 give uniform \(C^{1,\theta}\) estimates up to \(z=0\). The lower height bound implies \(f_{i,z}(0,\omega)\geq c\). The Hölder gradient estimate therefore makes a common collar noncritical. On it ordinary quasilinear boundary Schauder estimates yield a uniform \(C^{2,\gamma}\) bound for \(0<\gamma<\min(\alpha,\theta)\). Compactness gives the stated convergence, and smooth interior convergence of the metrics permits elliptic bootstrap on each compact subset of the collar away from \(z=0\).

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

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

Integration of \(\operatorname{div}(f_DX_D)=t(|df_D|)|df_D|\) over a truncation gives the first identity in Equation (50). The outer boundary term tends to zero because \(f_D=O(z)\) while the flux has a finite limit. The integrand is integrable at infinity: it is \(O(z^3)\) and the volume element is \(O(z^{-3})\,dz\,d\omega\).

Fix \(0<\lambda<1\). The coarse radial supersolution in Theorem 13 gives \(f_D\leq cD r^{-\xi}\) for a fixed \(0<\xi<1\), with \(c\) independent of \(D\). Hence \[ \mathop{\mathrm{Vol}}_g\{f_D>\lambda D\}\leq V_\lambda<\infty \tag{52}\] uniformly in \(D\). The function \(\sigma(1-t(\sigma))\) is bounded on \([0,\infty)\): it tends to zero both at zero and at infinity, since \(1-t(\sigma)=O(\sigma^{-2})\) at infinity. Thus for a constant \(C_0\), \[D\mathcal F_D \geq\int_{\{f_D>\lambda D\}}|df_D|\,dV_g-C_0V_\lambda.\]

We justify the coarea lower bound with the exact full-boundary convention. For almost every level \(a\in(0,D)\), the set \(\{f_D>a\}\) has finite perimeter, contains an inner collar of \(T\), and is bounded. Its complement has a unique component containing the end. Fill the other complementary components. This operation can only decrease the perimeter, and the remaining entire boundary separates the end from every component of \(T\). Approximate the filled set by smooth sets in a fixed compact region while preserving a smaller inner collar and the distant exterior; their perimeter converges to the finite-perimeter value. Such approximation follows by convolving the characteristic function in finitely many interior charts and using regular levels; the inner and outer collars are held fixed, and bounded complementary components are again filled. Their full boundaries are smooth admissible cuts. It follows that the original level perimeter is at least \(A_g(T)\). Coarea, valid for the locally Lipschitz \(f_D\), now gives \[\int_{\{f_D>\lambda D\}}|df_D|\,dV_g =\int_{\lambda D}^{D}\operatorname{Per}\{f_D>a\}\,da \geq(1-\lambda)D A_g(T).\] No smooth Sard assertion at critical points of a \(C^{1,\theta}\) solution is needed. We obtain \[(1-\lambda)A_g(T)-C_0V_\lambda/D \leq\mathcal F_D\leq A_g(T).\] Let \(D\to\infty\), then \(\lambda\downarrow0\). Finally, Equation (51) is Hölder’s Inequality on the sphere with its probability measure \(d\omega/(4\pi)\). ◻

Proof. Let \[W(\sigma)=\frac{3+k^2}{2k}.\] Differentiating \(\sigma=\sqrt t/k\) and \(k^2=1-t^{3/2}\) gives \[ W'(\sigma)=\tfrac32t(\sigma),\qquad W(\sigma)-\sigma W'(\sigma)=2k. \tag{55}\] Consequently \(W\) is convex and \(\mathcal P=Wg-\sigma W'e\otimes e\). The weak equation says that \(f\) is a local minimizer of \(\int W(|df|)\,dV_g\): convexity gives the minimizing inequality against every compactly supported additive variation. Composing with a compactly supported smooth diffeomorphism is another admissible local variation. Differentiating this inner variation yields \(\int\mathcal P:\nabla X\,dV_g=0\) for every compactly supported smooth vector field \(X\). The differentiation is valid for the locally Lipschitz minimizer by change of variables and dominated convergence. This proves distributional conservation.

Trace reversal in Equation (53) gives \[ L_{ee}=-\frac{3-k^2}{2k},\qquad L|_{e^\perp}=-kI. \tag{56}\] Both eigenvalues are negative and their quadratic constraint combination is \(6\). As \(\sigma\to0\), \(k=1+O(\sigma^3)\), so the definitions extend continuously across critical points, independently of \(e\).

Positivity of \(K-(\mathop{\mathrm{tr}}K)g\) implies \(\mathop{\mathrm{tr}}K\leq0\) and \(\mathop{\mathrm{tr}}_{e^\perp}K\leq0\). Thus \[\mathcal P:K=2k\mathop{\mathrm{tr}}K+ \frac{3(1-k^2)}{2k}\mathop{\mathrm{tr}}_{e^\perp}K\leq0.\] The flat Hamiltonian of \(K+L\) is \[R_g+(\mathop{\mathrm{tr}}(K+L))^2-|K+L|^2 =\mathcal H_g(K)-2\mathcal P:K,\] and its momentum is \(\mathcal J_g(K)\) by conservation. Negative definiteness of \(L\) gives the strict boundary expansion decrease.

We spell out the smoothing assertion because pointwise tensor approximation alone would not control momentum. In a coordinate chart let \(Q^{ij}=\sqrt{\det g}\,\mathcal P^{ij}\). Conservation means \[\partial_jQ^{ij}=-\Gamma^i_{jk}Q^{jk}.\] Convolution gives smooth symmetric densities converging uniformly to \(Q\), whose divergence differs from the required connection term by a commutator tending uniformly to zero. In a boundary chart \(x^3\geq0\) use instead \[Q_\epsilon(x)=\int\rho_\epsilon(y) Q(x+2\epsilon e_3-y)\,dy.\] Its kernel lies on the original side even at the face, and the same commutator estimate holds. After a partition of unity, additional divergence errors are sums of \(d\chi\,(Q_\epsilon-Q)\), which also tend uniformly to zero. Divide by the original volume density and trace reverse. The algebraic terms and the momentum therefore converge uniformly on the compact set. Strict DEC and strict negative expansion survive sufficiently fine smoothing. A separate smoothing scale may be chosen for each member of a sequence. ◻

Proof. Choose a spherical orthonormal frame \(\bar E_a\), independent of \(s\), and set \(E_a=s^{-1}\bar E_a\). At a point where the spherical connection vanishes, the connection of \(q\) is \[\begin{aligned} \nabla_{E_a}n&=\frac{\sqrt F}{s}E_a,& \nabla_{E_a}E_b&=-\frac{\sqrt F}{s}\delta_{ab}n,\\ \nabla_nn&=\frac1s(\nabla_\gamma\log\sqrt F)^aE_a,& \nabla_nE_a&=-\frac1s(\partial_a\log\sqrt F)n. \end{aligned}\] Away from this point the second identity also contains the spherical connection divided by \(s\). Thus the leaf shape operator is \(\sqrt F I/s\). For lapse \(N=F^{-1/2}\) the scalar Gauss identity is \[R_q=R_{s^2\gamma}-|A|^2-H^2-2n(H)-2N^{-1}\Delta_{s^2\gamma}N.\] Substituting \(H=2\sqrt F/s\) gives Equation (57). Moreover, \[(\mathop{\mathrm{tr}}B)^2-|B|^2=4hk+2k^2-2|G|^2-|U|^2,\] which proves Equation (58) by differentiating \(F\). For \(P=B-(\mathop{\mathrm{tr}}B)q\), its blocks are \(P_{nn}=2k\), \(P_{nT}=G\) and \(P_T=(h+k)I+U\). In the normal divergence, the two acceleration terms give \(-2G\cdot\nabla_\gamma\log\sqrt F/s\), while the shape terms give \(2\sqrt F(k-h)/s\). In the tangential divergence the shape terms give \(3\sqrt F G/s\) and the acceleration terms give \(((k-h)I-U)\nabla_\gamma\log\sqrt F/s\). Adding the ordinary block derivatives proves the two momentum identities. ◻

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

Proof of Theorem 10. If \(K-(\mathop{\mathrm{tr}}K)g\leq0\), replace \(K\) by \(-K\). Indeed \(\mathop{\mathrm{tr}}_TK\geq0\) for every two-plane, so this decreases the future expansion and preserves DEC. We may therefore assume \(K-(\mathop{\mathrm{tr}}K)g\geq0\).

Apply Lemma 11 and first fix one prepared data set, suppressing its index. Add the short inner collar from Lemma 12. Solve the saturating Dirichlet problem there with height \(D\), and restrict its stress to the original exterior. Lemma 22 gives strict flat-cosmological DEC and strict future trapping for \((g,K+L)\). The stress is smooth on an end, and it can be smoothed on the remaining compact region without changing either the end or the strict conditions.

Proposition [prop:str-af-bending], with arbitrarily small error, produces a smooth asymptotically Euclidean exterior satisfying all hypotheses in Equation (16), with \[E\leq p_0(g)-\tfrac12\fint C_D^3+\epsilon, \qquad A_q(S)\geq(1-\epsilon)A_g(S).\] Theorem 25 therefore gives, on letting \(\epsilon\downarrow0\), \[p_0(g)\geq\sqrt{\frac{A_g(S)}{16\pi}} +\tfrac12\fint C_D^3 \geq\sqrt{\frac{A_g(S)}{16\pi}} +\tfrac12(\mathcal F_D/(4\pi))^{3/2}.\] For each fixed added collar, let \(D\to\infty\) and use Lemma 21. Then let the collar width tend to zero and use Equation (26). This proves Equation (15) for the prepared data. Finally, remove the preparation using the mass and area convergence in Lemma 11.

Every limiting operation just used is a limit of scalar inequalities. No smooth limiting stress as \(D\to\infty\), or regular limiting metric at equality, is required for this argument. The fixed-chart proof did not use a sign or timelikeness assumption on the flux covector. If it is future timelike, a background hyperbolic isometry makes its time component equal to its Lorentz norm. The geometric and decay hypotheses, the compact support and sign of the constraint stress, and the cut area are unchanged. Lemma 6 then proves the final assertion. ◻

Proof. This is the total-region theorem of (Andersson and Metzger 2009, Definitions 7.1–7.2 and Theorem 7.3), together with its strict-barrier existence theorem and stability conclusion (Andersson and Metzger 2009, Theorems 3.1 and 4.1). The conventions in Section 2 of that paper allow an empty required inner boundary for the total-region theorem. Apply these results on each connected component of \(M\): its outer part is nonempty, and strict-barrier existence is used only when its own inner part is nonempty. A component with empty inner part contributes a frontier only if its collection of trapped domains is nonempty. Take the finite union of the resulting regions. Each free frontier is oriented toward the designated outer part. Their domains include the entire required inner boundary, which is not discarded as an empty relative frontier. ◻

Proof. Choose a countable dense set \(x_i\) and \(L\) larger than the diameter of the compact space. Set \[d_c(x)=\min\{L,\operatorname{dist}(x,E_c)\}, \qquad d_c\equiv L\quad\hbox{if }E_c=\varnothing .\] For every \(i\), the bounded nonincreasing function \(c\mapsto d_c(x_i)\) has at most countably many right discontinuities. Avoid their countable union. The functions \(d_c\) are uniformly \(1\)-Lipschitz, so convergence at the dense set implies uniform convergence by a finite net argument. Nesting gives one Hausdorff inclusion; if points in \(E_{c_j}\) stayed a positive distance from \(E_c\), evaluating the uniformly convergent distance functions at those points would contradict \(d_{c_j}=0\) there. If \(E_c\) is empty, uniform convergence to \(L\) is incompatible with a zero of any \(d_{c_j}\). ◻

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

If \(\lambda=0\), the kernel of \(L\) is spanned by \(\varphi>0\), and its adjoint kernel is spanned by \(\varphi^*>0\). These scalar elliptic principal-eigenfunction properties also follow by applying the maximum principle after dividing a kernel element by \(\varphi\), together with Fredholm index zero. The derivative of the augmented map \[(v,a)\longmapsto \left(\Phi(v)-a,\int_\Sigma v\,\,\mathrm dA\right)\] is \[(v,a)\longmapsto \left(Lv-a,\int_\Sigma v\,\,\mathrm dA\right).\] It is an isomorphism. For a prescribed first component \(f\), pairing with \(\varphi^*\) uniquely fixes \(a=-\int\varphi^*f/\int\varphi^*\); adding a unique multiple of \(\varphi\) then fixes the second component. The implicit-function theorem supplies \(v(s),a(s)\) with output \((0,s)\). Differentiation gives \(a'(0)=0\) and \(v'(0)=\varphi/\int_\Sigma\varphi>0\). Thus the graphs are ordered and have constant expansion \(c+a(s)\). If \(a(s)\le0\) for any sufficiently small \(s>0\), replacing this one frontier component by that outer graph enlarges the bounding region while keeping every free component at expansion at most \(c\). This contradicts maximality. The claimed bounds on the leaf parameter follow from positive velocity and compactness. ◻

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

For \(X\in T_xM\), solve the Jacobi boundary-value problem \[\nabla_{\dot\gamma}^2J_X+ \operatorname{Rm}(J_X,\dot\gamma)\dot\gamma=0,\qquad J_X(0)=X,\qquad J_X(z)=P_zX.\] In a parallel frame its integral equation is \[j_X(t)=X+tA_X-\int_0^t(t-s)\mathcal R(s)j_X(s)\,\,\mathrm ds,\qquad A_X=\frac1z\int_0^z(z-s)\mathcal R(s)j_X(s)\,\,\mathrm ds.\] For small \(z\), absorption gives \(\sup_t\lvert j_X(t)\rvert\le2\lvert X\rvert\) and \(\lvert A_X\rvert\le Cz\lvert X\rvert\). Moreover \(\langle J_X,\dot\gamma\rangle\) is affine with equal endpoint values. Hence \(A_X\) is perpendicular to \(\dot\gamma(0)\). We can prescribe the first jet of a local unit vector field \(V\) by \[V(x)=\dot\gamma(0),\qquad \nabla_XV(x)=A_X .\] To realize this jet, use a frame obtained by radial parallel transport from \(x\), choose affine coefficients with the prescribed first jet, and normalize their length. The perpendicularity just proved means that normalization preserves the first derivative. The second derivatives of this unit field have a uniform bound on a small ball.

For \(F_z(y)=\exp_y(zV(y))\), the Jacobi construction gives \[ F_z(x)=x',\qquad (\,\mathrm dF_z)_x=P_z,\qquad \lvert(\nabla\,\mathrm dF_z)_x\rvert\le Cz . \tag{94}\] For the last estimate, differentiate the exponential map twice, including the bounded jets of \(V\). At \(z=0\) the map is the identity and its covariant second derivative is zero; the first derivative in \(z\) is uniformly bounded on these jets. This construction never uses derivatives of the supporting function.

Every \(y\in E\) near \(x\) is moved a distance \(z\), so \(F_z(y)\in E_z\). Thus \(\psi=\phi\circ F_z\) supports \(E\) at \(x\). Equation (94) makes its gradient length exactly that of \(\phi\) at \(x'\), and its normal is \(\dot\gamma(0)\). The covariant chain rule reads \[\mathop{\mathrm{Hess}}\psi(X,Y)=\mathop{\mathrm{Hess}}\phi(P_zX,P_zY) +\,\mathrm d\phi\bigl((\nabla\,\mathrm dF_z)(X,Y)\bigr).\] Trace over \(\dot\gamma(0)^\perp\) and divide by the common gradient length. The first term is exactly the normalized tangential Hessian of \(\phi\); the second costs \(Cz\). Parallel transport changes the tangential trace of \(Q\) by at most \(Cz\). Therefore \(\lvert\mathcal E_Q[\psi](x)-\mathcal E_Q[\phi](x')\rvert\le Cz\). The assumed bound for \(\psi\) proves the result. In particular there is no error proportional to \(z\lvert\mathop{\mathrm{Hess}}\phi\rvert\). ◻

Proof. Take \(u>0\) sufficiently small that all new distance frontiers are in a fixed compact interior neighborhood, avoid \(\Gamma_+\), and satisfy \(c+Cu<c'\) in Lemma 30. The collar of \(\Gamma_-\) ensures that a nearest segment ending on a new distance frontier starts on the interior frontier of \(E\); no extension across the required boundary is needed.

Uniformly approximate \(\operatorname{dist}(\,\cdot\,,E)\) by a smooth function with error below \(u/32\), using a finite coordinate cover, mollification, and a partition of unity. A regular value between \(u/3\) and \(u/2\) gives a smooth relative neighborhood \(U\) with \[E_{u/4}\subset U\subset E_{3u/4}.\] Fill any component of its complement that meets no designated outer face. This may enlarge its interior beyond \(E_{3u/4}\), but only deletes free frontier components; all remaining frontier points still have distance between \(u/4\) and \(3u/4\) from \(E\).

Choose a smooth \(\eta\), equal to one on that compact free frontier, with \(0\le\eta\le1\) and support in \(\{\operatorname{dist}(\,\cdot\,,E)<u\}\). For a sufficiently large finite \(A>0\), set \[Q_A=Q-\tfrac12(c'+A\eta)g .\] The free frontier of \(U\) is strictly negative for \(Q_A\), while the outer faces remain strictly positive. Apply the strict-barrier part of Lemma 26 to each component of the complement of \(U\). Each meets an outer face by filling and has nonempty inner boundary by connectedness of \(M\) and \(U\ne\varnothing\). The resulting smooth frontier \(\Sigma\) encloses \(U\) and satisfies \[ \Theta_Q(\Sigma)=c'+A\eta\ge c'. \tag{95}\]

If \(\Sigma\) enters the support of the modification, let \(z=\min_\Sigma\operatorname{dist}(\,\cdot\,,E)<u\). We have \(z\ge u/4>0\), and \(E_z\) lies on the enclosed side of \(\Sigma\). Indeed, a point at distance less than \(z\) from \(E\) can be joined to \(E\) by a path of length less than \(z\); crossing \(\Sigma\) would give a point of \(\Sigma\) at smaller distance. At a minimizing point, a defining function of \(\Sigma\) is therefore an exterior support of \(E_z\). It has expansion at most \(c+Cz<c'\), contradicting Equation (95). Thus \(\eta=0\) on all of \(\Sigma\), giving the desired expansion. The inclusion \(E_{u/4}\subset U\) makes the enclosure strict. Filling remaining components without an outer face again only deletes whole smooth boundary components. ◻

Proof. Fix a compact set containing the common radial range of all the negative-threshold regions and \(\mathop{\mathrm{supp}}(\mathop{\mathrm{tr}}K)\). On this compact set \(M_d\) has a positive minimum. Restrict \(c_b,c_w\) sufficiently close to zero so that \(\max\{\lvert c_b\rvert,\lvert c_w\rvert\}\lvert\mathop{\mathrm{tr}}K\rvert\) uses less than one quarter of this minimum. The two successive right-continuity choices remain possible by Lemma 27. Fix the regions and their collars.

On each collar take a smooth cutoff \(\kappa(s)\), equal to one near the face and zero near its far end, and choose \(k>0\) such that \(1-ks>0\) throughout that collar. A black contribution \(a_0\kappa(s)(1-ks)\) and a white contribution \(-a_0\kappa(s)(1-ks)\), extended by zero and summed over disjoint collars, give the required \(C\) with \(C_b=a_0\), \(C_w=-a_0\), and \(k_B=a_0k\). The \(C^1\) norm tends to zero with \(a_0\), after the collars have been fixed. Since \[\lvert h+C\rvert \le\max\{\lvert c_b\rvert,\lvert c_w\rvert\}+C_b-C_w ,\] choose \(a_0\) small enough for the first two terms of Equation (98) to be at most \(M_d/2\). They vanish outside a compact set. Finally multiply a fixed smooth positive symbol weight, equal to \(r^{-3-\delta}\) on the end, by a small constant so that \(\rho\le M_d/2\) everywhere. ◻

Proof. Differentiate \(D=1+l^2|\,\mathrm df|^2\), use \(\,\mathrm dl=pl\,\,\mathrm dt\), and substitute \(H^f=l\mathop{\mathrm{Hess}}f/\sqrt D\) and \(w=l\nabla f/\sqrt D\). This gives (113). Differentiating the last formula for \(w\), and using \(1-v=d\), gives \[\nabla_iw_j =H^f_{ij}-H^f(w,\cdot)_i w_j+pd\,(\,\mathrm dt)_i w_j.\] Since \(A_d=I-w\otimes w^\flat\), this is (114). Equation (115) follows from \(u=e^{2t}l\sqrt D\). Taking the trace of (114) and adding \(\langle\nabla\log u,w\rangle\) gives \[\frac{\operatorname{div}(uw)}u =\mathop{\mathrm{tr}}H^f+2(p+1)ax.\] Now \(\mathop{\mathrm{tr}}H^f=F+(1-\chi)q_1-\mathop{\mathrm{tr}}K\), proving (117).

From \(Z=t+\frac18\log D\), Equation (113) yields \[4\nabla Z+K(w,\cdot)=(4+pv)\nabla t+S(w,\cdot).\] Its image under \(A_\chi\) has axial and transverse components \[4dx+\chi aq_1,\qquad (4+pv)y+aM,\] because \(\chi(4+pv)=4d\). On the other hand, \[Q_{ee}=q_1+2pa x,\qquad Q_{e\perp}=M+pa y, \qquad Q_\perp=P.\] The same two components of \(p_Q(w,\cdot)+4A_d\nabla t+wF\) are \(-aP_0+4dx+aF\) and \(a(M+pa y)+4y\). Equation (112) identifies these with the preceding display, proving (116). Smoothness follows either from the original definitions or from \[ A_\chi =I-\frac{p+4}{4+p|w|^2}\,w\otimes w^\flat. \tag{118}\] In particular no choice of \(e\) is needed at \(w=0\). ◻

Proof. The stationary calculation. On a product with coordinate \(s_0\), introduce the stationary Lorentz metric \[\mathbf g=-l^2(\,\mathrm ds_0-\,\mathrm df)^2+\bar g =-l^2\,\mathrm ds_0^2+2l^2\,\mathrm ds_0\,\mathrm df+g.\] The \(s_0\)-slices have induced metric \(g\), shift \(\beta=l^2\nabla f=Uw\), lapse \(U=l\sqrt D\), and unit normal \(n=U^{-1}(\partial_{s_0}-\beta)\). We choose the sign \(k(X,Y)=\mathbf g(\mathbf\nabla_Xn,Y)\). Since the slice metric is independent of \(s_0\), \[k=-\frac{\operatorname{sym}\nabla\beta}{U} =-H^f-p(\,\mathrm dt\otimes w^\flat+w^\flat\otimes\,\mathrm dt).\] Here \(\operatorname{sym}T=(T+T^{\mathsf T})/2\).

For completeness, put \(\vartheta=\mathop{\mathrm{tr}}k\). The contracted Gauss equation and the normal expansion equation, with this sign convention, read \[R_{\mathbf g}=R_g-2\mathop{\mathrm{Ric}}_{\mathbf g}(n,n)-|k|^2+\vartheta^2, \qquad \mathop{\mathrm{Ric}}_{\mathbf g}(n,n)=-n(\vartheta)-|k|^2+U^{-1}\Delta_gU.\] The lapse term in the second formula follows from the acceleration \(\mathbf\nabla_n n=\nabla\log U\); its divergence contribution is \(\Delta_g\log U+|\nabla\log U|^2=U^{-1}\Delta_gU\). Stationarity gives \(Un(\vartheta)=-\beta(\vartheta)\), while \(\operatorname{div}\beta=-U\vartheta\). Combining these equalities gives \[ UR_{\mathbf g} =U(R_g+[k,k]) -2\operatorname{div}\bigl(\nabla U+\vartheta\beta\bigr). \tag{120}\]

In the coordinate \(s=s_0-f\), the same metric is \(-l^2\,\mathrm ds^2+\bar g\). Direct contraction of its warped-product curvature gives \(R_{\mathbf g}=R_{\bar g}-2l^{-1}\bar\Delta l\). The determinant and inverse metric of \(\bar g\) give, for a scalar \(\varphi\), \[\bar\Delta\varphi =D^{-1/2}\operatorname{div}(\sqrt D\,A_d\nabla\varphi).\] Moreover, Equations (113) and (114) imply \[\nabla U-\frac Ul A_d\nabla l=-k(\beta,\cdot).\] For example, after division by \(U\) its left side is \[H^f(w,\cdot)+p\{v\,\,\mathrm dt+w^\flat\langle w,\nabla t\rangle\} =-k(w,\cdot).\] Substituting into (120) therefore yields \[ R_{\bar g}=R_g+[k,k]+\frac2U\operatorname{div}(p_k\beta). \tag{121}\]

The constraint substitution. The symmetric part of \(\nabla\beta\) is \(-Uk\), so (111) gives \[\operatorname{div}(p_K\beta) =8\pi UJ(w)-U[K,k].\] Use \(Q=K-k\), \(p_Q=p_K-p_k\), and \([Q,Q]=[K,K]+[k,k]-2[K,k]\) in (121). The result is \[ R_{\bar g} =16\pi(\mu+J(w))+[Q,Q] -\frac2U\operatorname{div}(Up_Q(w,\cdot)). \tag{122}\]

The conformal calculation. In dimension three, \[\frac12 e^{4t}R_{\hat g} =\frac12R_{\bar g}-4\bar\Delta t-4|\,\mathrm dt|_{A_d}^2.\] Replacing the weight \(U\) in the last flux of (122) by \(u=e^{2t}U\) adds \(2p_Q(w,\nabla t)\). Writing \(\bar\Delta t\) in the preceding divergence form, and using \(\log(u/\sqrt D)=2t+\log l\), gives \[-4\bar\Delta t-4|\,\mathrm dt|_{A_d}^2 =-\frac1u\operatorname{div}(4uA_d\nabla t) +4(p+1)|\,\mathrm dt|_{A_d}^2.\] By (116), the two fluxes obtained so far sum to \(V-uwF\). Its divergence contributes \[\frac1u\operatorname{div}(uwF) =w(F)+F\frac{\operatorname{div}(uw)}u.\] It follows that the quadratic term remaining after the displayed \(-F\mathop{\mathrm{tr}}K\) in (119) is \[ \frac12[Q,Q]+2p_Q(w,\nabla t)+4(p+1)|\,\mathrm dt|_{A_d}^2 +F\{F+(1-\chi)q_1+2(p+1)ax\}. \tag{123}\] The next calculation identifies it with \(\mathcal T\), completing the proof. ◻

Proof. Using \(P_0=F-\chi q_1\), direct contraction gives \[\begin{align*} \frac12[Q,Q] &=\frac12|P^{\mathrm{tf}}|^2-\frac14P_0^2 +|M+pa y|^2-P_0(q_1+2pa x),\\ 2p_Q(w,\nabla t)&=-2aP_0x+2aM\cdot y+2pv|y|^2. \end{align*}\] Consequently (123) is \[\begin{align*} \mathcal T={}&\frac12|P^{\mathrm{tf}}|^2 +|M+(p+1)a y|^2+[4(p+1)-v]|y|^2 +4(p+1)dx^2+2(p+1)a\chi q_1x \\ &\quad+\frac34F^2-\frac12\chi Fq_1 +(\chi-\chi^2/4)q_1^2 . \tag{126}\end{align*}\] Completion of its axial and trace squares is exact because \[\chi-\frac{\chi^2}{4} =\frac{(p+1)\chi^2v}{4d}+\frac{\chi^2}{12} +\frac{4d(9-d)}{3(4+pv)^2}.\] This proves the asserted formula for \(\mathcal T\).

For smoothness, rewrite \((1-\chi)q_1=\mathop{\mathrm{tr}}S-F\) in (123), and \(ax=\langle w,\nabla t\rangle\). Every term is then an invariant smooth expression in \(S,w,\,\mathrm dt,p\), with \(F=\mathop{\mathrm{tr}}_{A_\chi}S\) and \(A_\chi\) given by (118). At \(w=0\) its value is \[\mathcal T=\frac12|S|^2+\frac12(\mathop{\mathrm{tr}}S)^2 +4(p+1)|\,\mathrm dt|^2.\] This also defines the direction-independent value at a critical point of \(f\).

To prove (124), denote the translated variables in the axial, trace, and mixed squares by \[z=x+\frac{\chi a q_1}{4d},\qquad r_F=F-\frac{\chi q_1}{3},\qquad m_F=M+(p+1)a y.\] The square formula bounds \(|P^{\mathrm{tf}}|^2,|m_F|^2\), \((p+1)|y|^2\), \((p+1)dz^2\), and \(r_F^2\) by absolute multiples of \(\mathcal T\). Its last coefficient satisfies \[\frac{4d(9-d)}{3(4+pv)^2} \ge \frac{32d}{3(N+4)^2},\] so \(dq_1^2\le C(N+4)^2\mathcal T\). Since \(\chi\le d\le1\), one has \[dx^2\le 2dz^2+\frac{\chi^2v}{8d}q_1^2 \le2dz^2+\frac18dq_1^2, \qquad F^2\le2r_F^2+\frac29\chi^2q_1^2.\] These estimates control \(x,F\), and \(|P|^2=|P^{\mathrm{tf}}|^2+\frac12(F-\chi q_1)^2\). Finally, \[|M|^2\le2|m_F|^2+2(p+1)^2v|y|^2 \le C(N+1)\mathcal T.\] They prove (124).

If \(\,\mathrm dt=0\), Equation (126) becomes \[\mathcal T=\frac12|P^{\mathrm{tf}}|^2+|M|^2 +\frac34F^2-\frac12\chi Fq_1 +\chi(1-\chi/4)q_1^2.\] Using \(\frac12|\chi Fq_1|\le\frac14F^2+\frac14\chi^2q_1^2\) and \(0<\chi\le1\) proves both bounds in (125). ◻

Proof. Consider the graph of \(f\) in the Riemannian warped product \(g+l^2\,\mathrm ds^2\). Its induced metric is \(\bar g\), its unit normal has horizontal component \(-w\), and its vertical component has length \(\sqrt d\). Write \(T=l^{-1}\partial_s\) for the unit vertical vector. The ambient curvature formulas are \[\begin{align*} R_{\mathrm{amb}}&=R_g-2l^{-1}\Delta_gl,\\ \mathop{\mathrm{Ric}}_{\mathrm{amb}}(X,Y)&=\mathop{\mathrm{Ric}}_g(X,Y)-l^{-1}\mathop{\mathrm{Hess}}l(X,Y),\\ \mathop{\mathrm{Ric}}_{\mathrm{amb}}(T,T)&=-l^{-1}\Delta_gl, \end{align*}\] with zero mixed Ricci components. Hence half the ambient term \(R_{\mathrm{amb}}-2\mathop{\mathrm{Ric}}_{\mathrm{amb}}(n,n)\) in the graph Gauss equation is \[\frac12R_g-v\mathop{\mathrm{Ric}}_g(e,e) -\frac vl\mathop{\mathrm{tr}}_\perp\mathop{\mathrm{Hess}}l.\] At the contact \(\,\mathrm dl=0\); the covariant components of the graph second form are therefore \(H^f\), up to the harmless common sign coming from the choice of graph normal. Tracing with the graph inverse metric \(A_d\) gives \[\frac12\bigl((\mathop{\mathrm{tr}}_{A_d}A)^2-|A|_{A_d}^2\bigr)=\mathcal S(A).\] Indeed the axial square cancels, the mixed contribution is \(-d|A_{e\perp}|^2\), and the transverse contribution is \(\frac14(\mathop{\mathrm{tr}}_\perp A)^2-\frac12|A_\perp^{\mathrm{tf}}|^2\). Finally, at \(\,\mathrm dt=0\) the conformal scalar formula contributes \(-4\mathop{\mathrm{tr}}(A_d\mathop{\mathrm{Hess}}t)\). We have thus proved the exact formula \[\frac12e^{4t}R_{\hat g} =\frac12R_g-v\mathop{\mathrm{Ric}}_g(e,e)+\mathcal S(H^f) -\frac vl\mathop{\mathrm{tr}}_\perp\mathop{\mathrm{Hess}}l-4\mathop{\mathrm{tr}}(A_d\mathop{\mathrm{Hess}}t).\] Both final terms are nonpositive, proving (127). ◻

Proof. Subtracting \(\mathcal S(S)\) from (126), with \(\,\mathrm dt=0\), gives \[\begin{align*} \mathcal T-\mathcal S(S) ={}&|P^{\mathrm{tf}}|^2+(1+d)|M|^2 +\chi(1+d-\chi/2)q_1^2-dFq_1+\frac12F^2. \tag{129}\end{align*}\] The identities \[d\ge\chi,\qquad d-\chi=\frac{pv}{4}\chi,\qquad \frac{d^2}{\chi}=\frac{d(4+pv)}4\le\frac{N+4}{4}\] make its parameter dependence explicit. For the part with cross coefficient \(-\chi\), Young’s Inequality gives \[\frac12F^2-\chi Fq_1+\chi(1+\chi/2)q_1^2 \ge\frac18F^2+\frac56\chi q_1^2.\] The excess cross term satisfies \[|(d-\chi)Fq_1| \le\frac13\chi q_1^2 +\frac34\frac{(d-\chi)^2}{\chi}F^2 \le\frac13\chi q_1^2+\frac3{64}p^2vF^2.\] It follows from (125) that \[ \mathcal T-\mathcal S(S) \ge\frac18\mathcal T-\frac3{64}p^2vF^2. \tag{130}\]

Because \(H^f=S-K\), expansion of the quadratic polynomial \(\mathcal S\) gives \[|\mathcal S(S-K)-\mathcal S(S)| \le C|K|\bigl(|P^{\mathrm{tf}}|+|M|+|F|+d|q_1|\bigr)+C|K|^2.\] Here we used \(\mathop{\mathrm{tr}}P=F-\chi q_1\) and \(\chi\le d\). The estimate for \(d^2/\chi\) above and (125) bound this by \[\frac1{16}\mathcal T+C(1+N)|K|^2.\] Combining with (130) proves the claim. ◻

Proof. The error weights. Since \(\delta+2\gamma<3\), the end decay implies \[|\nabla K|\le C\sqrt{\rho_0\rho_1},\qquad |K|^2\le C\rho_0.\] Young’s Inequality therefore gives \[ a|\nabla K|\le C(\rho_0+v\rho_1). \tag{135}\] Any bounded error supported in \(E\) is bounded by \(C\rho_0\), because \(\rho_0\) has a positive minimum there.

The exact tensor-drift derivative. Put \(H=H^f\), \(k_0=K_{ee}\), \(m_0=K_{e\perp}\), and \(P_H=H|_{e^\perp}\). At the point in question, \[\nabla p=0,\qquad \nabla\log u=H(w,\cdot), \qquad \nabla w=H A_d.\] For \(c=(p+4)/(4+pv)\), Equation (118) reads \[A_\chi K(w,\cdot)=K(w,\cdot)-cK(w,w)w.\] Its exact differentiated form is \[\begin{align*} \frac{\operatorname{div}(uA_\chi K(w,\cdot))}{u} ={}&a\{(\operatorname{div}K)(e) -(1-\chi)(\nabla_eK)(e,e)\}\\ &+\chi(2\chi-1)H_{ee}k_0 +2\chi\langle H_{e\perp},m_0\rangle +P_H:K_\perp-(1-\chi)(\mathop{\mathrm{tr}}P_H)k_0. \tag{136}\end{align*}\] To see the cancellation explicitly, use a normal orthonormal frame at the point. The entries of \(\nabla w=H A_d\) are \[\nabla_e w_e=dH_{ee},\quad \nabla_e w_A=H_{eA},\quad \nabla_A w_e=dH_{Ae},\quad \nabla_A w_B=H_{AB}.\] The coefficient of \(H_{ee}k_0\), after adding \(\langle\nabla\log u,A_\chi K(w,\cdot)\rangle\), is \[d-3cvd-2(\partial_vc)v^2d+v\chi =\chi(2\chi-1).\] The mixed coefficient is \(1+d-2cv+v=2\chi\). The remaining transverse terms are the last two terms of (136); differentiating \(K\) produces its first line. This proves the formula.

In particular every axial Hessian term in this divergence contains a factor \(\chi\). Since \(H=S-K\) and \(\mathop{\mathrm{tr}}P=F-\chi q_1\), the second line of (136) has absolute value at most \[C|K|\bigl(|P^{\mathrm{tf}}|+|M|+|F|+\chi|q_1|\bigr)+C|K|^2 \le\eta\mathcal T+C_\eta|K|^2.\] Here \(\chi|q_1|\le\sqrt{\chi}|q_1|\) and (125) were used. The first line is bounded by \(Ca|\nabla K|\). Equation (135) proves (133), without an \(N\)-dependent constant.

The trace derivative in each stage. At fixed \((x,w,p)\), the height derivatives in (131) are respectively \[1,\qquad 1+\theta\,\partial_hD_0,\qquad 1+(1-\zeta)\partial_hD_0(x,w,h+\zeta b(x)),\] so each is at least one. At \(\,\mathrm dt=0\), the chain rule and \(h=\tau f\) give \[ w(F) =\partial_h\mathfrak F\,\tau a|\,\mathrm df| +(\nabla_x\mathfrak F)(w) +D_w\mathfrak F(\nabla_w w), \qquad \nabla_w w=a(dH_{ee}e+H_{e\perp}). \tag{137}\] There is no \(\partial_p\mathfrak F\) term because \(\nabla p=0\).

The compact corrections have uniformly bounded first derivatives. For the only noncompact correction, use \[\mathop{\mathrm{tr}}_{A_\chi}K=\mathop{\mathrm{tr}}K-cK(w,w),\qquad D_wc[\xi]=2(\partial_vc)\langle w,\xi\rangle.\] The bounds \[|c|\le C(1+N),\qquad |\partial_vc|\le C(1+N)^2\] give, on the stated height interval, \[|\nabla_x\mathfrak F| \le C(1+N)^2\bigl(\mathbf 1_E+|\nabla K|\bigr), \qquad |D_w\mathfrak F| \le C(1+N)^2\bigl(\mathbf 1_E+|K|\bigr).\] The base derivative here also includes the bounded derivative of \(b(x)\) inside \(D_0\). The second term of (137) is consequently bounded below by \(-C(1+N)^2(\rho_0+v\rho_1)\). For its third term, the comparison (125) and \(d^2/\chi\le(N+4)/4\) yield \[\begin{split} &C(1+N)^2(\mathbf 1_E+|K|)\, a\bigl(|H_{e\perp}|+d|H_{ee}|\bigr)\\ &\hspace{12mm}\le \eta\mathcal T+ C_\eta(1+N)^5(\mathbf 1_E+|K|^2). \end{split}\] For example, replace \(H\) by \(S-K\), use weighted Young Inequalities against \(|M|^2+\chi q_1^2\), and absorb the remaining \(K\)-terms into the same right side. Thus one may take \(m=5\), after enlarging the constant. The weight observations at the start of the proof and \(\partial_h\mathfrak F\ge1\) establish (132).

Finally, outside \(E\), all three right sides have the form \[F=h+\zeta\mathop{\mathrm{tr}}_{A_\chi}K\] with \(0\le\zeta\le1\), where \(\zeta=0\) in the first two stages. The eigenvalues of \(A_\chi\) lie in \((0,1]\), so \(|F|\le |h|+C|K|\le Cr^{-\gamma}\) if the stated end height bound holds. On \(E\), \(F\) is uniformly bounded by the compact hypotheses. This proves (134). ◻

Proof. The constant face value gives \(\mathop{\mathrm{Hess}}f|_{TB}=(\partial_\nu f)\mathrm{II}_g\), and hence \(\mathop{\mathrm{tr}}_{TB}H^f=w_\nu H\). Using Equation (116), or taking the normal component of the original flux directly, gives \[\frac{V_\nu}{u} =4d\partial_\nu t+\chi w_\nu q_1 =4d\partial_\nu t+w_\nu(F-P_B)-vH.\] This is the first identity. The normal in \(\bar g\) is \(\sqrt d\,\nu\); its mean curvature is \(\sqrt d\,H\). The latter follows also by subtracting the connection change from the tangential Hessian above. The conformal boundary formula for \(\hat g=e^{4t}\bar g\) yields the second identity. At \(\,\mathrm df=0\) it follows by smooth continuation, or by the same direct computation with \(d=1\). ◻

Proof. Choose a smooth \(j:\mathbb R\to[0,1]\), zero for \(t\le0\) and one for \(t\ge1\). For \(0\le\lambda\le1\), define \[V_\lambda=uA_\chi(4\nabla Z+\lambda K(w,\cdot)),\qquad B_K=A_\chi K(w,\cdot).\] Until the last scaling stage, the inner scalar condition is \[ \frac{(V_\lambda)_\nu}{u} =[1-(1-\lambda)j(t)] [\,\mathcal B_F-(1-\lambda)(B_K)_\nu\,]. \tag{139}\]

We specify the two fixed collar corrections. Choose \(\eta>0\) with \(6\eta<b_+-b_-\). On black collars take a directional cutoff \(\chi_-(w\cdot\nu_s)\), supported where \(w\cdot\nu_s<-1/2\) and equal to one at \(-1\), and a nonincreasing height cutoff \(\eta_-(h)\), equal to one for \(h\le b_-+\eta\) and zero for \(h\ge b_-+2\eta\). On white collars use \(\chi_+\), supported where \(w\cdot\nu_s>1/2\) and equal to one at \(1\), and a nondecreasing \(\eta_+\), zero for \(h\le b_+-2\eta\) and one for \(h\ge b_+-\eta\). Multiply by disjoint smooth collar cutoffs \(\kappa_B\), equal to one on the faces, and set \[D_0(x,w,h)= -A_0\sum_{\text{black collars}}\kappa_B\chi_-\eta_- +A_0\sum_{\text{white collars}}\kappa_B\chi_+\eta_+, \qquad A_0>2\sup_B\lvert H\rvert+1 .\] Thus \(\partial_hD_0\ge0\), \(D_0(x,0,h)=0\); it is nonpositive at low heights \(h\le b_-+\eta\), nonnegative at high heights \(h\ge b_+-\eta\), and zero in every compatible black outward or white inward direction, for all lengths \(0\le\lvert w\rvert\le1\). All its derivatives on bounded height ranges are independent of \(N\).

At the assigned face height, the fully corrected expression \(F=h+C+D_0\), evaluated at the limiting normal directions \(w=\pm\nu\), satisfies \[ F(x,+\nu,b)\ge P_B+H,\qquad F(x,-\nu,b)\le P_B-H . \tag{140}\] These evaluations use \(\lvert w\rvert=1\) and \(\chi=0\) in the smooth direction-free coefficients. Equation (99) gives equality in the compatible direction. In the opposite direction the correction of magnitude \(A_0\) gives the inequality.

Take a smooth compactly supported extension \(b(x)\), constant at its assigned face value on each smaller collar. Put \[D_1(x,w)=A_1\sum_B\kappa_B(x)\,w\cdot\nu_s,\qquad A_1>\sup_B\lvert H\rvert+1 .\] The collars are disjoint. Thus \(D_1(x,0)=0\) and \(D_1(x,\pm\nu)=\pm A_1\) on every face. The family is as follows.

1. For \(0\le\xi\le1\), put \(\lambda=1-\xi\), retain \(F=h+C\) and \(h|_B=b\), and use \(\operatorname{div}V_\lambda=u\Xi\) with Equation (139).

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

3. For \(2\le\xi\le3\), put \(\lambda=0\), \(\zeta=\xi-2\), prescribe \(h|_B=(1-\zeta)b\), and set \[ F=h+(1-\zeta)\bigl[C+D_0(x,w,h+\zeta b(x))\bigr] +\zeta\bigl[\mathop{\mathrm{tr}}_{A_\chi}K+D_1(x,w)\bigr]. \tag{141}\] Keep the same scalar equation and boundary condition.

4. For \(3\le\xi\le4\), retain this trace equation at \(\zeta=1\), and multiply both \(\Xi\) and the right side of Equation (139) by \(s_\xi=4-\xi\).

The prescriptions agree at their endpoints. The height derivatives are \(1\), \(1+\alpha\partial_hD_0\), and \(1+(1-\zeta)\partial_hD_0\), respectively. At a stage-(iii) face, \(h+\zeta b=b\), so its limiting directional value is the convex combination of the earlier corrected value and \(P_B+D_1(\pm\nu)\). Equation (140) therefore persists.

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

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

Choose \(r_0\) outside the supports of \(C,D_0,D_1,b\) and \(\mathop{\mathrm{tr}}K\). The last stage has \(h=0\). In the other stages the tail equation takes the form \[ \frac{l}{\tau\sqrt D}\mathop{\mathrm{tr}}_{A_\chi}\mathop{\mathrm{Hess}}h-h +\theta\,\mathop{\mathrm{tr}}_{A_\chi}K=0,\qquad 0\le\theta\le1, \tag{143}\] where \(\theta=1\) in the first two stages and \(1-\zeta\) in the third. Since \(\mathop{\mathrm{tr}}K=0\), \[\mathop{\mathrm{tr}}_{A_\chi}K=(\chi-1)K(e,e),\qquad 1-\chi=\frac{(p+4)v}{4+pv}.\] For the radial test \(\psi=r^{-\gamma}\), with its radial gradient axis, Equation (88) gives uniformly in \(0<\chi\le1\) \[\mathop{\mathrm{tr}}_{A_\chi}\mathop{\mathrm{Hess}}\psi =\gamma\bigl((\gamma+1)\chi-2\bigr)r^{-2-\gamma} +O(r^{-3-\gamma}) \le-c_\gamma r^{-2-\gamma}\] after increasing \(r_0\).

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

In the part \(r\ge R/2\), evaluate coefficients at a possible comparison contact, so that the given input \(Z\) and the test gradient agree with the solution’s. Put \(z=l\lvert\,\mathrm d\phi\rvert/\tau\), noting \(\lvert\,\mathrm d\phi\rvert\asymp C r^{-1-\gamma}\). If \(z\ge1\), the magnitude of the negative Hessian term is at least \(c r^{-1}\), whereas the adverse tensor term is at most \(C_Kr^{-2}\). If \(z\le1\), the former is at least \(c(l/\tau)C r^{-2-\gamma}\), while the latter is at most \[C_K(N+4)(l/\tau)^2 C^2 r^{-4-2\gamma}.\] Their ratio is at most \(C(N+4)z/r\le C(N+4)/r\). Thus a lower bound \(R_{\mathrm{ht}}(N)\), enlarged after \(r_0,C\) have been fixed, makes \(\phi\) a strict upper barrier. Its negative is a lower barrier by the same absolute estimates. Height monotonicity allows comparison with the actual input \(Z\). This proves the first bound in Equation (142). Tangential derivatives vanish on \(S_R\); comparison at their common outer value gives the stated normal derivative bound.

Finally, \(Z=0\) on \(S_R\), and the map \(t\mapsto Z(t,\,\mathrm df)\) is increasing. Its value at \(t=-\epsilon\) is negative whenever \[\ell^2\lvert\,\mathrm df\rvert^2<e^{8\epsilon}-1.\] The boundary slope bound makes this true once \[C^2\ell^2\tau^{-2}R^{-2-2\gamma}<e^{8\epsilon}-1 .\] This is a permissible further lower bound on \(R\) at fixed \(N,\epsilon\). ◻

Proof. At the contact \(\,\mathrm dt=0\), \(\mathop{\mathrm{Hess}}t\ge0\), \(p=N\), and \(\,\mathrm dp=0\). Equation (119) and Lemma 36 give \[\frac{\operatorname{div}V}{u} \ge 8\pi\mu-\tfrac12R_g+8\pi J(w)+v\mathop{\mathrm{Ric}}(e,e) +\mathcal T-\mathcal S(H^f)+w(F)-F\mathop{\mathrm{tr}}K .\] The constraint gives exactly \[8\pi\mu-\tfrac12R_g =\tfrac12\bigl((\mathop{\mathrm{tr}}K)^2-\lvert K\rvert^2\bigr).\] Lemma 37 retains \(\tfrac1{16}\mathcal T\), with an error \(C_N(\lvert K\rvert^2+vF^2)\). The first-derivative estimates for the current trace and the omitted flux, from Lemma 38, retain \(\tau a\sigma\) and each cost an arbitrarily small fixed fraction of \(\mathcal T\). They have polynomial remainders.

For clarity, every coefficient remainder has the following global control, with constants independent of \(N,R,\xi\): \[\begin{align*} \lvert K\rvert^2&\le C\rho_0,& vF^2&\le Cv\rho_1,& \lvert F\mathop{\mathrm{tr}}K\rvert&\le C\rho_0,\\ v\lvert\mathop{\mathrm{Ric}}\rvert&\le Cv\rho_1,& a(\lvert\nabla K\rvert+\lvert J\rvert) &\le C(\rho_0+v\rho_1). \end{align*}\] The first bound uses \(\delta<1\); the second follows from Lemma 42 and the tail expression \(F=h+\zeta\mathop{\mathrm{tr}}_{A_\chi}K\). The trace term has compact support. The curvature bound follows from \(\mathop{\mathrm{Ric}}=O(r^{-3})\). Finally, on the end \(\lvert\nabla K\rvert+\lvert J\rvert=O(r^{-3})\) and \[\lvert\nabla K\rvert+\lvert J\rvert \le C\sqrt{\rho_0\rho_1}, \qquad \delta+2\gamma<3 .\] Weighted Young’s Inequality gives the last displayed estimate. All compactly supported coefficient derivatives are bounded by a constant times \(\rho_0\), which has positive minimum on their fixed compact support.

Choose the two small derivative fractions so that their sum is less than \(1/32\), and subtract \((1-\lambda)\operatorname{div}(uA_\chi K(w,\cdot))/u\). This proves Equation (144), after renaming the positive universal coefficient. No fixed-\(N\) regularity constant enters this ledger. ◻

Proof. Enlarge \(R_{\mathrm{floor}}\) to include the outer-boundary requirement of Lemma 42. Thus an outer contact is impossible.

At an inner contact in one of the first three stages, \(j(t)=0\). Equation (139) cancels the omitted drift on both sides and gives \(V_\nu/u=\mathcal B_F\). Lemma 40 implies \[\partial_\nu t=-\tfrac14(N_B+H)<0 .\] This contradicts the nonnegative inward derivative at a boundary minimum.

At an interior contact, Equation (144) has right side at least \(2\delta_0\mathcal T+\tau a\sigma-C_NW_N\). Choose the polynomial \(\Pi_N\) so large that \[(\Pi_N-C_N)W_N\ge 2\rho+\rho_0 .\] This is possible since \(W_N\ge\rho_0\) and \(\rho/\rho_0\) is bounded. At the floor \(m_0=1\); comparison with the prescribed source would then give \[0\ge \delta_0\mathcal T+(1-\delta_0)\tau a\sigma +(\Pi_N-C_N)W_N-\rho \ge\rho+\rho_0>0,\] a contradiction.

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

Proof. For the full Hessian array the Fourier multiplier \(D^2\Delta^{-1}\) has \(L^2\) operator norm one, since \(\sum_{i,j}\xi_i^2\xi_j^2/|\xi|^4=1\). Its finite \(L^p\) bounds and interpolation give constants \(C_p\) with \(C_p\longrightarrow1\) as \(p\downarrow2\). In Euclidean coordinates the Frobenius norm of \(I-A\) is exactly \(1-b\). A sufficiently small perturbation of the metric therefore gives \(|I-A|\le1-c_0/2\). Choose \(p_0\in(2,3)\) so that \(C_{p_0}(1-c_0/2)<1\). For a compactly supported \(v\), \[\|D^2v\|_{p_0} \le C_{p_0}\|A^{ij}D_{ij}v\|_{p_0} +C_{p_0}(1-c_0/2)\|D^2v\|_{p_0}.\] Absorb the last term. Christoffel symbols add a bounded first-order term. Applying this inequality to cutoffs on nested balls and using \(\|Dv\|_p\le\varepsilon\|D^2v\|_p+C_\varepsilon\|v\|_p\) gives Equation (147); the usual nested-radius absorption removes the outer Hessian term. The same argument at \(p=2\) proves the last assertion. All coefficient comparisons are pointwise, so no derivatives of \(b\) or of its axis have been used. ◻

Proof. Rotate so that \(a=\partial_3\) and put \(v=Y_3\). Differentiating Equation (152) in the sense of distributions gives \[\partial_1^2v+\partial_2^2v+\partial_3(b\partial_3v)=0.\] This is a scalar uniformly elliptic divergence equation, even when \(b\) depends on all three variables. The scalar De Giorgi–Nash estimate gives a \(C^{0,\alpha_1}\) bound for \(v\) on a smaller ball, controlled by \(\|v\|_2\). Caccioppoli’s Inequality, applied after subtracting \(v(x)\), then gives for balls contained in that smaller region \[ \int_{B_r(x)}|Dv|^2\le Cr^{1+2\alpha_1} \|Y\|_{W^{2,2}(B_{3/4})}^2. \tag{154}\] Now \(\Delta Y=F=(1-b)v_3\). By Cauchy–Schwarz, \[ \int_{B_r(x)}|F|\le Cr^{2+\alpha_1} \|Y\|_{W^{2,2}(B_{3/4})}. \tag{155}\] Take a cutoff equal to one on \(B_{1/3}\), supported where these estimates hold, and let \(V\) be the Newton potential of the cut-off \(F\). Its gradient kernel is bounded by \(C|x|^{-2}\) and the derivative of that kernel by \(C|x|^{-3}\). If \(d=|x-y|\), the contribution to \(|DV(x)-DV(y)|\) from distances at most \(2d\) is bounded, by dyadic annuli and Equation (155), by \(C\sum_{2^{-j}\le 2d}(2^{-j})^{\alpha_1}\le Cd^{\alpha_1}\). At larger distances the bound is \(Cd\sum_{2^{-j}>d}(2^{-j})^{\alpha_1-1}\le Cd^{\alpha_1}\). The same bounds hold for the cut-off source, with an additional harmless large-scale term. Thus \(DV\) is \(C^{0,\alpha_1}\). The Newton kernel is locally in \(L^2\) in dimension three, so \(\|V\|_\infty\) is controlled by \(\|F\|_2\). Finally \(Y-V\) is harmonic on \(B_{1/3}\); its ordinary interior derivative estimates complete Equation (153). ◻

Proof. Otherwise take errors \(\varepsilon_j\to0\) and \(k_j\uparrow1\) with neither alternative, and subtract \(z_j(0)\) in interior balls. Lemma 46, together with the gradient bound, bounds \(z_j\) in \(W^{2,p_0}\) on smaller balls. Thus \(Ds_j\) is bounded in \(L^{p_0}\). Failure of Equation (156) gives \(\inf_{B_{1/6}}s_j\to0\). The weak Harnack Inequality with bounded divergence and scalar errors implies \(s_j\to0\) in measure on \(B_{1/3}\); one may use Theorem 5.2 and Corollary 5.4 of (Trudinger 1973), with the radii \(1/6\), \(1/3\), and \(5/6\).

We need more than this convergence in measure. Test Equation (151), after dropping its nonnegative Hessian term, with \(\eta^2(h-s_j)_+\), where \(\eta\) is supported in \(B_{1/3}\). Ellipticity and Young’s Inequality give, for fixed \(h>0\), \[ \int_{\{s_j<h\}}\eta^2|Ds_j|^2 \le Ch^2\int|D\eta|^2+o_j(1). \tag{158}\] On \(\{s_j\ge h\}\), convergence in measure and the uniform \(L^2\) bound for \(Ds_j\) give convergence to zero of its \(L^1\) norm. On the complementary set, Equation (158) and Cauchy–Schwarz bound the \(L^1\) norm by \(Ch+o_j(1)\). First let \(j\to\infty\), then \(h\downarrow0\). We obtain \(Ds_j\to0\) in \(L^1(B_{1/4})\). Testing the full Equation (151) with a nonnegative cutoff now yields \[\int_{B_{1/8}}|\mathop{\mathrm{Hess}}z_j|^2\longrightarrow0.\] Since \(g_j\to I\) and \(Dg_j\to0\), the same is true for \(D^2z_j\). Poincaré’s Inequality and the uniform Lipschitz bound show, after passage to a subsequence, uniform convergence on smaller balls to an affine function. Its slope has length one, because \(s_j\to0\) in measure and the gradients converge strongly in \(L^2\). This contradicts failure of Equation (157). At a boundary center the functions are odd, so their affine limit has zero constant and tangential terms. Increasing \(k_*\) if necessary ensures \(k_*>r_*\). ◻

Proof. Set \(Y=(z-L_0)/h\) and hold the original coefficients of the \(z\)-equation fixed when applying Lemma 46 to \(Y\). The covariant Hessian of \(L_0\) is \(-\Gamma DL_0\), so that lemma gives \(\|Y\|_{W^{2,p_0}(B_{7/8})}\le C\). Put \(a=DL_0/|DL_0|\) and \(A_0=I-(1-b)a\otimes a\). The difference between the original axis projector and \(a\otimes a\) obeys \[|e\otimes e-a\otimes a| \le C\min\{1,h|DY|\}+Ch\eta_0.\] This remains valid at a zero gradient, since then \(h|DY|=|DL_0|\); metric changes have been included in the second term. For \(q=2p_0/(p_0-2)>p_0\) and \(\vartheta=(p_0-2)/2>0\), \[\|\min\{1,h|DY|\}\|_{L^q(B_{3/4})} \le C h^{p_0/q}=Ch^\vartheta.\] Consequently Hölder’s Inequality, with \(1/2=1/q+1/p_0\), gives \[ A_0^{ij}D_{ij}Y=F_*,\qquad \|F_*\|_{L^2(B_{3/4})}\le C(h^\vartheta+\eta_0). \tag{160}\] Here \(F_*\) includes the matrix error times \(D^2Y\), the term \(G/h\), and \(A^{ij}\Gamma_{ij}^k(D_kL_0/h+D_kY)\). Thus neither \(DY\) nor any derivative of \(b\) has been assumed pointwise bounded.

There is a small zero-Dirichlet correction for Equation (160). On the convex ball \(B_{3/4}\), \[ \|D^2u\|_2\le\|\Delta u\|_2 \quad(u\in W^{2,2}\cap W^{1,2}_0). \tag{161}\] For smooth functions this follows by integrating twice by parts: \(\int((\Delta u)^2-|D^2u|^2)= \int_{\partial B_{3/4}}H_{\partial B_{3/4}}(\partial_\nu u)^2\ge0\), with the outward convex convention. Density proves the stated version. The map \[u\longmapsto\Delta_D^{-1}\bigl(F_*+(1-b)u_{aa}\bigr)\] is therefore a contraction, in the Hessian norm, with factor \(1-c_0\). Its fixed point \(u\) satisfies \(A_0:D^2u=F_*\) and \(\|u\|_{W^{2,2}}\le C\|F_*\|_2\). In dimension three the embedding \(W^{2,2}\hookrightarrow L^\infty\) gives \[\|u\|_\infty\le C(h^\vartheta+\eta_0).\] The function \(Y_0=Y-u\) solves the fixed-axis equation and has a uniform \(W^{2,2}\) norm. Lemma 47 applies.

Choose \(\alpha_2<\alpha_1\), then \(r_2\) so small that the Taylor error \(Cr_2^{1+\alpha_1}\) is at most \(\tfrac14r_2^{1+\alpha_2}\). Next choose \(\eta_0\), and finally \(h_0\), so that \(C(h_0^\vartheta+\eta_0)\) is at most the same quantity. With \(L_1=L_0+h(Y_0(0)+DY_0(0)\cdot x)\), these bounds prove Equation (159). If the center is on the boundary plane, \(a\) is its normal, \(b\) is even, and \(Y\) and \(F_*\) are odd. Uniqueness of the correction on the symmetric ball makes \(u\) odd. Hence \(Y_0\) is odd and its Taylor polynomial has only a normal linear term, as required. ◻

Proof of Theorem 45. Choose the constants of Lemma 49 first, and then decrease \(h_*>0\) so that \(h_*\le h_0\) and \(Ch_*\sum_{j\ge0}r_2^{j\alpha_2}<1/4\). Use this \(h_*\) in Lemma 48. Set \(M=L+Q\); the case \(M=0\) is immediate. Normalize coordinates at each center as above. Choose a uniform initial radius \(\rho_0\) so small that, after dividing \(z\) by \(M\) and scaling this radius to one, all errors required by Lemma 48 are at most \(\varepsilon_*\), and the forcing and first-order metric errors are also at most \(h_*\eta_0\).

As long as the drop alternative holds, the successive physical radii and gradient amplitudes are \[ \rho_j=\rho_0r_*^j, \qquad M_j=Mk_*^j, \qquad z_j(y)=\frac{z(x+\rho_jy)-z(x)}{\rho_j M_j}. \tag{162}\] The forcing scales by \(\rho_j/M_j\), which decreases because \(r_*<k_*\). The geometric errors decrease with \(\rho_j\). In a reflected patch the plane density is recomputed from Equation (150) with the newly normalized gradient; it is again \(O(\rho_j)\). Thus the class is preserved at every drop, including at centers close to, but not on, the plane.

If drops continue indefinitely, \(Dz(x)=0\), and the constant affine approximations on these balls have errors at most \(C\rho_jM_j\). Otherwise let \(m\) be the first index where the affine alternative holds. At physical radius \(S=\rho_0r_*^{m+1}\), divide by \(SM_m\) to obtain affine approximation with error \(h_*\) on the unit ball and slope in \([1/2,2]\). Apply Lemma 49 repeatedly. At step \(j\) the radius is \(r_2^j\) and the flatness is \(h_j=h_*r_2^{j\alpha_2}\). The gradient amplitude \(M_m\) is kept fixed during this iteration; the spatial dilation alone is changed. The forcing and metric errors decrease as \(r_2^j\), so their ratios to \(h_j\) decrease as \(r_2^{j(1-\alpha_2)}\). The total change of the normalized slopes is less than \(1/4\), preserving the required slope range. At boundary centers every affine approximation is odd.

Choose \[0<\alpha\le\min\left\{\alpha_2, \frac{\log k_*}{\log r_*}\right\}.\] The drop scales give \(M_j\le M(\rho_j/\rho_0)^\alpha\). After the first affine scale, \[\inf_{L\ \mathrm{affine}} \|z-L\|_{L^\infty(B_r(x))} \le C M_mS(r/S)^{1+\alpha_2} \le CM\rho_0^{-\alpha}r^{1+\alpha}\qquad(r\le S).\] Before that scale the constant approximations have the same last bound. Passing from the discrete radii to arbitrary radii loses only a fixed factor. Every center therefore has affine approximations with error \(CMr^{1+\alpha}\). Comparing consecutive approximations on the same ball bounds their slope differences by \(CMr^\alpha\); comparing overlapping balls gives the same bound between the limiting slopes at two centers. Smoothness identifies each limiting slope with \(Dz\). This proves the first estimate in Equation (146), also for the reflected function.

For the second estimate take an affine approximation \(L_{x,2r}\) on \(B_{2r}(x)\) and apply the exponent-two form of Lemma 46 to \(z-L_{x,2r}\). Its source is \(G\) plus \(A^{ij}\Gamma_{ij}^kD_kL_{x,2r}\), and its slope is bounded by \(CM\). Scaling that estimate back gives \[\|D^2z\|_{L^2(B_r(x))} \le C\left(r^{-2}\|z-L_{x,2r}\|_{L^2(B_{2r}(x))} +Mr^{3/2}\right) \le CM r^{1/2+\alpha}.\] The connection term converting \(D^2z\) to \(\mathop{\mathrm{Hess}}z\) has squared integral at most \(CM^2r^3\). This proves the claimed Hessian bound. Restricting the reflected estimate to the original side proves the boundary assertion. ◻

Proof. Write \(\mathcal L=-\mathop{\mathrm{div}}(aD)\), choose \(k\ge\max\{1,2C_0/\lambda\}\), and set \(V_s=e^{skZ}\) for \(s=\pm1\). The exact distributional chain rule is \[ \mathcal LV_s =\mathop{\mathrm{div}}(skV_sB)-skV_se -k^2V_s\bigl(aDZ\cdot DZ+B\cdot DZ\bigr). \tag{165}\] It is valid for the present weak equation: the exponential is locally Lipschitz and can be approximated uniformly and in \(W^{1,2}\) in the test, so the measure term converges as well. The inequality \(|B\cdot DZ|\le(\lambda/2)|DZ|^2+B_0^2/(2\lambda)\) and Equation (164) imply \[ \mathcal LV_s\le\mathop{\mathrm{div}}F_s+\nu, \qquad F_s=skV_sB, \qquad \nu=e^{kM}\left(kC_0+\frac{k^2B_0^2}{2\lambda}\right)dx +ke^{kM}\mu. \tag{166}\] In particular \(F_s\) is bounded and \(\nu\) is a positive measure with the same admissible growth, in addition to a bounded density.

The correction estimate. We record the small correction estimate on \(B_R\): \[ \mathcal Lw_s=\mathop{\mathrm{div}}F_s+\nu, \qquad w_s\in W^{1,2}_0(B_R), \qquad \|w_s\|_\infty\le\varepsilon_R=CR^\beta. \tag{167}\] For the bounded vector term, testing the Dirichlet equation with \((w-t)_+\) gives, for \(p>3\), \[\int|D(w-t)_+|^2 \le C\|F_s\|_p^2|\{w>t\}|^{1-2/p}.\] Sobolev’s Inequality then yields \[|\{w>h\}| \le C\|F_s\|_p^6(h-t)^{-6} |\{w>t\}|^{3-6/p},\qquad h>t.\] The exponent on the right is greater than one. Iteration, also for \(-w\), gives \(\|w\|_\infty\le CR^{1-3/p}\|F_s\|_p\le CR\).

For the measure term the positive scalar Dirichlet Green function satisfies \(G_{B_R}(x,y)\le C|x-y|^{-1}\). This is the scalar measurable-coefficient Green bound: extend \(a\) elliptically to \(\mathbb R^3\), use the fundamental-solution bound in Theorem 3.1 of (Hofmann and Kim 2007), and compare the Dirichlet Green function by the weak maximum principle. The scalar De Giorgi–Nash estimate supplies the local Hölder hypothesis of that theorem. Dyadic annuli and Equation (164) give \[\sup_x\int_{B_R}G_{B_R}(x,y)\,d\nu(y) \le C(R^2+R^\beta).\] This also constructs the needed weak correction without assuming measure solvability. Approximate the restricted positive measure by smooth positive measures, preserving the ball-growth bound. The resulting potentials have the displayed uniform supremum bound, and their energy is bounded by their supremum times the total mass. A weak \(W^{1,2}_0\) limit solves the measure equation and retains the bound. Adding the vector and measure corrections proves Equation (167).

Oscillation decay. Let \(M_R=\sup_{B_R}Z\), \(m_R=\inf_{B_R}Z\), and set \[ H_+=e^{kM_R}-e^{kZ}+w_++\varepsilon_R, \qquad H_-=e^{-km_R}-e^{-kZ}+w_-+\varepsilon_R. \tag{168}\] These functions are nonnegative weak supersolutions of \(\mathcal LH_\pm\ge0\). Each exponential deficit is bounded above and below by positive constants, depending on \(kM\), times \(M_R-Z\) or \(Z-m_R\), respectively. One of the two midpoint sets \[\{Z\le(M_R+m_R)/2\}\cap B_{R/3},\qquad \{Z\ge(M_R+m_R)/2\}\cap B_{R/3}\] has at least half the volume of \(B_{R/3}\). Apply weak Harnack to the corresponding \(H_\pm\). In Theorem 5.2 of (Trudinger 1973), take the inner radius \(R/6\): its averaging ball is \(B_{R/3}\) and its required nonnegativity ball is \(B_{5R/6}\). Thus the infimum on \(B_{R/6}\) is at least \(c(M_R-m_R)\). Removing the correction in Equation (168) costs at most \(2\varepsilon_R\). It follows that either the upper endpoint decreases or the lower endpoint increases by \(c(M_R-m_R)-C\varepsilon_R\), whence \[ \operatorname{osc}_{B_{R/6}}Z \le(1-c)\operatorname{osc}_{B_R}Z+CR^\beta. \tag{169}\] Iteration gives any sufficiently small positive exponent below both \(\beta\) and \(-\log(1-c)/\log6\).

Boundary reflection. Flatten a boundary face and put \(S=\operatorname{diag}(1,1,-1)\). For an even extension of \(Z\), extend \(a^-=Sa^+S\) and \(B^-=SB^+\). If \(q\) is the third component of the original total flux on \(x_3=0\), the reflected equation has the additional density \(2q\,\delta_{\{x_3=0\}}\). It is bounded by the prescribed total conormal bound, after the coordinate volume factors are included. For a constant Dirichlet value, extend \(Z\) minus that value oddly and use \(a^-=Sa^+S\), \(B^-=-SB^+\) and the odd bulk source. The normal flux is then continuous, with no plane term. The reflected matrix remains measurable and uniformly elliptic. A bounded plane density has ball mass \(O(r^2)\), which is allowed in Equation (164) because \(\beta\le1\). This proves the boundary assertion. ◻

Proof. Theorem 45 gives a uniform \(C^{1,\alpha}\) bound for \(f\) and \[ \int_{B_r(x)}|D^2f|^2\le Cr^{1+2\alpha}. \tag{172}\] Coordinate connection terms do not affect this estimate. Apply Lemma 50 to Equation (170) with \(\mu=C|D^2f|^2dx\) and \(\beta=\min\{2\alpha,1\}\). At a flux boundary include the bounded reflected plane density. We obtain a uniform Hölder bound for \(Z\). Smooth composition in the bounded variables now makes the leading matrix and source of the \(f\)-equation Hölder. The interior and constant-Dirichlet Schauder estimates give \(f\in C^{2,\theta}\) for some uniform \(\theta>0\).

Expand Equation (170) in nondivergence form. Its leading matrix is uniformly elliptic and \(C^{0,\theta}\). Differentiating its coefficients produces terms at most quadratic in \(DZ\), because \(D^2f\) is now bounded. Thus its right-hand side is bounded by \(C(1+|DZ|^2)\). Fix \(p>3\). Linear local \(W^{2,p}\) estimates, with either constant Dirichlet or the normal data in Equation (171), give on nested patches \(U_x\Subset V_x\Subset V_x^+\) \[ \|Z\|_{W^{2,p}(U_x)} \le C\left(1+\|DZ\|_{L^{2p}(V_x)}^2 +\|Z\|_{W^{1,p}(V_x^+)}\right). \tag{173}\] The patch size here is chosen from the already established Hölder modulus of the leading matrix. The normal boundary operator is uniformly oblique; the local a priori estimate is, for example, a smooth-boundary specialization of Theorem 2.3 of (Dong and Li 2022), followed by localization. Only its estimate is used, not solvability for a pure oblique problem. The boundary norm is controlled by \[ \|H(x,Z,f,Df)\|_{W^{1-1/p,p}(\partial V_x)} \le C\left(1+\|Z\|_{W^{1,p}(V_x^+)}\right). \tag{174}\] Indeed, extend the smooth expression for \(H\) through the collar, differentiate it there, use the already bounded first two derivatives of \(f\), and apply the trace theorem. This explains the last term in Equation (173) without assuming a gradient bound for \(Z\).

We give the absorption of the quadratic gradient term. On balls or half-balls of radius \(h\), the scaled interpolation estimate is \[ \|DZ\|_{L^{2p}(B_h)}^2 \le C\operatorname{osc}_{B_{2h}}Z\, \|D^2Z\|_{L^p(B_{2h})} +Ch^{3/p-2}(\operatorname{osc}_{B_{2h}}Z)^2. \tag{175}\] One way to obtain it is to subtract a constant, integrate \(\int\eta^{2p}|DZ|^{2p}\) by parts against \(Z\), and use Hölder and Young Inequalities on the terms containing \(D^2Z\) and \(D\eta\). At the boundary use a bounded extension operator on a half-ball, bounded also in \(L^\infty\); its lower derivative terms are absorbed by the same interpolation. Scaling gives exactly the second power of \(h\) displayed in Equation (175).

Write \([Z]_{C^{0,\theta}}\le H_0\). Cover \(V_x\) by radius-\(h\) patches of bounded overlap, with their doubles in \(V_x^+\). Take the \(\ell^p\) sum of Equation (175). There are \(O(h^{-3})\) patches, so \[ \|DZ\|_{L^{2p}(V_x)}^2 \le CH_0h^\theta\|D^2Z\|_{L^p(V_x^+)} +CH_0^2h^{2\theta-2}. \tag{176}\] Also, for every \(\eta>0\), \(\|Z\|_{W^{1,p}(V_x^+)} \le\eta\|D^2Z\|_{L^p(V_x^{++})}+C_\eta\) on a further uniformly enlarged patch. Such enlarged patches are covered by a bounded number of the \(U_y\). On a fixed smooth finite-domain solution let \(T=\sup_y\|Z\|_{W^{2,p}(U_y)}\), which is finite before making any uniform assertion. Equations (173) and (176) imply \[ T\le C(H_0h^\theta+\eta)T+C(h,\eta). \tag{177}\] The radius for the linear estimate was fixed first. Now choose the auxiliary covering radius \(h\), and then \(\eta\), so that the coefficient of \(T\) is less than \(1/2\). This proves a uniform \(W^{2,p}\) bound, and hence \(Z\in C^{1,1-3/p}\).

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

Proof. Consider any sequence \(N_i\to\infty\), \(R_i\to\infty\) of the indicated solutions. Lemma 42 gives, with fixed constants, \[ |h_i|\le C_0, \qquad |h_i|\le C(r^{-\gamma}-R_i^{-\gamma}) \quad\hbox{on the end},\qquad \frac34<\gamma<1. \tag{179}\] Extend each \(h_i\) across every white face by the constant \(b_+\), and extend the fixed metric and tensor smoothly into those short collars. These extensions of \(h_i\) are continuous; no uniform boundary modulus is required. Let \[\underline h(x)= \lim_{j\to\infty}\inf\{h_i(y):i\ge j, \operatorname{dist}(x,y)<j^{-1}\}.\] This is a bounded lower semicontinuous function satisfying the limiting end bound.

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

At these contacts \(df_i=\tau_i^{-1}d\phi\), whence \[d_i\le\frac{\tau_i^2}{\ell_i^2|d\phi(x_i)|^2}\longrightarrow0, \qquad a_i\longrightarrow1,\qquad\chi_i\longrightarrow0.\] The positive Hessian matrix in the trace equation gives \[\mathop{\mathrm{tr}}_{A_{\chi_i}}K+ \frac{l_i}{\tau_i\sqrt{D_i}} \mathop{\mathrm{tr}}_{A_{\chi_i}}\mathop{\mathrm{Hess}}_g\phi \le F_i(x_i).\] Here \(l_i/(\tau_i\sqrt{D_i})=a_i/|d\phi|\). The bound on the collar addition and \(C\le C_b\) therefore give \[ \frac{\mathop{\mathrm{tr}}_{(d\phi)^\perp}\mathop{\mathrm{Hess}}_g\phi}{|d\phi|} +\mathop{\mathrm{tr}}_{(d\phi)^\perp}K\le k'+C_b. \tag{180}\] The left side is the expansion of the level set, oriented toward increasing \(\phi\). Removing the fourth-order localization proves the same assertion for a non-strict lower test.

To pass from functions to all supports of a closed set, fix \(b_-<k<k'\) and put \(E_k=\{\underline h\le k\}\). Define \[q_j(s)=\bigl(1+\exp[-j^2(s-k-j^{-1})]\bigr)^{-1}, \qquad v_j=q_j(\underline h).\] The lower relaxed limit of \(v_j\) is zero on \(E_k\) and one off it. At an equality point this follows from \(q_j(k)=(1+e^j)^{-1}\to0\); off \(E_k\), lower semicontinuity gives a neighborhood whose heights exceed \(k\) by a fixed positive amount. Let \(\psi\) be a nonzero-gradient lower test of this limit at a zero value. Strict localization gives lower tests of \(v_j\) at points \(y_j\to x\) with \(v_j(y_j)\to0\). Their original heights are below \(k'\) for large \(j\), since \(q_j(k')\to1\). For each fixed such \(j\) the smooth inverse \(q_j^{-1}\) turns this into a lower test of \(\underline h\) with finite nonzero gradient. Apply (180) to that inverse test, taking the solution-sequence limit with \(j\) fixed, and only then let \(j\to\infty\). An increasing change of defining function preserves the level expansion: the additional Hessian term is a multiple of \(d\psi\otimes d\psi\) and has zero tangential trace. Consequently every such test \(\psi\) satisfies (180).

A smooth local exterior support of \(E_k\) has a defining function that is zero at contact and nonpositive on \(E_k\). Shrinking and scaling the neighborhood so its positive values are below one makes it exactly a lower test of the preceding zero–one function. Thus every exterior support has expansion at most \(k'+C_b\). In particular \(E_k\) cannot touch a white face: the reversed face is an exterior support with expansion \[-H+\mathop{\mathrm{tr}}_TK=-c_w>0,\] whereas \(k'+C_b<0\) after \(k'\) is chosen sufficiently close to \(b_-\). Equivalently, in a signed white collar with \(s\ge0\) in \(\Omega\), a small multiple of \(-s\) is the offending lower test. This rules out boundary layers without assuming that the Dirichlet values persist in the relaxed limit. The end bound makes \(E_k\) compact.

Adjoin the entire black region and fill complementary components which contain no designated outer barrier. An exterior support at a black frontier point also supports the smooth black region, so its expansion is at most \(c_b<k'+C_b\). At other frontier points the preceding argument applies. Filling produces no frontier point outside the old closed set; every support of the larger filled set is still a support of that old set. The support bound is therefore preserved. The original required inner collars are covered by the black region, and the outer barriers remain outside. Lemma 31 places the resulting set in \(\mathcal B_{c''}\) for any sufficiently close threshold \(c''>k'+C_b\).

For a fixed compact \(Q\) disjoint from \(\mathcal B_{c_b}\), right continuity gives \(\delta>0\) with \(\mathcal B_{c_b+\delta}\cap Q=\varnothing\). Choose \[b_-<k<k',\qquad k'+C_b<c''<c_b+\delta.\] Then \(E_k\cap Q=\varnothing\) for every sequence considered above. Failure of a uniform lower margin would give violating points in \(Q\), a convergent subsequence, and a relaxed limit value at most \(k\) there. This contradiction proves the lower assertion. Apply the argument to \((-h,-K,-C)\) for the upper assertion. In the white problem, filling retains all reversed black outer faces. If the white region is empty, a nonempty violating sublevel would still provide the seed for Lemma 31; right continuity at the empty region rules out such seeds at arbitrarily close higher thresholds. Multiple components and bounded complementary pockets cause no change.

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

Proof. Let \(s\) be a fixed outward leaf parameter in a black collar, and let \(n_s=\nabla s/|ds|\). For \(h_0=b_-+\psi(s)\) with \(\psi'>0\), evaluate the operator at a comparison contact, where the gradients of the test and solution agree. The floor yields \[ d\le C\frac{\tau^2}{\ell^2(\psi')^2},\qquad 1-a\le d,\qquad N\chi\ge c_1d\quad(N\ge4). \tag{182}\] The trace on the test is exactly \[ P_s+aH_s+\chi K(n_s,n_s) +a\chi\left( \frac{\mathop{\mathrm{Hess}}s(n_s,n_s)}{|ds|} +|ds|\frac{\psi''}{\psi'}\right), \tag{183}\] where \(P_s=\mathop{\mathrm{tr}}_{T_s}K\). In particular it is the leaf expansion \(H_s+P_s\), an \(O(d)\) error, and the displayed logarithmic-slope term. This calculation differentiates neither the input \(Z\) nor an unknown coefficient. For a negative slope the Hessian terms have \(a\) replaced by \(-a\).

Since \(H_s+P_s\ge c_b\) and \(C=C_b-k_Bs\), choose small positive slopes with \(\psi(0)=0\) and \(\psi(s)\le k_Bs/2\). Lemma 53 orders the test below the solution at the outer end of a fixed short collar. The collar addition vanishes in this compatible direction. The residual of the trace equation is at least \[\frac{k_Bs}{2}-Cd+a\chi|ds|\frac{\psi''}{\psi'}.\] Choose a nonnegative smooth \(\beta_N\) equal to \(AN\) on \([0,N^{-1}]\), supported in \([0,2N^{-1}]\), and with integral at most \(2A\). Set \[\psi'(s)=m\exp\!\left(\int_0^s\beta_N(q)\,dq\right).\] A fixed large \(A\) makes the final term dominate \(Cd\) on the first interval, by (182). Beyond that interval \(k_Bs/2\) dominates \(Cd\le C\ell\) for large \(N\). A sufficiently small fixed \(m>0\) ensures all height and outer-edge requirements, since the slope multiplier is at most \(e^{2A}\). This is a strict lower barrier with its \(h\)-slope in a fixed positive interval.

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

In stage three start at its prescribed face value and use a large positive-slope upper barrier and a large negative-slope lower barrier, both with logarithmic absolute-slope derivative \(-\beta_N\). The directional inequalities in Proposition 41 give the initial residual signs. Formula (141) evaluates its shifted height at the original face value, so these signs persist along the interpolation. At the unshifted face height all errors are \(O(s+d)\); changing the height supplies a margin at least the absolute height change. Large fixed slopes absorb the \(O(s)\) errors and order the outer edge, and the same logarithmic correction handles the initial \(O(d)\) errors. This proves the absolute upper slope bound without using separation in stage three. Finally combine the own-color lower collar bounds with Lemma 53 on the rest of a fixed compact set. The opposite bound on each collar follows from that lemma as well. A finite cover proves the asserted height range. ◻

Proof. Set \(W=l\bar\nabla f=l\nabla f/D=\sqrt d\,a e\). Its \(\bar g\)-norm is at most one, and \(uW=Lw\). For a covector \(\zeta\), \[|\zeta(W)|\le|\zeta|_{A_d} \le(1+N/4)^{1/2}|\zeta|_{A_\chi}.\] Direct differentiation gives \[\begin{align*} u^{-1}\operatorname{div}_g(uW) &=\sqrt d\bigl(\mathop{\mathrm{tr}}_{A_d}H^f+(dp+p+2)a\,\partial_e t\bigr),\\ D^{-1/2}\operatorname{div}_g(\sqrt D\,W) &=\sqrt d\bigl(\mathop{\mathrm{tr}}_{A_d}H^f+dp\,a\,\partial_e t\bigr). \end{align*}\] Both are bounded by \(P_N(1+\sqrt{\mathcal T})\) using Lemma 35. In particular their axial Hessian coefficient is \(d^{3/2}\), and their axial \(dt\) coefficient is at most \((2N+2)\sqrt d\); the other Hessian terms are transverse. No factor \(\ell^{-1}\) occurs. At a black or white face, respectively, \(uW_\nu=La\) or \(-La\), and \(a\ge1/2\) by (181). Apply the divergence theorem with the corresponding sign, the collar cutoff, and \(\phi\). Replacing \(u\) by \(\sqrt D\) gives the second inequality. The formulas also hold at zero gradients by continuity. ◻

Proof. On the penalty band \(-\epsilon\le t\le-\epsilon+N^{-1}\), \(l\asymp\ell\) and \(D=e^{8(Z-t)}\). Thus \(a\sigma\) tends uniformly to infinity as \(Z\to\infty\) on that band. The penalty weights are bounded on the entire exterior, so a sufficiently large \(M_0(N)\) ensures \[ \Xi\ge\delta_0\mathcal T+\rho +\tfrac12\delta_0\tau a\sigma \quad\hbox{on }\{Z>M_0\}. \tag{188}\] Outside the penalty band this is immediate. Testing the first stage divergence equation by \(j_0(Z)\) has no outer boundary term, since the outer value of \(Z\) is zero. The drift term is bounded by \[u j_0'|\langle dZ,K(w,\cdot)\rangle_{A_\chi}| \le 2u j_0'|dZ|_{A_\chi}^2+C u j_0'|K|^2.\] The latter integral is bounded by \(C_N\): the floor and the bounded \(Z\) strip bound \(u\) there, while \(K=O(r^{-2})\) has finite squared integral on the three-dimensional AF end.

At either color of face, the matching signs give \(-H+w_\nu(F-P_B)=-(1-a)H\). The omitted-drift term has normal component bounded by \(C\chi\le Cd\), and the scalar boundary multiplier lies in \([0,1]\). Hence, uniformly in the first stage, \[ |(V_\xi)_\nu|\le Cud=CL\sqrt d \le C\frac\tau\ell L. \tag{189}\] Integration and (188) give the left side of (186), up to fixed positive coefficients, bounded by \(C_N+C(\tau/\ell)\int_B L j_0\). Use (184). On the fixed collars the positive minimum of \(\rho\) implies \[1+\sqrt{\mathcal T}\le C(\rho+\delta_0\mathcal T).\] Thus the terms with \(j_0\) absorb for \(N\) large, by (178). The derivative term is bounded by \[u j_0'|dZ|_{A_\chi}^2 +C(P_N\tau/\ell)^2u j_0'.\] Its second summand has a bounded integral on the compact collar and the bounded \(Z\) strip. This proves (186).

Now constants may have arbitrary fixed-\(N\) dependence. The measures \(dV_\Gamma=\sqrt{1+\sigma^2}\,dV_g\) and \(\sqrt D\,dV_g\) are comparable because \(\ell\le l\le e\). Moreover \[e^{2Z}=e^{2t}D^{1/4},\qquad D^{1/4}\le C_N(1+\tau a\sigma),\qquad u=le^{2t}\sqrt D.\] The middle bound follows by separating \(l\sigma\le1\) and \(l\sigma>1\), on which \(a\ge1/\sqrt2\). On a fixed compact \(Q\), its positive minimum of \(\rho\) and (186) control the high-level part of (187). On \(Z\le M_0+1\) the floor bounds \(D\), \(\sigma\), and \(e^{2Z}\), so the remaining graph volume is bounded by a constant times the fixed base volume. This is a local weighted estimate, with no global graph-volume assumption. ◻

Proof. Put \(d_o=(1+\sigma^2)^{-1}\). The matrices \(A_{d_o},A_d,A_\chi\) are uniformly comparable at fixed \(N\), without an upper bound on \(Z\). Here \(|\nabla_\Gamma\omega|=|d\omega|_{A_{d_o}}\). Extend a fixed base patch to a compact smooth enlargement and isometrically embed it in Euclidean space by Nash’s Theorem (Nash 1956). Its product with the line immerses the ordinary graph isometrically. The base embedding contributes a bounded amount to the graph mean curvature vector. Its mean curvature in the product is \[H_\Gamma= \frac{\sqrt D}{l\sqrt{1+\sigma^2}} \bigl(\mathop{\mathrm{tr}}_{e^\perp}H^f+d_o H^f(e,e)\bigr).\] The prefactor is bounded by \(C_N\) and \(d_o/\sqrt d\le C_N\). Coercivity therefore gives \(|H_\Gamma|\le C_N(1+\sqrt{\mathcal T})\). The Michael–Simon Inequality (Michael and Simon 1973), in the form with an ordinary boundary integral stated explicitly by Brendle (Brendle 2021, Theorem 1), says \[\left(\int_\Gamma\phi^{3/2}\,dV_\Gamma\right)^{2/3} \le C\left[ \int_\Gamma(|\nabla_\Gamma\phi|+(1+|H_\Gamma|)\phi)\,dV_\Gamma +\int_{\partial\Gamma}\phi\,dA\right]\] for nonnegative supported \(\phi\). The graph over a compact smooth patch is embedded by the fixed Nash embedding. Enlarging the ambient Euclidean space if needed puts its codimension at least two, as in Brendle’s Theorem. That theorem applies first to \(\phi+\varepsilon>0\); letting \(\varepsilon\downarrow0\) and using \(\sqrt{a^2+b^2}\le a+b\) gives the displayed form. The boundary form follows also by extending each smooth immersed graph slightly across its boundary and letting a cutoff collar shrink: its derivative integral tends to the boundary integral and its added curvature integral tends to zero. This is done for each smooth graph before taking the uniform estimate. On an inner face \(f\) is constant, so \(dA=dA_g\). Equation (185) bounds that boundary integral by \(C_N\int_\Gamma[(1+\sqrt{\mathcal T})\phi+ |\nabla_\Gamma\phi|]\,dV_\Gamma\). Taking \(\phi=|\omega|^4\) and using Cauchy–Schwarz proves \[ \|\omega\|_{L^6(dV_\Gamma)}^2 \le C_N\int_\Gamma \bigl(|\nabla_\Gamma\omega|^2+ (1+\mathcal T)\omega^2\bigr)\,dV_\Gamma. \tag{190}\]

For a compact base cutoff \(\eta\) test the scalar equation by \(q=\eta^2e^{2kZ}/L\). Its differential is \[dq=L^{-1}e^{2kZ} [2\eta\,d\eta+\eta^2(2k\,dZ-(p+2)\,dt)].\] The principal exponential term contributes \(8k\eta^2e^{2kZ}|dZ|_{A_\chi}^2\) against the measure \(\sqrt D\,dV_g\). Coercivity gives \[4(p+2)|\langle dt,dZ\rangle_{A_\chi}| \le\tfrac14\delta_0\mathcal T+C_N|dZ|_{A_\chi}^2.\] For \(k\ge k_*(N)\) the second term absorbs. The drift terms obey \[\begin{align*} 2k|\langle K(w,\cdot),dZ\rangle_{A_\chi}| &\le k|dZ|_{A_\chi}^2+Ck|K|^2,\\ (p+2)|\langle K(w,\cdot),dt\rangle_{A_\chi}| &\le\tfrac14\delta_0\mathcal T+C_N|K|^2. \end{align*}\] The cutoff terms have the same bounds with a remainder \(C_Nk e^{2kZ}(\eta^2+|d\eta|_g^2)\). The penalty is bounded on the entire exterior and contributes to this remainder. Before the last inequality of (189), the boundary term is at most \(C\eta^2e^{2kZ}\sqrt d\le C\eta^2e^{2kZ}\). Using (185) with \(\phi=\eta^2e^{2kZ}\) bounds it by \[C_N\int_\Gamma e^{2kZ} \bigl[\eta^2(1+\sqrt{\mathcal T}+k|\nabla_\Gamma Z|) +\eta|d\eta|_g\bigr]\,dV_\Gamma.\] Young’s Inequality absorbs the \(\sqrt{\mathcal T}\) term and the term linear in the gradient; the latter costs \(C_Nk\) times the weighted volume. After comparison of the graph measures we have \[ \int_\Gamma e^{2kZ}\eta^2 (\mathcal T+k|\nabla_\Gamma Z|^2)\,dV_\Gamma \le C_N k\int_\Gamma e^{2kZ} (\eta^2+|d\eta|_g^2)\,dV_\Gamma, \quad k\ge k_*(N). \tag{191}\] Its constants are independent of \(k\) and \(R\).

For nested base patches \(U_r\subset U_{r'}\), combine (190) with (191) applied to \(\eta e^{kZ}\): \[\|e^Z\|_{L^{6k}(\Gamma|U_r)} \le\bigl[C_Nk^2(1+(r'-r)^{-2})\bigr]^{1/(2k)} \|e^Z\|_{L^{2k}(\Gamma|U_{r'})}.\] Iterate with \(k_j=3^j k_*\) and geometrically decreasing gaps. The sums \(\sum k_j^{-1}\) and \(\sum j/k_j\) converge, giving \[S(r):=\sup_{U_r}e^Z \le C_N(r'-r)^{-A_N} \|e^Z\|_{L^{2k_*}(\Gamma|U_{r'})}.\] Increase \(k_*\) to at least two. The local \(L^2\) estimate (187) and interpolation give \(S(r)\le C_N(r'-r)^{-A_N}S(r')^{1-1/k_*}\). For any \(\vartheta>0\), Young’s Inequality turns this into \[S(r)\le\vartheta S(r')+ C_{N,\vartheta}(r'-r)^{-A_Nk_*}.\] Use radii increasing geometrically to a fixed outer patch and choose \(\vartheta<2^{-A_Nk_*-1}\). The resulting series converges. The final remainder tends to zero because each individual smooth solution has a finite supremum on that closed outer patch. The bound obtained is independent of this individual supremum. A finite cover proves the lemma, including at inner faces. ◻

Proof. It remains to extend the first-stage compact bound to the tail; the tensor \(K\) need not be compactly supported. Everywhere, not only at a critical point of \(t\), differentiation gives \[ \nabla w=H^f A_d+p d\,dt\otimes w, \qquad d\log u=(p+2+pv)\,dt+H^f(w,\cdot). \tag{192}\] Also \[A_\chi=I-\frac{4+p}{4+pv}w\otimes w, \qquad \partial_p\frac{4+p}{4+pv}=\frac{4d}{(4+pv)^2}.\] In differentiating \(A_\chi\), the derivatives of \(v\) have the form \(2a d\,H^f(\cdot,e)+2p d a^2dt\). Thus every axial \(H^f(e,e)\) or axial \(dt\) term in \(u^{-1}\operatorname{div}_g(uA_\chi K(w,\cdot))\) has a factor \(d\) or \(\chi\). More explicitly its absolute value is bounded by \[P_N|K|\bigl( |H^f_{TT}|+|H^f_{eT}|+d|H^f_{ee}| +|dt_T|+d|dt_e|\bigr) +P_Na|\nabla K|.\] For the \(d\log u\) term this follows from \(A_\chi K(w,\cdot)(e)=\chi aK(e,e)\); for the other terms it follows directly from (192) and the displayed derivative of the scalar coefficient. Coercivity and Young’s Inequality imply \[ \left|u^{-1}\operatorname{div}_g(uA_\chi K(w,\cdot))\right| \le\tfrac12\delta_0\mathcal T +C_N(|K|^2+a|\nabla K|). \tag{193}\] Since \(K=O(r^{-2})\), \(\nabla K=O(r^{-3})\), and \(\tau a\sigma=\tau a^2\sqrt D/l\ge(\tau/e)a^2\), the second term on the right is, sufficiently far out at fixed \(N\), at most \[\tfrac14\rho+\tfrac14\delta_0\tau a\sigma.\] Indeed the cross term is absorbed by a fixed fraction of \(\tau a^2\) with remainder \(C_Nr^{-6}\), while \(r^{-4}=o(r^{-3-\delta})\) because \(\delta<1\). At an interior maximum of \(Z\) above \(M_0\), the principal divergence contributes \(4uA_\chi:\mathop{\mathrm{Hess}}Z\le0\). Equations (188) and (193) contradict the scalar equation. The compact upper bound and outer value zero therefore give a global first-stage bound.

In stages two and three the drift is zero and the source is positive above a fixed high level, so an interior maximum there is impossible. At an inner face \[Z\le t+\tfrac18\log(1+e^2C_1^2/\tau^2).\] Consequently high boundary \(Z\) implies \(t>1\), where the boundary multiplier makes the conormal derivative zero. The boundary point principle on the high superlevel excludes that maximum. These maximum principles apply to each smooth solution with its positive definite coefficients, so they require no prior uniform ellipticity bound. Stage four has \(f=0\) and \(t=Z\). Its positive scales have the same high-level argument; at scale zero its homogeneous mixed problem gives \(Z=0\). This is uniform as the scale tends to zero.

Finally \(D=e^{8(Z-t)}\) and \(l\ge\ell\) give \[|df|^2=l^{-2}(e^{8(Z-t)}-1) \le\ell^{-2}(e^{8(C_N+\epsilon)}-1).\] The height bound was already proved. The finite set of compact separation and collar choices, followed by the polynomial absorption, fixes \(N_0\). Enlarge \(R_{\min}(N)\) to contain all fixed patches and the end thresholds. The preceding arguments apply to floor-contact solutions as well. ◻

Proof. On the bounded variable range \(l,u,D\) are bounded above and away from zero and \(\chi\ge c_N>0\). The normalized trace equation is \[A_\chi:\mathop{\mathrm{Hess}}_g f=\frac{\sqrt D}{l} (F_\xi-\mathop{\mathrm{tr}}_{A_\chi}K).\] Its right side is bounded. Theorem 45 gives \(Df\) Hölder and \[\int_{B_r}|D^2f|^2\,dx\le C_Nr^{1+\beta_N} \quad\hbox{for some }\beta_N>0.\] The scalar principal coefficient in coordinates is \(4\sqrt{\det g}\,u A_\chi^{ij}\); the drift is bounded. By the smooth implicit change \(t=t(x,Z,Df)\), \(Dt=t_ZDZ+t_{Df}:D^2f\) plus controlled coordinate terms. The source, including every homotopy modification, is therefore bounded in absolute value by \(C_N(1+|DZ|^2+|D^2f|^2)\). The prescribed inner flux is bounded and the outer value of \(Z\) is zero. Reflection contributes at most a bounded plane density, which has mass \(O(r^2)\). Lemma 50 now gives a Hölder estimate for \(Z\).

All smooth bounded-range coefficients of the trace equation are then Hölder, so its Dirichlet Schauder estimate gives \(f\in C^{2,\theta}\). At a face \(Df\) is normal, and the scalar condition solves for \[\partial_\nu Z=G_\xi(x,Z,f,Df)\] with \(G_\xi\) smooth on the bounded range: \(A_\chi\nu=\chi\nu\). The expanded scalar equation has Hölder leading matrix and right side bounded by \(C_N(1+|DZ|^2)\). Proposition 51 applies. Its interpolation after subtracting a constant uses small Hölder oscillation of \(Z\) to absorb the quadratic gradient term in local \(W^{2,p}\) estimates. The normal datum’s trace norm is controlled by a lower-order \(W^{1,p}\) norm; the derivatives of \(f\) entering that datum are already bounded. Subsequent Schauder estimates give every finite order. The data have uniform smooth end-patch bounds, and the large outer spheres have uniform boundary-chart constants. The local supremum-of-patch-norm argument in Proposition 51 therefore makes all these estimates independent of \(R\). ◻

Proof. Take a smooth extension of \(M\) and a distance collar radius smaller than a normal injectivity radius and half the separation of distinct boundary components. At a face use the barriers given by its value plus or minus \[\psi(r)=k^{-1}\log(1+Br),\qquad \psi''=-k(\psi')^2.\] At slopes \(q=\psi'\ge1\), the upper barrier’s axial contribution is at most \(-ckq/2\), whereas its tangential curvature and source cost at most \(C(1+q)\). Choose \(k\) large, then a collar radius \(r_0\) sufficiently small, and finally \(B\) large with \(Br_0\ge1\) and \(\psi(r_0)>2\operatorname{osc}f\). Then \(q\ge1/(2kr_0)\) throughout the collar. The lower barrier has the reverse sign. It suffices to check their residuals at possible comparison contacts, where their heights are in the fixed solution range; height monotonicity gives comparison. These barriers exclude a positive maximum of \[f(x)-f(y)-\psi(\operatorname{dist}(x,y)) \quad\hbox{for }0<\operatorname{dist}(x,y)<r_0\] with one endpoint on \(\partial M\). Their value at \(r_0\) excludes the other edge of this two-point domain.

At a positive interior maximum, the endpoint gradients have common length \(q=\psi'(r)\) and are parallel transports along the short joining geodesic. Simultaneous endpoint variations in parallel transverse directions, using the second variation of distance, give \[\mathop{\mathrm{tr}}_{e_x^\perp}\mathop{\mathrm{Hess}}f(x)- \mathop{\mathrm{tr}}_{e_y^\perp}\mathop{\mathrm{Hess}}f(y)\le Cqr.\] The transverse eigenvalues of both operators are exactly one. Separate axial variations give \(f_{e_xe_x}(x)\le\psi''(r)\) and \(-f_{e_ye_y}(y)\le\psi''(r)\). Therefore \[-2C(1+q)\le Q(x)-Q(y) \le Cqr+(b_x+b_y)\psi''(r) \le Cqr-\frac{2ckq^2}{1+q},\] which contradicts the choices of \(k\) and \(r_0\). No derivative or continuity modulus of \(b_x-b_y\) was used. Interchanging the points and then letting their distance tend to zero yields the global Lipschitz bound \(|df|\le\psi'(0)\).

Interpolate the normalized equation by \(A_s=(1-s)I+sA_b\) and \(Q_s=(1-s)c_*f+sQ\). The axial lower bound, source growth, constant barriers, and strict height sign persist. The preceding gradient estimate is uniform in \(s\), so the equations are uniformly elliptic. Theorem 45 first gives \(C^{1,\theta}\), then Schauder gives \(C^{2,\theta}\). On this range \(Df\) and the prescribed \(Z\) are Lipschitz; the undifferentiated coefficients therefore have bounded \(C^{b'}\) norms. Schauder upgrades to \(C^{2,b'}\) and one further differentiation gives \(C^{3,b'}\). The linearization has positive definite principal matrix and negative zeroth-order coefficient \(-\partial_fQ_s\). The Dirichlet maximum principle and linear elliptic theory give its inverse. The implicit function theorem supplies openness of continuation from \(\Delta_g f=c_*f\); the uniform estimates and compactness supply closedness. Comparison at a maximum of the difference proves uniqueness, because the gradients agree and the height source is strictly increasing. The implicit function theorem, or compactness and uniqueness, gives the asserted dependence and uniform bounds. ◻

Proof. The implicit definition (102) is smooth and strictly increasing in \(t\) at fixed \(df\). The height estimates use only the trace equation, including its strict height sign, so they hold without a floor. For bounded \(Z\) and \(\sigma\to\infty\), necessarily \(t\to-\infty\), \(p=N\), and \(l=e^{Nt}\). The exact relation is \[e^{8Z}=e^{8t}+\sigma^2e^{(2N+8)t}.\] Uniformly on bounded \(Z\) ranges, \[ t=\frac{4Z-\log\sigma}{N+4}+o(1),\qquad \chi\asymp\sigma^{-8/(N+4)},\qquad \frac{\sqrt D}{l}\asymp\sigma. \tag{195}\] Since \(N\ge4\), the axial factor is at least \(c/(1+\sigma)\). After division by \(l/\sqrt D\) the source is \[Q=\frac{\sqrt D}{l}(F_\xi-\mathop{\mathrm{tr}}_{A_\chi}K), \qquad |Q|\le C(1+\sigma).\] All collar terms are bounded on the height range, and \(Q_f=(\sqrt D/l)\tau(F_\xi)_h>0\). The principal matrix is independent of \(f\) itself. The normalized interpolation to \(\Delta f=\tau f\) retains these conditions and the constant barriers. Apply Proposition 60. ◻

Proof. The fixed set in \(\overline{\mathcal O}\) is compact by the joint compactness and continuity, and it lies in \(\mathcal O\) by hypothesis. For each parameter, surround its compact fixed-point set by a bounded open neighborhood whose closure is contained in all sufficiently nearby sections. All nearby fixed points of \(\overline{\mathcal O}\) lie in this neighborhood: otherwise a sequence outside it would converge to a fixed point at the original parameter outside the neighborhood. Excision identifies the section degree with the degree on this one neighborhood, and ordinary homotopy invariance makes it constant locally. At a parameter without fixed points the same compactness argument makes all nearby degrees zero. A finite covering of the parameter interval proves the assertion. These are the compact-perturbation degree and excision properties of Leray–Schauder (Leray and Schauder 1934). ◻

Proof. Choose \(N_0\ge\max\{4,\lceil2/\epsilon\rceil\}\) large enough for Proposition 44 and all the preceding estimates. Enlarge \(R_{\min}(N)\) to tend to infinity and to enforce the outer height and gradient barriers. Explicitly the outer slope has the bound \(\sigma\le C_{\mathrm{out}}\tau^{-1}R^{-1-\gamma}\). It suffices also to require \[ R\ge\left( \frac{2C_{\mathrm{out}}\ell} {\tau\sqrt{e^{8\epsilon}-1}}\right)^{1/(1+\gamma)}. \tag{196}\] At outer \(Z=0\) the implicit expression for \(Z\) evaluated at \(t=-\epsilon\) is then strictly negative, so its monotonicity excludes an outer floor contact. The choices may depend arbitrarily on fixed \(N\); there is no restriction requiring a polynomial outer radius.

Fix \(N,R\) and let \[X=\{Z\in C^{1,b'}(\overline{\Omega_R}): Z|_{\partial_{\mathrm{out}}\Omega_R}=0\}.\] For an input \(Z\), solve \(f=S_\xi(Z)\) by Corollary 61 and compute all auxiliary variables and source terms from this pair. Denote the frozen drift summand by \(D_\xi\), the full frozen bulk source by \(E_\xi\), and the prescribed inner flux by \(q_\xi\). Define \(H_\xi(Z)=\widetilde Z\) by \[ \begin{cases} \operatorname{div}_g(4uA_\chi\nabla\widetilde Z+D_\xi) =E_\xi&\text{in }\Omega_R,\\ (4uA_\chi\nabla\widetilde Z+D_\xi)_\nu=q_\xi&\text{on }B,\\ \widetilde Z=0&\text{on the outer face}. \end{cases} \tag{197}\] Only the displayed principal gradient uses the output. On bounded input sets, \(4uA_\chi\) is positive definite and \(C^{1,b'}\), the drift is \(C^{1,b'}\), the bulk source is \(C^{0,b'}\), and the boundary datum is \(C^{1,b'}\). In particular \(Dt\) involves \(DZ,D^2f\), and no second derivative of the input \(Z\). The nonempty outer Dirichlet face gives the Poincaré Inequality, so the mixed linear problem is uniquely solvable by coercivity. The inward-normal convention changes only the sign of its boundary functional. Linear regularity gives \(\widetilde Z\in C^{2,b'}\). The compact embedding into \(X\) and continuous dependence make \(H\) a continuous compact homotopy on bounded sets, including its concatenation points. Its fixed points satisfy the original equations and bootstrap to smoothness.

After the trace solve set \[\mathcal O=\{(\xi,Z): \min_{\overline{\Omega_R}}t_\xi[Z]>-\epsilon, \|Z\|_{C^{1,b'}}<M\}.\] This is relatively open because the trace solve and implicit change are continuous. Choose \(M\) larger than the fixed-point bound supplied by Propositions 58 and 59, using their higher estimates if necessary to reach the chosen exponent \(b'\). A fixed point in \(\overline{\mathcal O}\) is smooth and has \(t\ge-\epsilon\), so these estimates exclude its norm boundary. Equation (196) and Proposition 44 exclude its floor boundary. Thus Lemma 62, with the parameter interval rescaled, applies. Notice that arbitrary inputs need not satisfy the floor, and the map need not preserve \(\mathcal O_\xi\).

At \(\xi=4\) the unique trace solution is \(f=0\), the drift vanishes, and both bulk source and inner flux in (197) are zero for every input. Its output is identically zero. The zero input has \(t=0> -\epsilon\) and lies inside the norm cutoff. Hence the final degree is one, and so is the physical degree. A physical fixed point exists in \(\mathcal O_0\). The asserted radius-independent local bounds are Proposition 59. ◻

Proof. Apply Proposition 63 to any \(R_i\to\infty\). Uniform classical estimates and a diagonal subsequence give smooth convergence on each compact subset, including the fixed inner faces. The equations and boundary conditions pass to the limit, as does \(t\ge-\epsilon\). We prove the end bounds already on the truncations so that their normalization does not escape to infinity.

Where \(C=0\) and \(\mathop{\mathrm{tr}}_gK=0\), the trace equation is \[ A_\chi:\mathop{\mathrm{Hess}}_g f+B_f\cdot df-c_f f=0, \qquad c_f=\tau\frac{\sqrt D}{l},\qquad B_f=-\frac{4+p}{4+pv}\frac l{\sqrt D}K(\nabla f,\cdot). \tag{200}\] Indeed \(\mathop{\mathrm{tr}}_{A_\chi}K=-(1-\chi)K(e,e)\) and \[\frac{\sqrt D}{l}(1-\chi)K(e,e) =\frac{4+p}{4+pv}\frac l{\sqrt D}K(\nabla f,\nabla f).\] At fixed \(N\) the equation has uniformly elliptic principal coefficient, bounded drift, and \(c_f\ge\tau/e>0\). For a sufficiently small \(c_N>0\), the positive function \(\exp[-c_N(r-r_0)]\) is a supersolution of its negative-principal operator beyond a fixed large \(r_0\): the positive height term dominates the \(O(c_N^2)\) radial and \(O(c_N/r)\) tangential Hessian terms and the bounded drift term. Multiply it to dominate the inner sphere data; it also dominates the zero outer data. Comparison for both signs gives exponential decay uniformly in the truncation. The already uniform smooth coefficient bounds, followed by the homogeneous linear estimates for (200), give exponential decay of every fixed derivative. The corresponding zero-Dirichlet estimates apply near the outer spheres. Thus \(Z-t\) and graph errors are exponentially small with derivatives.

Using tracefreeness of \(K\) on the tail, the quadratic form is now \[\mathcal T=\tfrac12|K|^2+4(p(Z)+1)|dZ|^2+O(e^{-c_Nr}),\] and \(A_\chi=I+O(e^{-c_Nr})\), \(u=L(Z)+O(e^{-c_Nr})\). The drift \(uA_\chi K(w,\cdot)\) is exponentially small as well. After decreasing \(c_N\) to absorb polynomial factors, the scalar equation becomes \[ \begin{aligned} \Delta_g Z+b_N(Z)|dZ|^2 &=\tfrac14\rho-\tfrac14\Pi_N m_0(Z)\rho_0 +\tfrac18\delta_0|K|^2+O(e^{-c_Nr}),\\ b_N(s)&=p(s)+2-\delta_0(p(s)+1). \end{aligned} \tag{201}\] All fixed-order local derivatives of the exponential error are controlled. The right side is \(O(r^{-3-\delta})\) on the bounded solution range, even before the penalty disappears. Define the increasing smooth function \[\Phi_N(0)=0,\qquad \Phi_N'(s)=\exp\!\left(\int_0^s b_N(v)\,dv\right).\] Its derivative is bounded above and below on that range, and \(\Delta_g\Phi_N(Z)=O(r^{-3-\delta})\). For \(0<\delta'<\delta<1\), \[-\Delta_g(r^{-1}-r^{-1-\delta'}) =\delta'(1+\delta')r^{-3-\delta'}+O(r^{-4}) \ge c r^{-3-\delta'}\] at large radius. Comparison of both signs, with zero outer values, gives \(|\Phi_N(Z)|\le C_Nr^{-1}\) uniformly in truncation, and hence \(Z=O(r^{-1})\).

In the limit, sufficiently far out \(t> -\epsilon+N^{-1}\), so the penalty vanishes. Rescale comparable-radius annuli to unit size. The equation for \(\Phi_N(Z)\) has a bounded rescaled source and uniform smooth metric coefficients. Local \(W^{2,p}\) estimates first give the scaled gradient bound. Its source is a smooth function of \(Z\) times the symbol weights \(\rho\), \(|K|^2\), plus the controlled exponential errors. Schauder estimates then give two symbol derivatives, proving (198). Since \(L(0)=1\), \(u=1+O(r^{-1})\), and the graph and drift errors are exponential, the first assertion of (199) follows. The divergence \(u\Xi\) is integrable: its polynomial tail terms are \(O(r^{-4})\) and \(O(r^{-3-\delta})\), and its remaining terms are exponential. The divergence theorem on annuli proves the flux limit. ◻

Proof. At the physical endpoint \(F=h+C\) and \(w(h)=\tau a\sigma\). The scalar identity (119) therefore gives \[\begin{align*} \tfrac12e^{4t}R_{\widehat g} ={}&8\pi(\mu+J(w))+(1-\delta_0)\mathcal T +(1-\delta_0)\tau a\sigma\\ &+w(C)-(h+C)\mathop{\mathrm{tr}}K-\rho +m_0(t)\Pi_N(\rho_0+v\rho_1). \end{align*}\] The physical height range from Lemma 54, \(|w|\le1\), and the preparation inequality for \(C,\rho\) make this nonnegative. That preparation is needed only on the compact support of \(\mathop{\mathrm{tr}}K\) and \(dC\); on the rest of the end it reduces to \(\rho\le8\pi(\mu-|J|)\). The boundary identity and prescribed flux give \(H+4\partial_\nu t=-N_B\). Since \(f\) is face-constant, \[H_{\widehat g}=e^{-2t}\sqrt d\,(H+4\partial_\nu t) =-e^{-2t}\sqrt d\,N_B<0.\] The pointwise inequality \(\widehat g\ge e^{-4\epsilon}g\) proves completeness, with the compact smooth boundary included.

Proposition 64 gives the metric decay. The conformal formula and exponential graph error give \[R_{\widehat g}=e^{-4t} (R_g-8\Delta_g t-8|dt|^2)+O(e^{-c_Nr}).\] Equation (201) gives \(\Delta_g t=O(r^{-3-\delta})\), proving the required scalar falloff. Exponential graph errors have zero ADM contribution. The conformal ADM formula and \(t=O_2(r^{-1})\) give \[E_{\widehat g}=E_g-\frac1{2\pi} \lim_{r\to\infty}\int_{S_r}\partial_{\nu_g}t\,dA_g.\] Replacing the normal and area by their Euclidean versions costs \(o(1)\), and the \(O(r^{-3})\) error in (199) has vanishing sphere integral. This gives the stated normalization and sign. ◻

Proof. The inward normal on \(B\) gives \[ \mathcal F_N=\int_\Omega u\Xi\,dV_g+\int_B V_\nu\,dA_g. \tag{202}\] Both integrals are finite by the end estimates. At a physical face, \(V_\nu/u=-(1-a)H-N_Bd\ge-Cd\). The signed slopes yield \(\sqrt d\le C\tau/\ell\), so its negative integral is at most \(C(\tau/\ell)\int_B L\). Apply (184) with a constant test and a fixed collar cutoff. On those collars \(\rho\) has a fixed positive minimum, and therefore this loss is at most \[P_N\frac\tau\ell \int_\Omega u(\rho+\delta_0\mathcal T)\,dV_g.\] For \(N\) large it absorbs in one quarter of those positive terms.

On the penalty band \(l,L\asymp\ell\), with constants independent of \(N\), and the exact definitions give \[u\le C(\ell+\ell^2\sigma),\qquad uv\le C\ell^2\sigma,\qquad ua\sigma=e^{2t}l^2\sigma^2\ge c\ell^2\sigma^2.\] Thus, increasing only a polynomial constant, \[\begin{align*} u m_0\Pi_N(\rho_0+v\rho_1) &\le P_N[\ell\rho_0+\ell^2\sigma(\rho_0+\rho_1)]\\ &\le\tfrac14\delta_0\tau ua\sigma +P_N\left[\ell\rho_0+ \frac{\ell^2}\tau(\rho_0^2+\rho_1^2)\right]. \end{align*}\] The three remaining weight integrals are finite. Their end orders are \(r^{-3-\delta}\), \(r^{-6-2\delta}\), and \(r^{-4\gamma}\), respectively, with \(4\gamma>3\). After inserting both estimates in (202), positive fractions of all three bulk terms remain. Dropping them proves the bound. Finally \(\ell^2/\tau=\tau/\ell=\ell^{1/2}\), which decays faster than every fixed polynomial in \(N\). ◻

Proof. For each fixed large \(N\), the metric of Lemma 65 has a strictly mean-negative inner boundary and strictly mean-positive large coordinate spheres. Apply the three-dimensional trapped-region theorem with zero tensor to obtain its full outermost minimal enclosing frontier. It is a compact smooth surface, possibly disconnected, and bounds the connected exterior containing the end. It is outer area-minimizing: its minimizing enclosure exists by the direct method between the fixed barriers; minimal-obstacle regularity and the maximum principle make that enclosure a smooth minimal bounding frontier. A strict enlargement would contradict maximality of the full minimal trapped region. These are also the outermost/outer-minimizing horizon conventions in the Riemannian Penrose Theorem (Bray 2001; Bray and Lee 2009).

Let this full cut be \(\Gamma_N\). Its exterior is complete with boundary, has one AF end of order one through two derivatives, and has nonnegative scalar curvature with pointwise falloff \(O(r^{-3-\delta})\). These satisfy the boundary version of the Riemannian Penrose Theorem (Bray and Lee 2009, Theorem 1.4); no extension across the original boundary is needed. Every cut in the prepared exterior is a full enclosing cut of the original flat obstacle. The pointwise metric inequality and the two-dimensional area scaling therefore give \[|\Gamma_N|_{\widehat g} \ge e^{-4\epsilon}|\Gamma_N|_g \ge e^{-4\epsilon}A_*.\] Lemmas 65 and 66 imply \[E_g+\frac{\varepsilon_N}{8\pi} \ge E_{\widehat g} \ge\sqrt{\frac{|\Gamma_N|_{\widehat g}}{16\pi}} \ge e^{-2\epsilon}\sqrt{\frac{A_*}{16\pi}}.\] First the end was exhausted at fixed \(N\). Now let \(N\to\infty\) with \(\epsilon\) fixed, and then let \(\epsilon\downarrow0\). Only numerical estimates are passed to the last two limits; no smooth limit of the comparison metrics as \(N\to\infty\) is used. This completes Theorem 25. ◻

Proof. Let \(\chi_R\) equal one on \(r\le R\) and zero on \(r\ge2R\), with uniform differentiated bounds in \(\log r\). Interpolate the metric to \(b\) by \(q_R=b+\chi_R(g-b)\), keeping the original metric on the compact part. Let \(P=K-(\mathop{\mathrm{tr}}_gK)g\). On the exterior of a fixed sufficiently large sphere \(r=L\), solve \[ (\Delta_{q_R}+\mathop{\mathrm{Ric}}_{q_R})X_R =\chi_R\operatorname{div}_gP- \operatorname{div}_{q_R}(\chi_RP),\qquad X_R|_{r=L}=0,\qquad X_R\longrightarrow0 . \tag{206}\] The vector Laplacian acts on one-forms. On this fixed exterior the Ricci endomorphism is bounded above by a negative multiple of the identity, uniformly for large \(R\). The source has \(C^{0,\alpha}_{\tau'}\) norm \(O(R^{\tau'-\tau_{\rm data}})\). Kato’s Inequality compares \(|X_R|\) with the scalar operator \(-\Delta+2+o(1)\). For a radial power, \[(-\Delta_b+2)r^{-a} =\bigl(2+2a-a^2+O(r^{-2})\bigr)r^{-a}.\] The coefficient is positive for \(a<1+\sqrt3\), so \(r^{-\tau'}\) is a common supersolution after enlarging \(L\).

For clarity, existence in this argument follows on finite annuli from the coercivity of \(-\Delta-\mathop{\mathrm{Ric}}\) with zero Dirichlet values. The maximum comparison just given and local boundary estimates are independent of the outer annulus radius. Dirichlet exhaustion, followed by local Schauder estimates, gives \[X_R=O_2(\varepsilon_Rr^{-\tau'}),\qquad \varepsilon_R=R^{\tau'-\tau_{\rm data}}\longrightarrow0.\] On \(r>2R\) the metric is exactly \(b\) and the source vanishes. The same comparison with \(r^{-\kappa}\), and differentiation on hyperbolic balls, gives \(X_R=O_\infty(r^{-\kappa})\) there, with constants allowed to depend on \(R\).

Choose a fixed cutoff equal to zero near \(r=L\) and one beyond a slightly larger sphere, and add the cutoff of \[2\operatorname{sym}\nabla X_R-(\operatorname{div}X_R)q_R\] to \(\chi_RP\). Denote the resulting stress by \(P_R\), and let \(K_R\) be its inverse trace reversal in \(q_R\). The identity \[\operatorname{div}_{q_R} \bigl(2\operatorname{sym}\nabla X_R -(\operatorname{div}X_R)q_R\bigr) =(\Delta_{q_R}+\mathop{\mathrm{Ric}}_{q_R})X_R\] shows that \[\operatorname{div}_{q_R}P_R =\chi_R\operatorname{div}_gP+E_R ,\] where \(E_R\) is supported in a fixed annulus and tends to zero in every needed compact norm. The data are unchanged near \(S\). The family \(K_R\) is bounded by a common smooth weight of order \(-\tau'\), and its exact tail has order \(-\kappa\).

Here is the error estimate needed for the energy condition. Let \(w_0>0\) be smooth and equal to \(r^{-3-\delta}\) on the tail. There is a number \(e_R\to0\) such that \[ \chi_R(R_g+6)+Q_{q_R}(K_R) -2|\operatorname{div}_{q_R}P_R|_{q_R} \ge -e_Rw_0 . \tag{207}\] Indeed the expression with \(Q_{q_R}(K_R)\) replaced by \(\chi_RQ_g(K)\), and with the last norm in \(g\), is nonnegative by the original dominant energy condition. Its errors are the fixed-annulus term \(E_R\), products involving \(X_R\) and \(K\), the quadratic cutoff error \((\chi_R^2-\chi_R)Q_g(K)\), and the change of norm between \(g\) and \(q_R\). The noncompact terms are bounded by a coefficient tending to zero times \(r^{-2\tau'}\). Since \(2\tau'>3+\delta\), these are \(o(1)w_0\). The fixed-annulus terms have the same conclusion by compact positivity of \(w_0\). This argument uses no extra pointwise decay of the original density \(\mu\).

Let \(D\ge |K_R|_{q_R}\) be a common smooth positive weight of order \(-\tau'\), and let \(w_2>0\) be a smooth weight equal to \(r^{-\tau'}\) far out. Choose \(\eta_R\to0\) sufficiently slowly that \(e_R+\varepsilon_R=o(\eta_R)\). Solve \[ \begin{split} (-\Delta_{q_R}+3)v_R={}& \frac18\{\chi_R(R_g+6)-(R_{q_R}+6)\}\\ &+D\sqrt{|dv_R|^2+\eta_R^2w_2^2}+\eta_Rw_0,\\ \partial_\nu v_R|_S={}&-\eta_R,\qquad v_R\longrightarrow0 . \end{split} \tag{208}\] The scalar forcing satisfies \[\left\|\tfrac18\{\chi_R(R_g+6)-(R_{q_R}+6)\}\right\|_{C^{0,\alpha}_{\tau'}} \le C\varepsilon_R=o(\eta_R),\] by the differentiated cutoff estimates above. Here and below \(\nu\) at an inner boundary points into the exterior. This is a scalar equation with a bounded first-order drift and a positive zeroth-order coefficient. A complete continuation argument is as follows. On a finite truncation impose zero outer values and multiply the right side and the inner normal derivative by a parameter \(a\in[0,1]\). Subtract \(a\eta_R\) times a fixed collar function with normal derivative \(-1\); the remaining function has homogeneous Neumann data. At its extrema the gradient term is bounded by the fixed collar derivatives and the regularizing source. The Neumann maximum principle therefore gives a uniform \(O(\eta_R)\) bound. On the end, \(r^{-\tau'}\) is a supersolution for the equation with any of its bounded decaying drifts, because \(3+2\tau'-(\tau')^2>0\). Hence \[v_R=O_2(\eta_Rr^{-\tau'}).\] The estimates of Lemma 119, first absorbing the bounded first-order term by interpolation and then applying the one-sided Schauder estimate, give closedness of continuation. Its linearization has positive zeroth order and the same mixed boundary conditions, so the maximum principle and linear elliptic theory give openness. Exhaustion now gives the displayed solution. On the exact tail the sources decay faster than \(r^{-\kappa}\); comparison improves the bound to that power. The regularizer \(w_2\) bounds all differentiated coefficients on rescaled balls. Bootstrap gives the stated estimates of all orders at fixed \(R\).

Set \(g_R'=e^{4v_R}q_R\), \(K_R'=e^{2v_R}K_R\). Direct conformal calculation gives \[\begin{align*} 8\pi J_R'&=e^{-2v_R} \{\,\operatorname{div}_{q_R}P_R+4K_R(\nabla v_R,\cdot)\,\}, \tag{209}\\ e^{4v_R}(R_{g_R'}+6)&=\chi_R(R_g+6) +8D\sqrt{|dv_R|^2+\eta_R^2w_2^2}+8\eta_Rw_0 +6(e^{4v_R}-1-4v_R)-8|dv_R|^2 . \tag{210}\end{align*}\] The norm in the first equation contributes another factor \(e^{-2v_R}\). Equations (207)–(210), \(D\ge|K_R|\), and \(2\tau'>3+\delta\) imply strict dominant energy with a positive multiple of \(\eta_Rw_0\). At the boundary \[\theta_+(g_R',K_R') =e^{-2v_R}\{\theta_+(g,K)+4\partial_\nu v_R\} =e^{-2v_R}\{\theta_+(g,K)-4\eta_R\}<0 .\] Moreover, the four terms following \(\chi_R(R_g+6)\) in Equation (210) are \(O(\eta_R)w_0\).

We include the mass argument because uniform metric convergence alone would not prove it. The original constraint and the weighted integrability assumption imply \(\int V_0|R_g+6|\,dV_g<\infty\). For any static potential \(V_a\), its scalar-linearization flux has divergence \(V_aDR_b(g-b)\). The difference between \(DR_b(g-b)\) and \(R_g+6\) is quadratic in the metric error and its first two derivatives. Its weighted radial integral is bounded by \[C\int^\infty r^{2-2\tau'}\,dr<\infty .\] Equation (210) makes the curvature tail uniformly small, and the quadratic error has the same uniform integrable bound. Compare the fluxes on a fixed large sphere, where the data converge, and then send that sphere to infinity. Since \(|V_a|+|dV_a|\le C V_0\), this proves convergence of all four components. It also proves their existence for each approximating metric. The improved tail bounds and the constraints give all the weighted integrability requirements for the new data. If the original data have only the stated finite regularity, the constructed data are already smooth outside a compact set: there the vector equation is homogeneous on \(b\), and all coefficients and sources of the scalar equation are smooth after bootstrap. On the remaining compact set, smooth the metric in \(C^2\) and the tensor in \(C^1\), with a cutoff leaving that smooth tail unchanged. The strict energy margin and strict boundary expansion have positive minima in absolute value on their respective compact sets. Continuity of scalar curvature in \(C^2\), momentum divergence in \(C^1\), and boundary expansion in these norms therefore preserves both inequalities if the smoothing error is sufficiently small. Choose that error to tend to zero with \(R\). This gives smooth approximants with unchanged asymptotic estimates and mass components.

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

Proof. First smoothly replace \(r^2\) by a constant on a sufficiently large compact set. On a fixed compact region, taking \(A_0\) large makes all derivatives of the logarithm of the resulting square root small, proving Equation (212) there. For the exact hyperbolic metric on the tail, the tangential and radial Hessian eigenvalues divided by \(\lambda\) are \[\frac{1+r^2}{A_0^2+r^2}, \qquad 1-\frac{A_0^4-A_0^2}{(A_0^2+r^2)^2}.\] Each is at least \[|d\log\lambda|_b^2 =\frac{r^2(1+r^2)}{(A_0^2+r^2)^2}.\] Their gaps are positive multiples of \(r^{-2}\) for \(A_0>1\). Also \[\frac{\Delta_b\lambda}{\lambda}-3 =-\frac{(A_0^2-1)(3A_0^2+2r^2)} {(A_0^2+r^2)^2}.\] The errors from \(g-b=O_2(r^{-\kappa})\) are smaller than these gaps because \(\kappa>2\). This proves the assertions after enlarging the fixed compact transition. The last smallness condition follows by making \(s=0\) throughout a sufficiently large compact set: on its complement \(r^{-\kappa}=o(r^{-(3+\delta)/2})\). ◻

Proof. At an interior extremum of \(f\), \(w=0\), so \(A\), \(D_0\), and \(D_1\) contribute nothing to the trace equation. In stage three it reads \[\mathop{\mathrm{tr}}(l\mathop{\mathrm{Hess}}f)=\tau A_Rf+(1-\zeta)(C-\mathop{\mathrm{tr}}K).\] In the first two stages the same formula has \(\zeta=0\). Since \(A_R\ge1\), the maximum principle and the boundary values give \[|\tau f|\le \max\{|b_-|,|b_+|,\sup|\mathop{\mathrm{tr}}K|+\sup|C|\}.\] No derivative of a collar correction has been used. The necessary correction amplitudes are uniformly bounded: on a black face \(H=c_b-\mathop{\mathrm{tr}}_TK\), and the white face has the analogous bound. All supports can be kept in a fixed compact enlargement by shortening the individual collars.

Take \(r_0\) outside that enlargement and the fixed lapse transition. Compare \(f\) with \[\pm\frac{B_1}{\tau} \left\{\frac{r^{-\beta}}{A_R(r)} -\frac{T^{-\beta}}{A_R(T)}\right\}\] on a truncation at \(r=T\). At a contact the axis is radial. Put \(k=\beta+\partial_{\log r}\log A_R\), so \(\beta\le k\le\beta+1<4\) and its derivative is nonnegative. The Hessian term, together with the radial part \(2\chi A(w)\) coming from \(s=d\log\lambda\), has upper bound \[a\{-2+\chi(k-2)+o(1)\}\] for the positive decreasing test. This is bounded above by a fixed negative multiple of \(a\), uniformly for \(0\le\chi\le1\). The derivative of k has favorable sign.

The only additional spatial-profile term occurs where n varies. Positive t helps, because \(pt\,d\log n\) points outward. For negative t put \(X_0=-nt\). The adverse term is at most \(C\chi aX_0\). If \(X_0>C_1\log(r/\tau)\), then \(l_0=e^{-X_0}\) and the tested gradient gives \[\chi aX_0\le C X_0e^{-X_0}\frac{\lambda}{\tau A_R}r^{-\beta} =o(r^{-\beta})\] by taking the fixed \(C_1\) large. In the complementary regime \(\chi a\le C/\sqrt n\), so \[\chi aX_0\le C\frac{\log r+\log N}{\sqrt n}=o(1)r^{-\beta},\] uniformly on the varying region because \(q>2\beta\). There the shift at \(T\) is negligible after \(T\) is sufficiently large at fixed profiles.

The tensor term is \(O(r^{-\kappa})\), dominated by the positive test height since \(\beta<\kappa\), except near \(T\). Near \(T>2R\), n is constant, and on the test \(a\ge c_Nr^{-\beta}\) whenever the tested slope is small: \(\lambda/A_R\) and \(l_0\) have positive lower bounds at fixed profiles. Thus the negative Hessian contribution dominates the tensor term there as well. For larger slope it is stronger. The negative test gives the reverse inequality. Choose \(B_1\) to dominate the already proved bound on the fixed sphere \(r_0\); this uses no collar derivatives. Height monotonicity proves Equation (250). At the outer face the barrier derivative tends to zero in the quantity \(l|df|\), at fixed profiles. The relation \(Z=t+\log(1+l^2|df|^2)/8=0\) then excludes \(t=-\epsilon\), once \(T\) is sufficiently large.

We now specify the compact choices without circular dependence. Use this common B to choose a compact set Q outside which \(Br^{-\beta}|\mathop{\mathrm{tr}}K|\le\rho\). This is possible because \(\beta+\kappa>3+\delta\). Choose the two negative thresholds sufficiently close to zero, then their collars, and finally sufficiently small offset amplitudes, so that \[|(h+C)\mathop{\mathrm{tr}}K|+|\nabla C|\le\rho \quad\hbox{on Q whenever } b_-\le h\le b_+ .\] The constants controlling derivatives of the now fixed collars need not be uniform in this choice. The next lemma proves precisely the required physical height interval on Q. Outside Q the end bound proves Equation (251), after making the support of C lie in Q. Finally enlarge the fixed constant-n region so that Equation (250) rules out heights near \(b_-\) and \(b_+\) beyond it. This radius is fixed before N and R. ◻

Proof. We verify the extension of Lemma 53 rather than assume an asymptotically flat hypothesis. A putative failure supplies \(N_i\to\infty\) and outer radii tending to infinity. On the relevant compact range \(A_R=1,n=N\), and \(l/\tau\ge c\ell/\tau=cN^{B_0/4}\to\infty\). At a fixed nonzero-gradient lower test \(\phi\) of a lower relaxed height limit, \[d_i,\chi_i\longrightarrow0,\qquad a_i\longrightarrow1,\qquad \frac{l_i}{\tau_i\sqrt{D_i}}\longrightarrow|d\phi|^{-1}, \qquad 2\chi_iA(w_i)\longrightarrow0.\] The last term is the only new trace term on this compact range, where \(A=s\) is bounded. The limiting inequality is therefore \[\frac{\mathop{\mathrm{tr}}_{(d\phi)^\perp}\mathop{\mathrm{Hess}}\phi}{|d\phi|} +\mathop{\mathrm{tr}}_{(d\phi)^\perp}K\le k'+C_b\] at heights below \(k'\), with \(b_-<k'<0\) close to \(b_-\). The low-height \(D_0\) term is nonpositive. The lower relaxed passage, increasing logistic reparametrizations, and Lemma 31 now apply exactly under the hypotheses established in Lemma 53. In particular each height is extended constantly as \(b_+\) across a white face before taking the relaxed limit. Low test contacts are therefore interior, even if a boundary layer is present. The reversed white face has expansion \(-c_w>0\), and excludes contact of the resulting low closed sublevel set. The end bound makes that set compact. After adjoining the full black region, right continuity of the black threshold family excludes its meeting any compact set away from that region. Reversing signs proves the upper separation, including the empty-white-region case.

On an actual boundary collar \(A=0\), \(n=N\), \(A_R=1\), and \(\lambda=\lambda_0\) is constant. For a test \(h=b_-+\psi(s)\) of fixed positive slope, \[d\le C(\tau/\ell)^2=C N^{-B_0/2}, \qquad N\chi\ge c d.\] The trace residual is the leaf expansion plus \(O(d)\) and the term \(a\chi|ds|(\log\psi')'\). Use \[\psi'=m_1\exp\!\left(\int_0^s\gamma_N\right) \quad\hbox{or}\quad \psi'=M_2\exp\!\left(-\int_0^s\gamma_N\right),\] where \(\gamma_N\) is a fixed large multiple of N on \([0,1/N]\), vanishes beyond \(2/N\), and has bounded integral. The first term dominates \(O(d)\) on the short interval. Outside it the offset margin is a fixed multiple of s and dominates d when \(B_0>2\). Compact separation orders the lower barrier on the far end of the collar; fixed large slopes order the upper barrier. The third-stage directional inequalities give the same upper slope barriers of both signs. This verifies all hypotheses and all changed small parameters in the proof of Lemma 54; it does not require the special flat choice \(\tau=\ell^{3/2}\).

For the trace inequalities use the signed field \(\lambda_0Lw\), whose boundary flux has magnitude at least \(\lambda_0L/2\). Direct differentiation on these collars gives \[\frac{\operatorname{div}(\lambda_0Lw)}u =\sqrt d\{\mathop{\mathrm{tr}}_{A_d}H^f+(dp+p+2)a x_t\}.\] With the L weight removed, the same formula has only \(dp\,a x_t\) in its gradient term. Coercivity of Equation (217), equivalently Lemma 35, bounds these expressions by \(CN^m(1+\sqrt{\mathcal T})\). The gradient action is bounded by \(|d\phi|_{A_d}\), and \(A_d\le(1+N/4)A_\chi\). Integration with fixed collar cutoffs proves Equation (252). No inverse power of \(\ell\) occurs in this calculation. ◻

Proof. At an interior contact \(dt=0,\mathop{\mathrm{Hess}}t\ge0\). The Gauss equation of the graph in the Riemannian warped product \(g+l^2ds_0^2\) gives \[ \frac12e^{4t}R_{\hat g} \le \frac12R_g -v\left(\mathop{\mathrm{Ric}}(e,e)+\frac{\mathop{\mathrm{tr}}_\perp\mathop{\mathrm{Hess}}_xl}{l}\right) +\mathcal S\left(S-K+d^{-1}A(w)w\otimes w\right), \tag{253}\] where \[\mathcal S(B)=\frac14(\mathop{\mathrm{tr}}_\perp B)^2 -\frac12|B_{\perp\perp}^{\rm tf}|^2 +dB_{ee}\mathop{\mathrm{tr}}_\perp B-d|B_{e\perp}|^2.\] To obtain this formula, the spatial lapse derivatives contribute \(w\otimes A+A\otimes w\) to the graph second form, and its vertical connection contributes \(d^{-1}A(w)w\otimes w\). The remaining \(p\mathop{\mathrm{Hess}}t\) lapse curvature and conformal Hessian terms are nonpositive at a contact. The rank-one term changes \(\mathcal S\) by only \(vA(w)\mathop{\mathrm{tr}}_\perp(S-K)\).

We give the derivative ledger that controls the additional terms. At a floor contact, \(p=n\), \(d_xp=dn\), and \[A=s-\epsilon n\,d\log n,\qquad \nabla w=H^fA_d+dA\otimes w,\qquad d\log u=H^f(w,\cdot)+(1+v)A .\] Positive-order derivatives of \(\log n\) are bounded. Thus \(\mathop{\mathrm{Hess}}_xl/l\), \(A\), and the needed spatial derivatives have fixed-degree polynomial bounds in \(n\), uniformly in position. The factor \(\epsilon<1\) introduces no larger growth. Moreover \[\partial_p\frac{4+p}{4+pv} =\frac{4d}{(4+pv)^2},\qquad dv=2d\{H^f(w,\cdot)+vA\}\] at contact. Therefore the derivatives of \(A_\chi\) always carry a \(d\) factor in the uncontrolled axial Hessian direction. In \(d\log u\), that direction contracts with \(\chi\) in the flux. The same is true for the new drift \(A_\chi(A(w)w)=\chi A(w)w\). A spatial derivative of \(A\) in this drift is contracted axially and retains \(\chi\).

The algebraic floor comparison of Lemma 37, applied to \(S\), and Equation (253) therefore give, with universal \(c>0\), small prescribed \(\eta>0\), and fixed finite powers \(m\), the bounds \[\begin{split} \mathcal T-\mathcal S \left(S-K+d^{-1}A(w)w\otimes w\right) &\ge c\mathcal T-Cn^m(\rho_0+v),\\ w(F)&\ge\tau A_Ra\sigma-\eta\mathcal T-C_\eta n^m(\rho_0+v),\\ u^{-1}|\operatorname{div}(u\mathcal D)| &\le\eta\mathcal T+C_\eta n^m(\rho_0+v). \end{split}\] Here \(|K|^2+|\nabla K|^2\lesssim\rho_0\), and the height estimate bounds the squared end height by the same weight. For example \(a|\nabla K|\) is bounded by \(v+|\nabla K|^2\). The derivative of \(h=\tau A_Rf\) gives the displayed positive term and \(h\,w(d\log A_R)\), whose absolute value is at most \(Ca|h|\le C(v+\rho_0)\) for this floor estimate. Derivatives of \(D_0,D_1,b,C\) are fixed compact coefficients; the derivative in h is nonnegative and at least one. These observations enumerate all new spatial terms beyond the flat floor calculation, including \(d_xp\), which is not zero here.

Combining these bounds with Equation (220) and the constraint gives \[ u^{-1}\operatorname{div}V_\vartheta \ge 2\delta_0\mathcal T+\tau A_Ra\sigma -Cn^m(\rho_0+v) \tag{254}\] for fixed sufficiently small \(\delta_0\). The cosmological constants cancel in deriving this lower bound: \(8\pi\mu-R_g/2-3=Q_g(K)/2\). In the prescribed source the remaining explicit term \(3(e^{4t}-1)\) is negative at the floor, where \(d_0=\delta_0\) and \(m_0=1\). Choose \(P_0>m+1\) and N sufficiently large that its penalty dominates the error and \(2\rho+\rho_0\). Equation (254) then contradicts the scalar equation.

Outer contacts were excluded in Lemma 77. At an inner contact \(j=0\), so Equation (247) restores the full flux condition. Since \(A=0\) on the collar, its face identity is \[d(H+4\partial_\nu t)=-N_Bd.\] It forces \(\partial_\nu t<0\), contrary to a minimum. In the final scalar stage \(f=0,t=Z\). At a floor minimum the unscaled source is negative after the same penalty is enlarged; at positive scale the inner derivative is negative as well. At zero scale the homogeneous mixed problem gives \(Z=0\) by multiplication by Z. This proves separation for all stages. ◻

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

Proof. The profiles are fixed throughout this proof. On the ultimate tail \(n=n_\infty\) is constant and \(A=s\), and \(A_R\) is proportional to \(r\). The height bound of Lemma 77 therefore gives \(\lambda f=O(r^{-\beta})\). On each hyperbolic unit patch rescale \(f\) by the central value of \(\lambda\). The trace equation is uniformly elliptic there by Proposition 80. Its inhomogeneous term at \(f=0\) is \(O_j(r^{-\kappa})\); the remaining zeroth-order term is a bounded coefficient times the rescaled \(f\). The local linear estimates for this fixed smooth solution consequently give \(\lambda f=O_j(r^{-\beta})\), using \(\beta<\kappa\). The equations for \(w\) and \(D\) give the next two estimates. The same argument holds on the outer truncations with uniform boundary estimates at their zero Dirichlet face.

Here is the scalar expansion before assuming decay of \(Z\). Since \(t\) is bounded and \(n_\infty\) is fixed, the substitution \(t=Z+O_j(r^{-2\beta})\) has uniformly bounded differentiated coefficients. In the flux, \[A_\chi=I+O_j(r^{-2\beta}),\qquad u^{-1}du=s+(p+2)dZ+O_j(r^{-2\beta}).\] The tensor part of \(S\) has order \(O_j(r^{-\beta})\). In Equation (217), the part quadratic in \(dZ\) is \(4(p+1)|dZ|^2\) to error \(O_j(r^{-2\beta})\). Every mixed term between \(S\) and \(dt\) has an additional factor \(a\): its two possible forms are \(a\langle M,y\rangle\) and \(a q_1x_t\). Thus these terms, including their differentiated bounds, have order \(r^{-2\beta}\). The tensor drift \(K(w,\cdot)\) has order \(r^{-\kappa-\beta}\). The leading lapse term \(4\langle s,dZ\rangle\) from \(u^{-1}\operatorname{div}(4u\nabla Z)\) cancels the explicit lapse term on the right side of the scalar equation.

The capillary term is \(O_j(r^{-2\beta})\). Even before its support disappears, the penalty contributes \(O_j(r^{-3-\delta})+O_j(r^{-2\beta})\), because \(n_\infty\) is fixed and all derivatives of \(t\) are bounded. Since \(2\beta>4+\delta\), the resulting equation is \[ \Delta_gZ+C_N(Z)|dZ|^2 =\frac34(e^{4Z}-1)+O_j(r^{-3-\delta}),\qquad C_N(z)=p(z)+2-d_0(z)(p(z)+1). \tag{259}\] Here \(p(z)=n_\infty p_*(n_\infty z)\) and \(d_0(z)\) denotes the corresponding composed coefficient. This expansion requires only the bounded differentiated range already proved.

Define the increasing function \[\Phi(z)=\int_0^z\exp\!\left(\int_0^\xi C_N(\eta)\,d\eta\right)d\xi .\] On the bounded solution range both \(\Phi'\) and its reciprocal are bounded. Writing \(Y=\Phi(Z)\), Equation (259) becomes \[(\Delta_g-c(Z))Y=O_j(r^{-3-\delta}),\qquad c(z)=\frac{3(e^{4z}-1)\Phi'(z)}{4\Phi(z)},\qquad c(0)=3.\] The quotient is positive and bounded away from zero on this range. For \(\gamma=1\), \(\Delta_g r^{-\gamma}=-r^{-\gamma}+o(r^{-\gamma})\); hence a large multiple of \(r^{-1}\) bounds both signs of \(Y\) by the maximum principle, first on truncations whose outer value is zero and then in their limit. In particular \(Z\to0\), so \(c(Z)\to3\). For any \(0<\gamma<3\), \[(\Delta_g-c(Z))r^{-\gamma} =(\gamma^2-2\gamma-3+o(1))r^{-\gamma},\] whose leading coefficient is negative. The same comparison proves \(Y,Z,t=O(r^{-\gamma})\). Unit-patch estimates for the transformed equation, followed by differentiation and the already bounded coupled coefficients, prove the differentiated versions.

Choose \(\gamma>(3+\delta)/2\). Lemma 75 now applies to Equation (259) and gives the scalar-curvature estimate, also after differentiation. In particular the penalty vanishes sufficiently far out. Proposition 74 applies with \(s=\gamma>3/2\): all the exponent inequalities there follow from \(\kappa>2\) and \(\beta>2\). This proves existence of the four limits and both stated flux identities. ◻

Proof. Use the primitive of Proposition 72. For brevity write \[\mathfrak C=\tau A_Ra\sigma =\frac{\tau A_R}{\lambda l_0}\frac{v}{\sqrt d}, \qquad b_c=\frac{1-e^{-2t}}2\quad(t\ge0).\] The exact divergence is \[ u^{-1}\operatorname{div}V_* =d_0(\mathcal T+\mathfrak C)+\rho -m_0n^{P_0}(\rho_0+v)+\mathcal R, \tag{260}\] where \(\mathcal R\) is Equation (232). We estimate each of its terms. Every Young constant below depends only on a chosen fixed fraction of the positive terms; it does not depend on \(N,R\), or a classical bound for the solution.

On \(r\le2R\) first set aside the additive term \(4\Delta G_c\) in the ordinary-volume divergence. The bounded logarithmic derivatives of \(n,\lambda\) give \[|\Delta G_c|\le C\lambda/n,\qquad \int_{r\le2R}|\Delta G_c|\,dV_g \le\frac C N\left(1+\int_1^{2R}r^{2-q}\,dr\right) \le\frac C N .\] Also, uniformly in this whole region, \[ \frac{(1+\log N+\log(1+r))^2}{n\rho}=o_N(1), \tag{261}\] since \(q>3+\delta\). This estimate includes the capped transition, where \(n\asymp NR^q\).

Proof of Theorem 68. Fix one prepared data set and the compact black–white exterior used in the construction. Every full enclosing cut in this smaller exterior is a full enclosing cut in the prepared original exterior. Since \(t\ge-\epsilon\) and the graph term is nonnegative, \(\widehat g\ge e^{-4\epsilon}g\). On each tangent two-plane its area density is therefore at least \(e^{-4\epsilon}\) times the \(g\)-density. Taking the infimum over all full cuts gives \[A_{\min}(\partial\widehat\Omega;\widehat g) \ge e^{-4\epsilon}A_{\min}(S;g).\] This comparison uses the entire intrinsic boundary of each cut, including the original boundary when the exterior itself is chosen.

By Proposition 80 and Lemma 81, \((\widehat\Omega,\widehat g,0)\) has one complete AH end with the required tensor-norm decay and weighted scalar-curvature integrability, all four finite metric fluxes, scalar curvature at least \(-6\), and strictly mean-negative boundary. Thus the time-symmetric case of Theorem 10 applies in the designated chart, irrespective of the causal character of its flux covector. Together with Proposition 82, it gives \[p_0(g)+o_N(1)\ge p_0(\widehat g) \ge \sqrt{\frac{e^{-4\epsilon}A_{\min}(S;g)}{16\pi}} \left(1+\frac{e^{-4\epsilon}A_{\min}(S;g)}{4\pi}\right).\] Send \(N\to\infty\); then \(\epsilon=B_0\log N/N\to0\). Finally remove the strict preparation using Lemma 69, which preserves the four metric flux limits and the full-cut infimum. This proves Equation (203). ◻

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

A future timelike covector admits a proper future Lorentz transformation taking it to \((m_{\rm AH},0,0,0)\). Apply Theorem 68 in this chart. The geometry of the exterior, its future boundary signs, and its intrinsic full-cut area infimum have not changed. This proves the assertion. Only a hyperbolic change of spatial end coordinates was used. ◻

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

The local obstacle regularity theorem (Huisken and Ilmanen 2001, Regularity Theorem 1.3(iii)) applies here to pure perimeter, a smooth obstacle, a smooth metric and ambient dimension three. It gives \(C^{1,1}\) boundaries and smooth minimal free parts. Part (iii) is used for this individual regularity. For a fixed \(0<\alpha_0\le1/2\), the local \(C^{1,\alpha_0}\) estimates in Part (ii) are uniform: the obstacle bounds and positive metric \(C^1\) bounds are uniform for the family \(g_s\).

Only away from \(S\) do we need density estimates. Here is a local proof with uniform constants. In a sufficiently small ball centered at a boundary point, compare a minimizing set \(E\) with \(E\setminus B_\rho\) and \(E\cup B_\rho\), filling any unnecessary pockets afterwards. The ball misses the obstacle. For \(f(\rho)=\mathop{\mathrm{Vol}}(E\cap B_\rho)\), minimality and coarea give, for almost every \(\rho\), \[P(E;B_\rho)\le f'(\rho),\qquad P(E\cap B_\rho)\le2f'(\rho).\] Uniform local isoperimetry implies \(f'(\rho)\ge c f(\rho)^{2/3}\), hence \(f(\rho)\ge c\rho^3\). The same argument applies to the complement. Relative isoperimetry then gives \(P(E;B_\rho)\ge c\rho^2\); the comparison above and the uniform sphere-area bound give \(P(E;B_\rho)\le C\rho^2\). These constants are uniform on compact sets disjoint from \(S\). Perimeter compactness now gives subsequential indicator convergence. Lower semicontinuity, together with testing a fixed minimizer, gives convergence of perimeters, including for varying metrics.

Here is the measure-continuity detail needed for differentiation. In a fixed orthonormal frame let \(D\chi_{E_j}\) be the vector normal measures. Their weak convergence and convergence of total variations imply weak convergence of their variation measures: any weak limit dominates \(|D\chi_E|\), and equality of total masses eliminates the excess. Approximate the limiting unit normal in the square-integral norm of this measure by continuous vector fields. For such a field \(v\), expand \[\int|\nu_j-v|^2\,d|D\chi_{E_j}| =\int(1+|v|^2)\,d|D\chi_{E_j}| -2\int v\cdot dD\chi_{E_j}.\] Both terms converge. Continuous approximation and uniform continuity then give convergence for every continuous function of position and unit normal, and hence for functions of the unoriented tangent plane. A partition of unity proves the assertion globally. Uniform metric convergence converts varying-metric perimeters to this fixed-metric argument.

On every rectifiable cut of bounded area, the area-element expansion is uniform: \[ \mathop{\mathrm{Area}}_{g_s}(T)=\mathop{\mathrm{Area}}_g(T)+s\,a'_T(H) +O(s^2)\mathop{\mathrm{Area}}_g(T). \tag{274}\] A fixed minimizing cut gives the upper one-sided bound in (273). For the lower bound take minimizers \(T_s\) for \(g_s\), pass along any sequence \(s\downarrow0\) to a member \(T\in\mathcal M\), and use \[\frac{A(s)-A(0)}s \geq a'_{T_s}(H)+O(s)\longrightarrow a'_T(H).\] Compactness and continuity of \(a'_T(H)\) prove the formula.

Finally, the union and intersection of two filled minimizing sets remain obstacle competitors. Perimeter submodularity gives \[P(E_1\cup E_2)+P(E_1\cap E_2) \leq P(E_1)+P(E_2)=2A.\] Both terms on the left are at least \(A\), so both are minimizers. Different tangent planes at an intersection would give a corner in one of them, contradicting interior regularity. To check continuity on the union, take points on a sequence of minimizers converging away from the obstacle. The two phase-volume density bounds prevent disappearance of the limiting point under indicator convergence, so it lies on a limiting minimizer. The uniform local \(C^{1,\alpha_0}\) estimates and strict perimeter convergence then give tangent-plane convergence there. The common-plane property makes this independent of the chosen sequence. ◻

Proof. Solve (276) on spherical truncations with outer Dirichlet value zero. Continue both the source and the inner Neumann value from zero. The gradient term can be written \(b\cdot d\phi+f\), where \[|b|\leq D,\qquad 0\leq f\leq Dw_2+w .\] The operator is consequently \(-\Delta-b\cdot\nabla+3\). A fixed collar function supplies the prescribed outward Neumann derivative, which is \(+1\); adding a sufficiently large constant dominates its bounded compactly supported derivatives and the source. The maximum principle gives a bound independent of the truncation and continuation parameter. Beyond a fixed sphere, \[(3-\Delta_b)r^{-\beta} =(3-\beta)(1+\beta)r^{-\beta}+O(r^{-\beta-2}).\] The decaying drift and metric errors preserve its positive leading coefficient. Comparison with a multiple of this function, using the already bounded value on the fixed sphere and zero outer data, gives \(|\phi|\leq C r^{-\beta}\).

Lemma 119, followed by Sobolev embedding and the one-sided Schauder estimate, gives the same weighted bound through two derivatives. At the fixed inner face use the corresponding Neumann estimates. The linearized equation has bounded drift, zeroth-order coefficient \(3\), homogeneous Neumann data at the inner face and homogeneous Dirichlet data at the outer face. Its kernel is zero by the maximum principle, so the usual elliptic Fredholm alternative makes it invertible. The a priori estimates close the method of continuity on each truncation. Exhaustion and local compactness then produce the asserted decaying solution, smooth on compact sets including the one-sided boundary.

The additional source is \(O(r^{-2\beta})=o(w)\), proving the ratio claim. Replacing \(\Delta_g\) by \(\Delta_b\) costs \(O(r^{-\tau-\beta})\). Both this error and \(r^{-2\beta}\) have finite integrals with weight \(V_0\,dV_b\), as does \(w\). The scalar linearization identity \[\operatorname{div}_b\mathbb U(V,4\phi b) =8V(3-\Delta_b)\phi,\qquad \mathop{\mathrm{Hess}}_bV=Vb,\] therefore proves existence of all the conformal fluxes. Replacing \(b\) by \(g\) in \(4\phi b\) costs order \(r^{-\tau-\beta}\), beyond the mass order.

For the exact conformal transformation \(g_s=e^{4s\phi}g\), \(K_s=e^{2s\phi}K\), direct differentiation gives \[\mathcal C'_Q=-4\phi\mathcal C+ 8((3-\Delta)\phi+K(v,\nabla\phi)).\] The first term vanishes on \(\mathcal Z\). The expansion transforms as \(\theta_s=e^{-2s\phi}(\theta+4s\partial_\nu\phi)\); since \(\theta=0\) on \(S\), its derivative is \(-4\). ◻

Proof. Compactify \(\mathcal Z\) by one point, collapsing all rays whose base points escape compact sets; if it was compact, add an isolated point. The ball fibers are compact. The functions \(\mathcal C'/w\) extend continuously with value \(8a(H,P)\). On the disjoint compact union of this space, \(S\), and \(\mathcal M\), consider the linear test map \[ (H,P)\longmapsto \big(\mathcal C'/w,\ -\theta',\ c\,a'_T(H)-16\pi p_0'\big). \tag{279}\] Lemma 85 supplies continuity on the last piece.

Suppose its image contained a function positive everywhere. Write its direction as \(aQ+(H_0,P_0)\). Positivity at the additional point gives \(a>0\). Realize the direction by \[g_s=e^{4as\phi}(g+sH_0),\qquad K_s=e^{2as\phi}(K+sP_0).\] These metrics remain positive and uniformly comparable to \(g\). On the far end, retain the nonnegative background constraint with its positive conformal scaling factor. The remaining terms have the following bounds, uniformly over the unit ball: \[\begin{array}{c|c} \text{term}&\text{size}\\ \hline \text{strict conformal gain}&8asw\\ \text{non-strict linear changes}&s\,o(w)\\ \text{new quadratic and cross terms}&O(s^2w). \end{array}\] For clarity, the non-strict end metric and tensor tests are \(O(r^{-3})\), their background constraint terms are \(O(r^{-4})+O(r^{-3-\tau})\), and their products with the strict direction are \(O(r^{-3-\beta})\). The strict direction’s own quadratic errors are \(O(r^{-2\beta})\). All are controlled by \(w\). Applying the exact conformal law after the intermediate variation gives these bounds without differentiating the nonnegative original constraint into an uncontrolled error. One may compose the two isometries of unit balls in this calculation: at an active unit ray a difference of isometric identifications is tangent to the unit sphere and is annihilated by \(J\); at a vacuum ray \(J=0\).

On the remaining compact ball bundle, positivity of the derivative on the closed active set persists on a neighborhood of it. The original constraint has a positive minimum on the compact complement. Taylor’s formula thus gives the DEC for sufficiently small \(s>0\) everywhere. The boundary has strictly negative future expansion. The same estimates are integrable with weight \(V_0\), and the flux divergence identity proves differentiability of all four fluxes. The covector remains future timelike by continuity. Thus (265) applies to this path.

Positivity on \(\mathcal M\), however, says \[c\min_{T\in\mathcal M}a'_T(H)-16\pi p'_0>0.\] By Lemma 85 and (264), the right derivative at zero of the difference in (265) is negative. That difference is initially zero and is nonnegative for \(s>0\), a contradiction.

The image of (279) is therefore disjoint from the open positive cone of continuous functions. Hahn–Banach separation gives a nonzero continuous functional, nonnegative on nonnegative functions and annihilating this linear image. The Riesz Representation Theorem gives finite nonnegative measures on its three compact pieces. The objective piece has positive mass: otherwise evaluation on \(Q\), strictly positive on both other pieces, would contradict annihilation. Normalize that mass to one. Divide the finite measure on finite active rays by \(w\) and call it \(\mathfrak L\); absorb eight times the end-point atom into \(z\). Rearranging the identity gives (278). ◻

Proof. On a compact neighborhood of the support, choose a Gaussian Lorentzian collar \[\mathbf g_\epsilon=-dt^2+g_\epsilon(t),\qquad g_\epsilon(0)=g,\quad \partial_tg_\epsilon(0)=2K.\] Prescribe its second time jets so that \[\frac1{8\pi}(G_{\mathbf g_\epsilon}-3\mathbf g_\epsilon)_{ij} =D_{\epsilon,ij},\qquad D_\epsilon=\frac{J\otimes J}{\mu+\epsilon} \quad\hbox{at }t=0.\] The spatial Einstein tensor depends on the freely chosen normal derivative of \(K\) by a nonzero multiple of trace reversal. In dimension three trace reversal has eigenvalues \(1\) and \(-2\); it is invertible. The remaining terms are fixed spatial jets. Thus these smooth second jets can be prescribed, and a quadratic time extension with a cutoff realizes a smooth Lorentzian metric on a sufficiently short collar. The normal and mixed components of the displayed tensor are \(\mu,J\), by the constraints.

Set \(D=J\otimes J/\mu\) where \(\mu>0\), and set it to zero at vacuum. The DEC gives \[|D_\epsilon|\leq\mu,\qquad |\nabla D_\epsilon|\leq2|\nabla J|+|\nabla\mu|.\] At a zero of the smooth nonnegative function \(\mu\), its differential vanishes; \(|J_i|\leq\mu\) then gives \(dJ_i=0\) there also. The displayed derivative bound proves that \(D\) is \(C^1\) across vacuum with zero derivative there. Splitting any compact set into \(\{\mu\geq\delta\}\) and \(\{\mu<\delta\}\) proves \(D_\epsilon\to D\) in \(C^1\).

Vary the slice by graphs \(t=a s(x)\), differentiating at \(a=0\). The metric derivative is \(2sK\). At a nonvacuum active ray, \(\mu=|J|\), \(v=-J^\sharp/\mu\), and \(\zeta=n+v\) is null. The limiting prescribed stress has the form \[\mu^{-1}\gamma\otimes\gamma,\qquad \gamma(n)=\mu,\quad\gamma|_{T\Omega}=J,\quad\gamma(\zeta)=0.\] The covector \(\gamma\) is nonnegative on future causal vectors. Parallel transport \(\zeta\) in any spatial direction using the spacetime connection. Its contraction with \(\gamma\) is nonnegative and has a zero, so its derivative there is zero. Consequently the spatial covariant derivatives of the limiting stress, contracted in one slot with \(\zeta\), vanish.

Let \(T_\epsilon=(G_{\mathbf g_\epsilon}-3\mathbf g_\epsilon)/(8\pi)\). Contracted Bianchi, before taking the limit, gives \[(\nabla_nT_\epsilon)(n,\zeta) =\sum_i(\nabla_{e_i}T_\epsilon)(e_i,\zeta).\] The right side tends to zero by the compact \(C^1\) convergence just proved. Thus the required normal derivative of the constraint combination tends to zero too. Differentiating its arguments causes no additional limiting term: the first argument is paired against a tensor with \(\zeta\) in its kernel, and the second argument remains null; at activity \(\gamma\) is proportional to \(\zeta^\flat\), so nullity kills its differentiated contraction. At vacuum the tensor and its spatial derivatives vanish, and Bianchi gives the same conclusion for every vector in the closed unit ball. All expressions use only the fixed compact jets and \(D_\epsilon,\nabla D_\epsilon\); the convergence is uniform, including near vacuum. ◻

Proof. Apply Lemma 88 in (278). Mass, expansion and end-point terms are zero. The constraint measure is finite over the fixed compact support, so the limit gives \[\int_{\mathcal M}\int_T s\,\mathop{\mathrm{tr}}_TK\,dA_g\,d\omega(T)=0 \qquad(s\in C_c^\infty(\operatorname{int}\Omega)).\] Off \(S\), form the positive aggregate area measure \(\mathfrak b=\int_{\mathcal M}dA_T\,d\omega(T)\). Lemma 85 makes \(F(x)=\mathop{\mathrm{tr}}_{\Pi(x)}K(x)\), for the common tangent plane \(\Pi(x)\), a single continuous function on the union of the cuts locally away from \(S\). The last identity is \(F\mathfrak b=0\). Fubini on a countable compact exhaustion shows that, for \(\omega\)-almost every cut, \(\mathop{\mathrm{tr}}_TK=0\) almost everywhere off \(S\). Such a cut is smooth minimal there, so continuity gives \(H_T=\mathop{\mathrm{tr}}_TK=0\) throughout its free part.

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

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

Proof. For a pure tensor variation \(P\), integration of momentum by parts gives the distributional coefficient \[2\big[u(\tau_Kg-K)-\operatorname{sym}\nabla X +(\operatorname{div}X)g\big].\] Its trace gives \(\operatorname{div}X=-u\tau_K\), and substitution gives (281). A simultaneous conformal test \((4\psi g,2\psi K)\), using complementarity to cancel its background scaling term, gives \[8\langle u,(-\Delta-\Lambda_{\rm c})\psi\rangle +8\langle X,K(\nabla\psi,\cdot)\rangle=0.\] This is (282). Diverging the shift equation and using its trace gives \[ \Delta X_j+\mathop{\mathrm{Ric}}_{jk}X^k =(\tau_K\delta_j{}^i-2K_j{}^i)\nabla_i u +u(\nabla_j\tau_K-2\nabla^iK_{ij}). \tag{284}\] Together with the lapse equation this is an elliptic system with diagonal scalar-Laplacian principal part and smooth first-order couplings. Measures belong locally to a negative Sobolev space. The local elliptic estimate on nested compact cutoffs improves their Sobolev order by one, because the couplings have at most one derivative. Iteration and Sobolev embedding yield smoothness.

For completeness, the metric equation is computed next. Put \(P_K=K-\tau_Kg\). The compact constraint pairing is the variation of \[\int\big[u(R-2\Lambda_{\rm c}+\tau_K^2-|K|^2) +2X^i\nabla^j(P_K)_{ij}\big]\,dV_g\] together with \(-8\pi\int H(X,J^\sharp)\,dV_g\). The latter is exactly the vector-transport correction. Changing the fixed background volume density to the varying one adds zero, since its background integrand is \(16\pi(u\mu_{\Lambda_{\rm c}}+J(X))=0\). Integrating the shift term once gives \(-\int P_K^{ij}(\mathcal L_Xg)_{ij}\,dV_g\). At fixed covariant \(K\), contravariant \(X\), and \(u\), use \[\begin{split} R'&=-\langle\mathop{\mathrm{Ric}},H\rangle+\nabla^i\nabla^jH_{ij} -\Delta\mathop{\mathrm{tr}}H,\\ \tau_K'&=-K^{ij}H_{ij},\qquad (|K|^2)'=-2(K^2)^{ij}H_{ij},\\ (P_K^{ij})'&=-H^i{}_aK^{aj}-H^j{}_aK^{ia} +(K^{ab}H_{ab})g^{ij}+\tau_KH^{ij},\\ (\mathcal L_Xg)'&=\mathcal L_XH,\qquad (dV_g)'=\tfrac12(\mathop{\mathrm{tr}}H)dV_g. \end{split}\] Integration of \(\mathcal L_XH\) and collection of coefficients gives \[\begin{align*} 0={}&\mathop{\mathrm{Hess}}u-(\Delta u)g-u\mathop{\mathrm{Ric}}+2u(K^2-\tau_KK) +\frac u2(R-2\Lambda_{\rm c}+\tau_K^2-|K|^2)g \\ &+\mathcal L_XK-(\operatorname{div}X)K -\big(X(\tau_K)+K^{ab}\nabla_aX_b\big)g -8\pi X_{(i}J_{j)}. \tag{285}\end{align*}\] The shift equation gives \[\operatorname{div}X=-u\tau_K,\quad K^{ab}\nabla_aX_b=-u|K|^2,\quad \mathop{\mathrm{tr}}(\mathcal L_XK)=X(\tau_K)-2u|K|^2.\] Taking the trace of (285) and using complementarity yields \[ \Delta u+X(\tau_K) =\frac u2(R+\tau_K^2+|K|^2-4\Lambda_{\rm c}). \tag{286}\] Substitution gives (283).

To exhibit finite type, write \(B_{ij}=\nabla_{(i}X_{j)}=-uK_{ij}\). Commuting covariant derivatives gives \[\begin{split} \nabla_i\nabla_jX_k ={}&\nabla_iB_{jk}+\nabla_jB_{ik}-\nabla_kB_{ij}\\ &+\tfrac12\big([\nabla_i,\nabla_j]X_k -[\nabla_i,\nabla_k]X_j-[\nabla_j,\nabla_k]X_i\big). \end{split}\] All commutators are curvature times \(X\). Derivatives of \(B\) use only \(u,du\) and the first derivatives of \(K\). Equation (283) similarly expresses the Hessian of \(u\) in first jets. Therefore, for \(\mathcal J=(u,X,\nabla u,\nabla X)\), \[ \nabla\mathcal J=\mathcal A(g,K,\Lambda_{\rm c})\,\mathcal J \tag{287}\] for a smooth bundle-valued linear coefficient. ODE uniqueness along paths proves continuation. At a zero of \(u\geq0\), \(du=0\); domination \(|X|\leq u\) also gives \(X=dX=0\). Nonzero moments consequently have positive lapse everywhere. ◻

Proof. The jet at \(p\) cannot be zero by continuation. On any fixed compact set, a finite family of coordinate neighborhoods and paths from \(p\) has bounded lengths and uniformly bounded coefficients in (287). Grönwall’s Inequality gives uniform first-jet bounds there. The equations and local elliptic estimates give bounds of every order on smaller compact sets. Diagonal compactness gives the limit. Its jet at \(p\) still has norm one, and causality is a closed pointwise condition. Lemma 90 supplies positive limiting lapse. ◻

Proof. In a smooth normal collar of \(S\), the coefficient in (287) is smooth up to the face. Solve its linear ODE down each collar segment from a fixed interior surface. Smooth dependence on its footpoint gives a smooth extension; ODE uniqueness identifies it with the original field in the collar.

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

We next derive end behavior without imposing growth on the measures. Express the first-jet system in a \(b\)-parallel radial frame, with \(\eta=\operatorname{arsinh}r\). At \((b,0)\) its lapse system is \(\mathop{\mathrm{Hess}}_bu=ub\) and its shift system is hyperbolic Killing transport. Their radial propagators have norm at most \(C e^{|\eta-\eta'|}\). This follows directly from the \(2\)-by-\(2\) blocks \(y''=y\) and the constant blocks, or from the ambient linear functions and Lorentz generators on the hyperboloid. The actual coefficient differs by \(O(e^{-\tau\eta})\); only two metric derivatives and one tensor derivative occur. Variation of constants gives \[e^{-\eta}|\mathcal J(\eta)| \leq C+C\int_{\eta_0}^{\eta} e^{-\tau s}e^{-s}|\mathcal J(s)|\,ds.\] Grönwall therefore gives \(\mathcal J=O(r)\), uniformly in angle.

The lapse and tangential-shift radial equations now read \[y''-y=O(r^{1-\tau}),\] and the normal-shift derivative is \(O(r^{1-\tau})\). Variation of constants, with \(3/2<\beta<\min\{2,\tau\}\), consequently gives continuous angular coefficients \[\begin{align*} u&=r f_0(\omega)+O(r^{1-\beta}),& \partial_{\eta}u&=r f_0(\omega)+O(r^{1-\beta}),\\ X_T&=r W_0(\omega)+O(r^{1-\beta}),& X^{\nu_b}&=a_0(\omega)+O(r^{1-\beta}). \end{align*}\] Convergence of the leading coefficients is uniform; no angular differentiability is required at this stage. The homogeneous \(r^{-1}\) mode is included in the remainder because \(\beta<2\).

Use first a conformal prototype \(\psi=r^{-3}f(\omega)\) in the multiplier identity and integrate to a large sphere. Its interior pairing vanishes by the adjoint equations, its \(S\) terms vanish, and its end-point coefficient is zero. All pairings converge: the moments grow at most as \(r\), while the prototype derivatives are \(O(r^{-4})+O(r^{-3-\tau})\) or better. The sole nonvanishing boundary expression is \[-8\int_{r=R} (u\partial_{\nu_b}\psi-\psi\partial_{\nu_b}u)\,dA_b.\] Terms containing \(KX\psi\) have integrated order \(O(R^{-\tau})\); replacing \(g\) by \(b\) has the same vanishing order. The displayed limit is \(32\int_{\mathbb S^2}f_0f\,d\omega\). The left side is \(16\pi p'_0=32\int_{\mathbb S^2}f\,d\omega\) by the defining metric flux. Arbitrary \(f\) gives \(f_0=1\).

For a mixed tensor prototype, the only boundary term is twice its trace-reversed stress flux against \(X\). Its limit is \(2\int_{\mathbb S^2}\langle W_0,W\rangle_\sigma\,d\omega\), whereas its metric mass and area derivatives vanish. Hence \(W_0=0\). The tangential trace of the shift equation on coordinate spheres now gives \[\operatorname{div}_{S_R}X_T+H_{S_R}X^{\nu_b} =O(R^{1-\beta}).\] Interpret this on the unit sphere. The divergence term tends to zero distributionally, since integration against a fixed test function gains the factor \(R^{-1}\), and \(H_{S_R}\to2\). Thus \(a_0=0\). The already uniform limits identify it pointwise.

We have obtained the value estimates in (266), since \(V_0-r=O(r^{-1})\). The coupled elliptic system for \((u-V_0,X)\) has uniformly controlled unit-patch coefficients and source \(O_{0,\alpha'}(r^{1-\tau})\). Local estimates applied to these value bounds yield the first and second derivative Hölder bounds in (266), with a smaller \(\alpha'\) if necessary. The prototype normalization makes the field nonzero; the last assertion of Lemma 90 gives \(u>0\). ◻

Completion of the proof of Theorem 84. Since \(u>0\), (270) is a smooth Lorentzian metric on the open exterior product. Its coefficients are independent of \(t\), so \(\xi=\partial_t\) is Killing. Its future normal is \(n=u^{-1}(\partial_t-X)\); the future-normal convention fixed after Equation (6) gives \[\tfrac12\mathcal L_n g=-\frac1{2u}\mathcal L_Xg=K\] by (267). Thus both induced tensors are the original ones.

We verify its curvature without a spacetime existence assumption. The Gauss–Codazzi and lapse identities, obtained by differentiating the displayed metric, give for spatial slots \[\mathop{\mathrm{Ric}}_{\mathbf g,ij} =\mathop{\mathrm{Ric}}_{g,ij}+\tau_KK_{ij}-2(K^2)_{ij} -u^{-1}\big((\mathcal L_XK)_{ij}+(\mathop{\mathrm{Hess}}u)_{ij}\big),\] and \[R_{\mathbf g}=R_g+\tau_K^2+|K|^2 -2u^{-1}(\Delta u+X(\tau_K)).\] Equation (286) with \(\Lambda_{\rm c}=-3\) gives \(R_{\mathbf g}=-12\). Equation (269) then gives \[u(G_{\mathbf g}-3\mathbf g)_{ij}=-8\pi X_{(i}J_{j)}.\] The normal and mixed components of this tensor are the constraints, \(8\pi\mu\) and \(8\pi J_i\). Complementarity and causality imply \[0=u\mu+J(X)\geq u\mu-|J||X| \geq u(\mu-|J|)\geq0.\] Where \(\mu>0\), equality forces \(|X|=u\), \(|J|=\mu\), and \(J=-\mu X^\flat/u\). Where \(\mu=0\), \(J=0\). Substitution into all the normal, mixed and spatial components gives \[G_{\mathbf g}-3\mathbf g =8\pi\mu u^{-2}\xi^\flat\otimes\xi^\flat,\qquad \mu(u^2-|X|^2)=0.\] Since the matter term is tracefree, this is exactly (271). It allows nonzero matter on the null set; its exclusion is a separate argument in the next section.

It remains to determine the boundary data. There are no boundary constraint atoms by Lemma 92, and the integration normal at \(S\) is \(-\nu\). A compact pure tensor test \(P\), arbitrary at the boundary, has integrated boundary term \[-2\int_S\big(P(X,\nu)-X^\nu\mathop{\mathrm{tr}}P\big)\,dA_g.\] Its expansion derivative is \(\mathop{\mathrm{tr}}_SP\). Separating mixed and tangential-trace components in (278) gives \[X_T=0,\qquad d\eta=2X^\nu\,dA_g.\] For a compact simultaneous conformal test \((4\psi g,2\psi K)\), its area derivative is \(4\int_S\psi\,dA_g\) and its expansion derivative is \(4\partial_\nu\psi\). Integrating the lapse equation gives \[-4c\int_S\psi\,dA_g =8\int_S\big[u\partial_\nu\psi -(\partial_\nu u+K(X,\nu))\psi\big]\,dA_g -4\int_S\partial_\nu\psi\,d\eta .\] The boundary value and normal derivative of \(\psi\) can be prescribed independently. They give \(d\eta=2u\,dA_g\), \(X=u\nu\), and \(\partial_\nu u+K(X,\nu)=c/2=\kappa_h\).

Finally let \(L_{\rm MOTS}\) be the normal expansion linearization at \(S\), with principal part \(-\Delta_S\). Extend any \(s\nu\), \(s\in C^\infty(S)\), to a compact one-sided field \(Y\) and test with \((\mathcal L_Yg,\mathcal L_YK)\). In the open exterior its active-ray derivative is zero: under diffeomorphism transport it differentiates a nonnegative scalar at a zero, and the difference from isometric transport is annihilated at activity. No boundary constraint measure remains. The mass derivative is zero, while the area and expansion derivatives are \(\int_SHs\,dA_g\) and \(L_{\rm MOTS}s\). The multiplier identity therefore gives \[c\int_SHs\,dA_g=2\int_SuL_{\rm MOTS}s\,dA_g,\qquad 2L_{\rm MOTS}^*u=cH.\] Outer area minimization gives \(H\geq0\), by the first area variation for arbitrary nonnegative outward speeds. The nonnegative function \(u|_S\) is consequently a supersolution of an elliptic operator with principal part \(-\Delta_S\). The strong minimum principle applies: after reversing the operator and subtracting a sufficiently large zeroth-order constant, the usual nonpositive zeroth-order condition holds because \(u\geq0\). On connected \(S\), either \(u>0\) everywhere or \(u=0\) everywhere. This completes all assertions. ◻

Proof. Since \(X=u\nu\) on \(S\), the function \(N\) vanishes there. The normal component of Equation (288) gives \(\langle\nabla_\nu X,\nu\rangle=-uK(\nu,\nu)\), and hence \[\partial_\nu N =2u\partial_\nu u-2u\langle\nabla_\nu X,\nu\rangle =2u\bigl(\partial_\nu u+K(X,\nu)\bigr)=2u\kappa_h.\] All tangential derivatives of \(N\) vanish on \(S\). This proves Equation (298) and the positive-lapse collar assertion. If \(u=0\) on \(S\), then \(X=0\) there and its tangential covariant derivatives vanish. The normal-normal and normal-tangential components of \(\operatorname{sym}\nabla X=0\) give \(\nabla_\nu X=0\) on \(S\). Smooth division by \(s\) and by \(s^2\) gives Equation (299); the boundary lapse equation gives \(a|_S=\kappa_h\). Compactness makes the collars uniform. Finally, Equation (289) gives \(N=V_0^2(1+O_2(r^{-\tau'}))\), so the entire sufficiently distant end lies in \(P\). The remaining zero set is a closed subset of a compact set separated from \(S\). ◻

Proof. The Killing equation makes \(D\) antisymmetric. Killing differentiation gives \(\Box_{\mathbf g}\xi_\beta=-\mathop{\mathrm{Ric}}_{\mathbf g\,\beta\gamma}\xi^\gamma\). Since \(\mathbf g(\xi,\xi)=-N\), it follows that \[\Box_{\mathbf g}N =-2|D|_{\mathbf g}^2+2\mathop{\mathrm{Ric}}_{\mathbf g}(\xi,\xi) =-2|D|_{\mathbf g}^2+6N.\] In the last equality the additional term in Equation (291) vanishes by \(\mu N=0\). For a stationary scalar the spatial block of \(\mathbf g^{-1}\) is \(C\), and the spacetime volume density is \(u\sqrt{\det g}\). This proves Equation (300).

For the second identity, write \(\vartheta=\,\mathrm dt-\mathcal A\). The Killing one-form is \(\xi^\flat=-\lambda^2\vartheta\), so \(\xi^\flat\wedge\,\mathrm d\xi^\flat=-\lambda^4\vartheta\wedge\mathcal F\). The differentiated Killing equation, contracted with the volume form, gives \[\,\mathrm d*_{\mathbf g}(\xi^\flat\wedge\,\mathrm d\xi^\flat) =2*_{\mathbf g}\bigl(\xi^\flat\wedge\mathop{\mathrm{Ric}}_{\mathbf g}(\xi,\cdot)\bigr) =0.\] The right side vanishes because \(\mathop{\mathrm{Ric}}_{\mathbf g}(\xi,\cdot)\) is proportional to \(\xi^\flat\). Evaluating the Hodge star using the unit time form \(\lambda\vartheta\) turns the left side, up to a fixed orientation sign, into \(\,\mathrm d(\lambda^3 *_h\mathcal F)\). Thus \[\operatorname{div}_h\bigl(\lambda^3 C^{ik}C^{j\ell} \mathcal F_{k\ell}\bigr)=0.\] Use \(\lambda^3\,\mathrm dV_h=uN\,\mathrm dV_g\) to change the divergence density. The connection term on the free index of an antisymmetric two-tensor vanishes for either torsion-free connection. The resulting equation is exactly \(\nabla_iQ^{ij}=0\). ◻

Proof. On a sufficiently small fixed neighborhood of the compact set \(Z\), both \(u\) and \(B\) are bounded below by positive constants. The tensor \(C\) acts as the identity on \(T\) and has eigenvalue \(N/u^2\) in direction \(e\). Smooth nonnegativity of \(N\) implies \[ |\,\mathrm dN|_g^2\leq C_0N \tag{304}\] on compact subsets after slightly enlarging the neighborhood. Indeed, choose a uniform coordinate radius and a bound \(L\) for \(\mathop{\mathrm{Hess}}_gN\) large enough that \(|\,\mathrm dN|/L\) is less than that radius. Taylor’s Inequality along the negative gradient geodesic at length \(|\,\mathrm dN|/L\) yields \(0\leq N-|\,\mathrm dN|^2/(2L)\).

Put \(n=(\partial_t-X)/u\), and complete \(e\) to a spatial orthonormal frame \(e,E_1,E_2\). Since \(\xi=un+Be\), differentiation of its norm gives \[uD_{in}=-\tfrac12N_i-BD_{ie}.\] Writing \(v_a=D_{ae}\), expansion of the temporal and spatial contractions therefore gives the exact formula \[ -2|D|_{\mathbf g}^2 =\frac{|\,\mathrm dN|_g^2}{u^2} +\frac{4B}{u^2}\sum_{a=1}^2N_av_a -\frac{4N}{u^2}\sum_{a=1}^2v_a^2-4D_{12}^2. \tag{305}\] All coefficients and the components of \(D\) are bounded on the fixed neighborhood. Equations (300) and (304) consequently imply \[ \frac1u\operatorname{div}_g(uC\,\,\mathrm dN) \leq C_1(N+|\,\mathrm d_TN|_g). \tag{306}\] The additional cosmological term \(6N\) has the harmless first order shown.

Multiply Equation (306) by \(u\eta^2/(N+\varepsilon)\) and integrate. Writing \(E_\varepsilon\) for the integral in Equation (302), before taking the supremum, integration by parts gives \[E_\varepsilon -2\int_M\frac{u\eta C(\,\mathrm d\eta,\,\mathrm dN)}{N+\varepsilon}\,\mathrm dV_g \leq C_1\int_Mu\eta^2\,\mathrm dV_g +C_1\int_M\frac{u\eta^2|\,\mathrm d_TN|}{N+\varepsilon}\,\mathrm dV_g.\] Cauchy’s Inequality bounds the cutoff term by \(E_\varepsilon/4+C\int u|\,\mathrm d\eta|_C^2\), and the last term by \(E_\varepsilon/4+C\int u\eta^2\), because \(|\,\mathrm d_TN|\leq|\,\mathrm dN|_C\). This proves the uniform bound. Fatou’s Lemma on \(P\), together with the positive lower bound for \(u\), proves Equation (303). ◻

Proof. Lower the free index of \(Q^{kj}\mathcal A_j\) using \(g\). Direct substitution of \[\mathcal F=N^{-1}\,\mathrm dX^\flat-N^{-2}\,\mathrm dN\wedge X^\flat, \qquad C X^\flat=(N/u^2)X\] gives \[ W_i:=g_{ik}Q^{kj}\mathcal A_j =u^{-1}\left\{X^j(\,\mathrm dX^\flat)_{ij} -B^2(\,\mathrm d_T\log N)_i\right\}. \tag{307}\] This expression is orthogonal to \(X\). At a zero of \(X\) its second numerator is interpreted as \(B^2\,\mathrm dN-\,\mathrm dN(X)X^\flat\), so the identity does not require a choice of \(e\) away from the neighborhood used above.

Let \(\chi\) be a product of three nonnegative cutoffs, excluding first \(Z\), then the original boundary, then infinity. Testing \(\nabla_iQ^{ij}=0\) by \(\chi\mathcal A_j\) gives \[ \frac12\int_P\chi uN C^{ik}C^{j\ell} \mathcal F_{ij}\mathcal F_{k\ell}\,\mathrm dV_g =-\int_P\langle W,\,\mathrm d\chi\rangle_g\,\mathrm dV_g. \tag{308}\] The integrand on the left is nonnegative.

Keep the boundary and infinity cutoffs fixed. Let the first cutoff change from zero to one on \(\varepsilon<N<2\varepsilon\), with derivative bounded by \(C/\varepsilon\). Orthogonality in Equation (307) makes its error bounded by \[C\int_{\{\varepsilon<N<2\varepsilon\}} \bigl(|\,\mathrm d_T\log N|+|\,\mathrm d_T\log N|^2\bigr)\,\mathrm dV_g\] on a fixed compact set. This tends to zero by Lemma 96: the shell indicators tend pointwise to zero on \(P\) and are dominated by an integrable function there. The argument also applies if \(Z\) has positive volume.

For the boundary cutoff use the normal coordinate \(s\). If \(u|_S>0\), Equation (298) gives \(N\asymp s\), \(X=u|_S\nu+O(s)\), and \(\,\mathrm d_TN=O(s)\). Thus \(W\) is bounded. Its orthogonality to \(X\) implies \(W(\nu)=O(s)\), so a cutoff with derivative \(O(\delta^{-1})\) on \(\delta<s<2\delta\) has error \(O(\delta)\). If \(u|_S=0\), Equation (299) gives \[u\asymp s,\quad X=O(s^2),\quad \,\mathrm dX^\flat=O(s), \quad N\asymp s^2,\quad \,\mathrm dN=O(s),\] and Equation (307) gives \(W=O(s^2)\), with the same conclusion.

Finally, Equation (289) gives \(W=O(r^{1-2\tau'})\) on the end. A cutoff on \(R<r<2R\) has bounded hyperbolic gradient and the shell has volume \(O(R^2)\). Its error is \(O(R^{3-2\tau'})\), which vanishes since \(\tau'>3/2\). Successively removing these cutoffs and using Fatou’s Lemma in Equation (308) makes its nonnegative integrand vanish everywhere. Since \(uN>0\) and \(C\) is positive definite on \(P\), one has \(\mathcal F=0\) there. ◻

Proof. Locally write \(\mathcal A=\,\mathrm df\) and \(T_0=t-f\). Then \(\mathbf g=h-\lambda^2\,\mathrm dT_0^2\). Equation (291) is vacuum on \(P\). The only mixed-time Christoffel symbols of this static product are \[\Gamma^{T_0}_{iT_0}=\lambda^{-1}\partial_i\lambda, \qquad \Gamma^i_{T_0T_0}=\lambda h^{ij}\partial_j\lambda.\] Thus its spatial Ricci tensor is \(\mathop{\mathrm{Ric}}_h-\lambda^{-1}\mathop{\mathrm{Hess}}_h\lambda\), and its time-time component is \(\lambda\Delta_h\lambda\). Comparing with \(-3\mathbf g\) gives the first two equations in Equation (293); tracing gives \(R_h=-6\). These equations are intrinsic and hence hold globally on \(P\).

The end belongs to a single component of \(P\). Any other component has compact closure in the original manifold with boundary. The continuous function \(\lambda\) is zero on every finite boundary point of that component, including a possible point on \(S\). It therefore attains a positive maximum at an interior point, contrary to \(\Delta_h\lambda=3\lambda\). There is no other component. ◻

Proof. Near \(Z\), smooth nonnegativity and compactness give \(N\leq C\operatorname{dist}_g(\cdot,Z)^2\). If \(\rho=\operatorname{dist}_g(\cdot,Z)\), then \(|\,\mathrm d\rho|_g\leq1\) almost everywhere and \[|\,\mathrm d\log\rho|_k^2 =N C(\,\mathrm d\log\rho,\,\mathrm d\log\rho) \leq N\rho^{-2}\leq C.\] Hence approach to \(Z\) has infinite optical length. In the zero boundary lapse case the same calculation uses \(s\) and \(N=O(s^2)\).

For positive boundary lapse it is essential to allow arbitrary tangential motion. In the \(g\)-normal collar decompose \(X=X^\nu\nu+X_T\). Since \(|\,\mathrm ds|_g=1\), the exact inverse-metric formula is \[ |\,\mathrm ds|_k^2 =N\left(1-\frac{(X^\nu)^2}{u^2}\right) =\frac{N}{u^2}\bigl(N+|X_T|_g^2\bigr)=O(s^2). \tag{309}\] Here \(N\asymp s\), \(X_T=O(s)\), and \(u\) is bounded below. Therefore \(|\,\mathrm ds(v)|\leq Cs|v|_k\) for every vector \(v\), and any curve approaching \(S\) has infinite optical length. This proves the assertion without restricting to normal curves.

After truncation, \(N\) is bounded above, so \(k\geq c g\). Any optical Cauchy sequence is consequently Cauchy for \(g\) and has a limit in the compact original region with the large sphere attached. The logarithmic estimates exclude a limit in \(Z\) or in \(S\). Every other limit is an ordinary smooth point of the truncated metric, including its large-sphere boundary. This proves completeness. A smooth manifold with boundary and its intrinsic Riemannian distance is a locally compact length space; completeness makes it proper. For a covering metric completeness follows by projecting a Cauchy sequence and then taking an evenly covered small neighborhood of its limit. Sufficiently short connecting paths remain in that neighborhood, so the sequence eventually lies in a single sheet and converges there. ◻

Proof. Suppose such a path has optical length \(\ell>0\). In the abstract static product \((\mathbb R\times L,\widehat{\mathbf g})\), where \(\widehat{\mathbf g}=-\lambda^2\,\mathrm dT^2+h\), its lift with increasing \(T\) from \(0\) to \(\ell\) is a null geodesic up to reparametrization. One can see this directly from the null energy relation \(\lambda^2\dot T=E>0\): optical arclength and affine parameter are related by \(\,\mathrm ds/\,\mathrm dT=\lambda^2/E\), and the spatial geodesic equation becomes the optical geodesic equation. Use an affine parameter on a compact interval \([a,b]\) and write its tangent as \(k_0\).

This null curve has no future timelike competitor with endpoints in the fixed-time surfaces \(\{0\}\times\Sigma_0\) and \(\{\ell\}\times\Sigma_1\). Indeed every such competitor satisfies \[\ell=\int_a^b\dot T\,\,\mathrm ds >\int_a^b\frac{|\dot x|_h}{\lambda}\,\,\mathrm ds \geq\operatorname{dist}_{\lambda^{-2}h}(\Sigma_0,\Sigma_1)=\ell.\] We produce a competitor by second variation.

Choose a parallel orthonormal screen pair \(E_1,E_2\) along the null geodesic, with \(\langle E_A,k_0\rangle=0\), initially tangent to \(\{0\}\times\Sigma_0\). At the terminal endpoint, add suitable multiples of \(k_0\) to remove their time components. Orthogonality to \(k_0\) then places them in \(T\Sigma_1\). Extending these multiples smoothly along the curve gives fields \(V_A=E_A+b_Ak_0\) tangent to the endpoint surfaces, still orthonormal, with \(\langle\nabla_{k_0}V_A,\nabla_{k_0}V_A\rangle=0\). Let \(H_0,H_1>0\) denote the specified mean curvatures. The spatial component of \(k_0\) is \(-c_0\nu_0\) initially and \(c_1\nu_1\) terminally, for positive \(c_0,c_1\). The static time slices have zero second fundamental form. The endpoint terms in the energy second variation therefore have trace \(-c_0H_0-c_1H_1\).

More explicitly, choose variations with initial and terminal curves in the fixed-time surfaces and first variational fields \(V_A\). The second variation of \(\frac12\int\langle\dot\gamma,\dot\gamma\rangle\) is \[I(V_A,V_A)= \int_a^b\left( |\nabla_{k_0}V_A|^2 -\langle R(V_A,k_0)k_0,V_A\rangle\right)\,\mathrm ds +\left[\langle\nabla_{V_A}V_A,k_0\rangle\right]_a^b.\] At an endpoint, the surface acceleration has normal component \(-\mathrm{II}_{\nu_j}(V_A,V_A)\). This proves the stated signs. Generator multiples do not change the curvature term, and summing the screen curvatures gives \(\mathop{\mathrm{Ric}}_{\widehat{\mathbf g}}(k_0,k_0)=0\). Consequently \[ \sum_{A=1}^2I(V_A,V_A)=-c_0H_0-c_1H_1<0. \tag{310}\] Choose one of these fields with negative index and a smooth variation \(\gamma_\varepsilon\) realizing it. The first variation of squared speed vanishes pointwise, because \(\langle\nabla_{k_0}V,k_0\rangle=0\). Put \(q(s)=\partial_\varepsilon^2|\dot\gamma_\varepsilon|^2|_{\varepsilon=0}\); its integral is \(2I(V,V)<0\). Let \(\overline q\) be its negative average. Choose a parallel auxiliary null vector \(L_0\) with \(\langle L_0,k_0\rangle=-1\), and add to the variation a second-order displacement \(\frac12\varepsilon^2 f(s)L_0\), where \[f(a)=0,\qquad f'(s)=\tfrac12(q(s)-\overline q).\] Then \(f(b)=0\), so the endpoints are unchanged; the second variation of squared speed changes by \(-2f'\) and becomes the constant \(\overline q<0\). Uniform Taylor expansion on \([a,b]\) now makes the varied curve timelike for all sufficiently small nonzero \(\varepsilon\). It remains future directed because \(\dot T>0\) on the original compact segment. Its endpoint contacts remain transverse and directed into and out of \(L\); away from small endpoint intervals the original curve lies a positive distance from the boundary. Thus the variation remains in \(L\) and is the forbidden timelike competitor. ◻

Proof. Suppose its universal cover is nontrivial. The ordinary end is a product of a sphere and a ray and is simply connected. Its inverse image therefore consists of at least two disjoint copies. Truncate every copy at the same large radius \(R\). Their boundary faces are compact spheres. By Equation (289) and the formula for \(h\), their mean curvatures for the normals pointing into the removed tails satisfy \[H_h=\frac{2\sqrt{1+R^2}}{R}+o(1)>0.\] The truncated cover is connected: attaching or removing a product tail cannot join different components of its complement. By Lemma 99 its intrinsic optical metric is complete and proper.

Fix one face. The union of all the other faces is a closed set, since it is the inverse image of the truncation sphere with the fixed component removed: local finiteness in evenly covered neighborhoods makes each component open as well as closed within that inverse image. The remaining union is disjoint from the fixed compact face. Properness implies that the positive distance between that face and the union is attained. A minimizing path has no intermediate contact with a face: a later contact with the initial face or an earlier contact with any other face would shorten the selected distance. To check its endpoint directions without assuming optical convexity, let \(\rho\) be the inward \(k\)-distance to the initial face in its smooth normal collar, and parametrize the path by unit optical speed from \(t=0\). The bound \(|d\rho|_k=1\) gives \(\rho(\gamma(t))\le t\); replacing the initial segment by a normal collar segment gives the reverse inequality by minimality. Hence \(\rho(\gamma(t))=t\) and \(\dot\gamma=\nabla_k\rho\) near the endpoint. Apply the same argument backwards at the final face. The path is therefore an interior geodesic with transverse orthogonal endpoint contacts. The static equations lift to the cover, so Lemma 100 gives a contradiction. ◻

Proof. The compactness of \(Z\) and positivity of \(u\) give \(X\ne0\) on a neighborhood of \(Z\). On \(P\), closedness of \(\mathcal A\) gives \(X^\flat\wedge\,\mathrm dX^\flat=0\). This extends by continuity to the relative boundary of \(P\). In an open region where \(N=0\), Equation (305) and Equation (300) give \(D_{12}=0\). Since the spatial restriction of \(\,\mathrm d\xi^\flat\) is \(\,\mathrm dX^\flat=2D_{ij}\,\mathrm dx^i\otimes\,\mathrm dx^j\) in components, the restriction of \(\,\mathrm dX^\flat\) to \(X^\perp\) vanishes there also. These cases exhaust the neighborhood, proving Frobenius integrability.

Choose the transverse coordinate oriented so that \(X^\flat=B_1\,\mathrm dy\) with \(B_1>0\), and transport coordinates within the leaves orthogonally using \(g\). This gives the metric decomposition in Equation (311). On the positive set, \(\,\mathrm d(B_1\,\mathrm dy/N)=0\) says that every leaf derivative of \(N/B_1\) vanishes. The same derivatives vanish on the interior of the zero set and, by continuity, on its remaining boundary. Thus \(N/B_1=a(y)\) throughout the coordinate neighborhood. Nonnegativity gives \(a=a'=0\) at a zero. A boundary leaf is approached by leaves with \(a>0\).

We compute the limit of the static equations from those positive leaves. On them, \[h=q_y+P_1^2\,\mathrm dy^2,\qquad P_1=\left(D_1^2+\frac{B_1}{a}\right)^{1/2}, \qquad \lambda=b_1\sqrt a.\] The second fundamental form of a leaf in \(h\) is \(k_{AB}=(2P_1)^{-1}\partial_yq_{AB}=O(\sqrt a)\), and \(P_1^{-1}\partial_yk_{AB}=O(a+|a'|)\). The tangential Ricci formula is therefore \[(\mathop{\mathrm{Ric}}_h)_{AB} =\mathcal K_y q_{AB}-P_1^{-1}(\mathop{\mathrm{Hess}}_{q_y}P_1)_{AB} +O(a+|a'|) \longrightarrow \mathcal K_y q_{AB}-b_1^{-1}(\mathop{\mathrm{Hess}}_{q_y}b_1)_{AB}.\] The tangential normalized Hessian has the limit \[\lambda^{-1}(\mathop{\mathrm{Hess}}_h\lambda)_{AB} =b_1^{-1}(\mathop{\mathrm{Hess}}_{q_y}b_1)_{AB}+O(a+|a'|) \longrightarrow b_1^{-1}(\mathop{\mathrm{Hess}}_{q_y}b_1)_{AB}.\] These estimates follow directly by inserting \(P_1=b_1a^{-1/2} (1+aD_1^2/B_1)^{1/2}\); all coefficients in their error terms are bounded smooth functions on a fixed coordinate patch. The tangential static Ricci equation gives the first equation in Equation (312).

For the scalar equation let \(v=\sqrt{\det q_y}\), \(f=\log b_1\), and \(w=(1+aD_1^2/B_1)^{1/2}\). The normal divergence contribution to \(\lambda^{-1}\Delta_h\lambda\) is exactly \[ \frac1{b_1^2wv}\partial_y \left[vw^{-1}\left(af_y+\frac{a'}2\right)\right]. \tag{313}\] At \(a=a'=0\) it tends to \(a''/(2b_1^2)\). The tangential contribution tends to \(\Delta_{q_y}b_1/b_1+|\,\mathrm db_1|_{q_y}^2/b_1^2\). Hence \(\Delta_h\lambda=3\lambda\) gives \[b_1\Delta_{q_y}b_1+|\,\mathrm db_1|_{q_y}^2+\frac{a''}2=3b_1^2,\] which is the second equation in Equation (312). ◻

Proof. An invariant transverse line. Write \(f=\phi_1\) and use the smooth complementary vector \[\nu_p=\frac{\alpha_p^\sharp}{\lvert\alpha_p\rvert_g^2}, \qquad \alpha_p(\nu_p)=1.\] Relative to \(T_pM=E_p\oplus\mathbb R\nu_p\), the forward derivative has the block form \[ Df_p= \begin{pmatrix} A_p&B_p\\0&e^2\end{pmatrix}, \qquad \lVert A_p\rVert\leq e. \tag{316}\] Here \(B_p\in E_{f(p)}\) and \(\sup_{\mathcal K}\lvert B_p\rvert_g<\infty\). Fix a full orbit \(p_n=f^n(p)\), \(n\in\mathbb Z\). A transverse line at \(p_n\) with generator \(\nu_{p_n}+v_n\), \(v_n\in E_{p_n}\), is invariant precisely when its slopes satisfy \[ v_{n+1}=e^{-2}(A_{p_n}v_n+B_{p_n}). \tag{317}\] Each affine map in this recurrence contracts differences by at most \(e^{-1}\). Starting with slope zero at \(p_{n-k}\) and letting \(k\to\infty\) therefore gives a bounded solution at \(p_n\); the solutions at different indices obey the same recurrence. Equivalently, one may sum the absolutely convergent series obtained by repeated substitution. Its bound is \[ \sup_{n\in\mathbb Z}\lvert v_n\rvert_g \leq\frac{\sup_{\mathcal K}\lvert B\rvert_g}{e^2-e}. \tag{318}\] The bounded solution is unique: the difference of two such solutions at index \(n\) is at most \(e^{-k}\) times their uniformly bounded difference at index \(n-k\).

Set \(u_n=\nu_{p_n}+v_n\) and \(U_n=\mathbb R u_n\). Then \[\alpha_{p_n}(u_n)=1,\qquad Df_{p_n}u_n=e^2u_{n+1}.\] Compactness, nonvanishing of \(\alpha\), and Equation (318) give uniform equivalence between the ambient norm and the norm \[\lvert(s,w)\rvert_n=\max\{\lvert s\rvert,\lvert w\rvert_g\}, \qquad su_n+w\in U_n\oplus E_{p_n}.\] In particular these transverse lines have angles uniformly bounded away from zero relative to \(E\).

Uniform charts and backward products. Put \(q_j=p_{-j}\) for \(j\geq0\). Choose centered charts \[\Psi_j(s,w)=\exp_{q_j}(s u_{-j}+w),\qquad w\in E_{q_j}.\] They and their inverses have uniformly bounded derivatives on suitable fixed radii. This follows from compactness and the uniformly bounded linear coordinate changes just constructed. No derivatives of \(U_n\) as a function of \(p_n\) are used. On a fixed smaller radius \(\rho>0\), the backward time-one maps in these charts have the form \[ \Psi_{j+1}^{-1}\circ\phi_{-1}\circ\Psi_j(z) =\begin{pmatrix}c&0\\0&T_j\end{pmatrix}z+R_j(z), \qquad c=e^{-2}, \tag{319}\] where \(T_j=D\phi_{-1}|_{E_{q_j}}\) maps \(E_{q_j}\) onto \(E_{q_{j+1}}\). The remainders satisfy, in the displayed norms, \[ R_j(0)=0,\qquad DR_j(0)=0,\qquad \sup_{j\geq0,\,\lvert z\rvert\leq\rho} \lVert D^2R_j(z)\rVert\leq M_0 \tag{320}\] for some finite \(M_0\). We increase \(M_0\) to a positive number if necessary. Equation (314) gives \[ \bigl\lVert T_j^{-1}T_{j+1}^{-1}\cdots T_i^{-1}\bigr\rVert \leq e^{i-j+1}\qquad(i\geq j). \tag{321}\] Indeed that product is the forward leaf derivative from \(q_{i+1}\) to \(q_j\).

The two Green sums. Fix \(\beta=3/2\) and consider the Banach space of sequences \(z_j=(s_j,w_j)\in\mathbb R\oplus E_{q_j}\) with norm \[\lVert z\rVert_\beta =\sup_{j\geq0}e^{\beta j}\max\{\lvert s_j\rvert,\lvert w_j\rvert_g\}.\] For a forcing sequence \(r_j=(r_j^s,r_j^E)\) taking values in \(\mathbb R\oplus E_{q_{j+1}}\), use the analogous norm with weight \(e^{\beta j}\). Define \[\begin{align*} (Gr)^s_j &=\sum_{i=0}^{j-1}c^{j-1-i}r_i^s,\tag{322}\\ (Gr)^E_j &=-\sum_{i=j}^{\infty} T_j^{-1}T_{j+1}^{-1}\cdots T_i^{-1}r_i^E. \end{align*}\] The first sum is empty when \(j=0\). Both sums converge in the indicated weighted norm. More generally, for \(1<\beta<2\), \[ \lVert G\rVert\leq\Gamma_\beta, \qquad \Gamma_\beta=\max\left\{ \frac{e^\beta}{1-e^{\beta-2}},\, \frac{e}{1-e^{1-\beta}}\right\}. \tag{323}\] For the first sum, setting \(k=j-1-i\) bounds its weighted norm by \(e^\beta\sum_{k\geq0}e^{(\beta-2)k}\lVert r\rVert_\beta\). For the second, Equation (321) gives the bound \(e\sum_{k\geq0}e^{(1-\beta)k}\lVert r\rVert_\beta\).

Prescribe a small initial transverse coordinate \(\xi\in\mathbb R\) and put \(\mathcal H(\xi)_j=(c^j\xi,0)\). The desired orbit sequence is a fixed point of \[ z=\mathcal H(\xi)+G\mathcal R(z),\qquad \mathcal R(z)_j=R_j(z_j). \tag{324}\] Choose \(0<\delta<\rho\) so small that \[ \vartheta_0:=\Gamma_\beta M_0\delta\leq\tfrac12, \qquad \lvert\xi\rvert\leq\tfrac12\delta. \tag{325}\] On the closed ball \(\lVert z\rVert_\beta\leq\delta\), Equation (320) yields \[\lVert\mathcal R(z)\rVert_\beta\leq\tfrac12 M_0\delta^2, \qquad \lVert\mathcal R(z)-\mathcal R(\widetilde z)\rVert_\beta \leq M_0\delta\lVert z-\widetilde z\rVert_\beta.\] Consequently the right side of Equation (324) maps that ball into its ball of radius \(3\delta/4\) and contracts distances by at most \(\vartheta_0\). The contraction principle gives a unique fixed point there, with \[ \lVert z(\xi)\rVert_\beta \leq\frac{\lvert\xi\rvert}{1-\vartheta_0}\leq2\lvert\xi\rvert. \tag{326}\] Substituting the sums in Equation (322) shows that this fixed point satisfies Equation (319) at every step, with \(s_0=\xi\). It therefore represents an actual backward orbit, rather than a sequence of approximate orbit points.

Dependence on the initial coordinate. The same contraction estimate gives \[\lVert z(\xi+h)-z(\xi)\rVert_\beta \leq\frac{\lvert h\rvert}{1-\vartheta_0}.\] The map \(\mathcal R\) is continuously differentiable on this sequence ball: its derivative acts pointwise by \(DR_j(z_j)\), and the uniform second-derivative bound in Equation (320) bounds the remainder in the weighted norm. In particular \(\lVert G D\mathcal R(z(\xi))\rVert\leq \vartheta_0\). Inverting by a Neumann series gives \[ Dz(\xi)h =\bigl(I-GD\mathcal R(z(\xi))\bigr)^{-1}\mathcal H(h). \tag{327}\] For clarity, this is a genuine Fréchet derivative: Taylor’s estimate and the preceding Lipschitz bound give \[\bigl\lVert z(\xi+h)-z(\xi)-Dz(\xi)h\bigr\rVert_\beta \leq\frac{\Gamma_\beta M_0}{2(1-\vartheta_0)^3}\lvert h\rvert^2.\] The derivative is continuous by the same uniform estimates. At \(\xi=0\), the fixed point is zero and \(D\mathcal R(0)=0\), so \(Dz(0)h=((c^jh,0))_{j\geq0}\).

Define \(\gamma_p(\xi)=\Psi_0(z_0(\xi))\). Its initial derivative is \(u_0\), hence \(\alpha_p(\gamma_p'(0))=1\). Since \(s_0=\xi\), it is an embedded graph in this chart on a smaller interval. Uniform comparison between chart and ambient distances, together with Equation (326), proves Equation (315) at integer times. The smooth flow has uniformly bounded derivatives near \(\mathcal K\) for times in \([-1,0]\); decreasing the chart radius if necessary gives the same estimate for all intervening times. Compactness makes all constants uniform in \(p\). Shrinking \(\epsilon\) then keeps the whole backward arc in any prescribed neighborhood of \(\mathcal K\). ◻

Proof. The intrinsic vector field and the compact flat set. On a foliated chart define \[f=\log b,\qquad Y=\nabla_\Sigma f,\qquad L=\lvert Y\rvert_g^2.\] On an overlap of cooriented charts the transverse coordinates satisfy \(\widetilde y=\vartheta(y)\) with \(\vartheta'>0\) after restriction to an overlap plaque. Consequently \[\widetilde B=\frac{B}{\vartheta'(y)},\qquad \log\sqrt{\widetilde B} =\log\sqrt B-\tfrac12\log\vartheta'(y).\] The last term has zero leaf derivative. Thus \(Y\) and \(L\) are globally defined smooth fields on \(U\), even though \(f\) need not be a global function. The vector field \(Y\) is tangent to the foliation.

At a zero \(y_0\) of \(a\), nonnegativity gives \(a'(y_0)=0\), and direct differentiation of Equation (328) gives \[ \mathop{\mathrm{Hess}}_gN=B\,a''(y_0)\,\mathrm dy\otimes\mathrm dy \quad\hbox{on the plaque }\{y=y_0\}. \tag{330}\] In particular flatness, meaning \(a''(y_0)=0\), is independent of the chosen transverse coordinate. The boundary property is also constant along a plaque: \(N>0\) on that chart exactly where \(a(y)>0\). The set \[ F=\partial P\cap\{\mathop{\mathrm{Hess}}_gN=0\} \tag{331}\] is therefore closed in \(Z\), compact, and locally a union of whole plaques. It is preserved by the local flow of \(Y\).

Choose a neighborhood \(U_0\) of \(Z\) with compact closure in \(U\) and multiply \(Y\) by a smooth compactly supported cutoff equal to one near \(\overline{U_0}\). Extend the resulting field by zero to \(M\) and denote its complete flow by \(\phi_t\). Compact support ensures completeness. On \(F\) this flow agrees with the original leafwise flow for all times: local plaque invariance and closedness of \(F\) show that \(\{t:\phi_t(p)\in F\}\) is both open and closed in \(\mathbb R\) for every \(p\in F\). In all subsequent uses outside \(F\) we keep the flow inside \(U_0\), where the cutoff equals one.

The logistic identity. On a flat plaque, tracing the first equation in Equation (329) gives \(\Delta_\Sigma b=b(K_\Sigma+3)\). The second then gives \[6b^2=\Delta_\Sigma(b^2) =2b^2(K_\Sigma+3)+2\lvert\mathrm d_\Sigma b\rvert_q^2.\] It follows that \[ K_\Sigma=-L,\qquad \mathop{\mathrm{Hess}}_\Sigma f=\frac{3-L}{2}q- \mathrm d_\Sigma f\otimes\mathrm d_\Sigma f, \qquad Y(L)=3L(1-L). \tag{332}\] For the last identity, use \(Y(L)=2\mathop{\mathrm{Hess}}_\Sigma f(Y,Y)\).

Since \(F\) is compact and invariant, \(L\) is bounded on every complete orbit in \(F\). A logistic solution with \(L(0)>1\) has the explicit form \[L(t)=\frac{L(0)e^{3t}}{1-L(0)+L(0)e^{3t}}\] and blows up in finite negative time. Thus \(0\leq L\leq1\) on \(F\). Suppose for contradiction that \(F\) is nonempty. On any flat plaque \(L\) cannot vanish identically: that would give \(Y=0\) and \(\mathop{\mathrm{Hess}}_\Sigma f=0\), whereas Equation (332) would give \(\mathop{\mathrm{Hess}}_\Sigma f=\tfrac32 q\). Hence some point of \(F\) has \(L>0\). The forward logistic orbit through this point has \(L(t)\to1\). Compactness supplies an accumulation point with \(L=1\), so \[ F_1=F\cap\{L=1\} \tag{333}\] is a nonempty compact set invariant under the full flow.

Leaf and transverse derivative rates. Along \(F_1\), Equation (332) becomes \[ \mathop{\mathrm{Hess}}_\Sigma f=q-\mathrm d_\Sigma f\otimes\mathrm d_\Sigma f, \qquad \lvert Y\rvert_g=1. \tag{334}\] For a leaf-tangent vector \(v(t)=D\phi_t(p)v(0)\), \(p\in F_1\), the usual derivative of its squared length within the leaf is \[\frac{\mathrm d}{\mathrm dt}\lvert v(t)\rvert_g^2 =2\mathop{\mathrm{Hess}}_\Sigma f(v(t),v(t)) =2\bigl(\lvert v(t)\rvert_g^2- \langle Y,v(t)\rangle_g^2\bigr).\] This lies between zero and \(2\lvert v(t)\rvert_g^2\). Integration gives the two tangent inequalities in Equation (314). Furthermore, throughout \(U_0\), tangency of \(Y\) implies \(Y(y)=0\) and \[ \mathcal L_Y\alpha=Y(B)\,\mathrm dy=2L\alpha, \qquad Y(N)=2LN. \tag{335}\] Here \(Y(\log B)=2Y(f)=2L\); the second identity also holds where \(N=0\), by the smooth factorization. Since \(L=1\) along \(F_1\), the first identity gives \(\alpha_{\phi_t(p)}\circ D\phi_t(p)=e^{2t}\alpha_p\). All hypotheses of Lemma 103 therefore hold with \(\mathcal K=F_1\) and \(E=\ker\alpha\).

The two incompatible decay bounds. Fix \(p\in F_1\) and apply Lemma 103. By compactness and continuity choose a neighborhood \(V\) of \(F_1\), with closure in \(U_0\), on which \(L\leq5/4\). Shrink the resulting transverse arc so that all its backward trajectories remain in \(V\). If one of its points \(q=\gamma_p(\xi)\) had \(N(q)>0\), the second equation in Equation (335) would give, for every \(t\geq0\), \[ N(\phi_{-t}(q)) =N(q)\exp\left(-\int_0^t2L(\phi_{-s}(q))\,\mathrm ds\right) \geq N(q)e^{-5t/2}. \tag{336}\] In particular the trajectory remains positive at each finite time.

On the other hand, \(N\) and \(\mathrm dN\) vanish at every point of \(F_1\), because \(N\) is smooth and nonnegative. Its Hessian is uniformly bounded on a compact neighborhood of \(F_1\). Taylor’s Theorem in uniform normal-coordinate balls therefore gives \[0\leq N(x)\leq C_0d_g(x,z)^2 \quad\text{when }z\in F_1\text{ and }x\text{ is sufficiently close to }z.\] Apply this with \(z=\phi_{-j}(p)\), \(x=\phi_{-j}(q)\) and use Equation (315). Uniformly for all integers \(j\geq0\), \[ N(\phi_{-j}(q))\leq C_1\lvert\xi\rvert^2e^{-3j}. \tag{337}\] Equations (336) and (337) imply \(N(q)\leq C_1\lvert\xi\rvert^2e^{-j/2}\) for every \(j\), a contradiction. Thus \(N\) vanishes on the whole small arc.

In a foliated chart at \(p\), transversality gives \(\frac{\mathrm d}{\mathrm d\xi}y(\gamma_p(\xi))|_{\xi=0}\ne0\). After restriction, the transverse coordinate of the arc covers an open interval containing \(y(p)\). Equation (328) and positivity of \(B\) then imply that \(a\) vanishes on that interval. This gives an open neighborhood of \(p\) on which \(N=0\), contradicting \(p\in\partial P\). Hence \(F\) is empty. Finally \(a''\geq0\) at any zero of the nonnegative function \(a\), so Equation (330) proves the stated strict positivity at every boundary zero. ◻

Proof. Apply Lemma 104 with \(\alpha=X^\flat\); its hypotheses are precisely the compactness in Lemma 94 and the factorization and equations in Lemma 102. Every boundary zero therefore satisfies \(a''>0\). It is therefore an isolated zero in the transverse coordinate, and its neighborhood consists of a smooth two-sided zero leaf with \(P\) on both sides. There can be no open null region. Indeed, if the interior of \(Z\) were nonempty, it would have a boundary in the connected manifold \(M\), since \(P\) is nonempty. Such a boundary point lies in \(\partial P\) and would have the just-described neighborhood with no open zero region, a contradiction. Thus all of \(Z\) equals \(\partial P\), and it is a compact embedded hypersurface. The local isolation and compactness give only finitely many components. The nonvanishing global form \(X^\flat\) coorients each of them.

On a component of \(Z\), the local positive functions \(B_1/a''\) patch to a global smooth function. To check this, another oriented transverse coordinate has the form \(\widetilde y=\phi(y)\) with \(\phi'>0\). At a zero leaf, \[\widetilde B_1=B_1/\phi',\qquad \widetilde a=\phi'a,\qquad \widetilde a''=a''/\phi',\] where the last derivatives use \(\widetilde y\) and \(a=a'=0\). The quotient is therefore invariant. Since \(a''\) is constant along each plaque, Equation (312) gives \[\Delta_{q_y}\left(\frac{B_1}{a''}\right) =6\frac{B_1}{a''}-1.\] The maximum and minimum principles on the compact component imply \(B_1/a''=1/6\). Its leaf derivative is zero; thus \(b_1\) is constant along each plaque, and the first equation in Equation (312) gives \(\mathcal K_y=-3\).

Cut the original manifold along these finitely many two-sided components and truncate its ordinary end. The resulting compact orientable three-manifold \(L\) is connected and has interior homotopy equivalent to \(P\): the end and each of the original and cut boundary collars are product collars, which may be shortened without changing homotopy type. Proposition 101 therefore gives \(H_1(L;\mathbb Z)=H^1(L;\mathbb Z)=0\). Poincaré–Lefschetz duality and the boundary long exact sequence give \[H_2(L,\partial L;\mathbb Z)\cong H^1(L;\mathbb Z)=0, \qquad 0=H_2(L,\partial L;\mathbb Z)\longrightarrow H_1(\partial L;\mathbb Z)\longrightarrow H_1(L;\mathbb Z)=0.\] Consequently every component of the orientable boundary is a sphere. In particular each cut copy of a component of \(Z\) is a sphere, incompatible with its constant Gaussian curvature \(-3\) by Gauss–Bonnet. Thus \(Z\) must have been empty. ◻

Proof. In the zero-lapse case put \(d=a^2-s^2|W|_g^2\) in Equation (299). Then \[h=g+s^2d^{-1}W^\flat\otimes W^\flat, \qquad \lambda=s\sqrt d, \qquad \mathcal A=d^{-1}W^\flat, \qquad d|_S=\kappa_h^2.\] These are smooth in the original collar and give \(h|_{TS}=g|_{TS}\) and \(\partial_{\nu_h}\lambda=\kappa_h\). The static Hessian equation extends to \(S\) and gives \(\mathop{\mathrm{Hess}}_h\lambda=0\) there. Its tangential restriction is \(\kappa_h\mathrm{II}_h\), proving \(\mathrm{II}_h=0\).

If \(u|_S>0\), Equation (298) makes \(N\) a smooth original defining function. In coordinates \((N,y^1,y^2)\), smooth division of the tangential components gives \[X^\flat=a_0(N,y)\,\mathrm dN+N\beta_A(N,y)\,\mathrm dy^A, \qquad a_0(0,y)=\frac1{2\kappa_h}.\] The change \(N=\lambda^2\) gives the metric coefficients \[\begin{align*} h_{\lambda\lambda}&=4\lambda^2g_{NN}+4a_0^2, &h_{\lambda A}&=2\lambda(g_{NA}+a_0\beta_A), &h_{AB}&=g_{AB}+\lambda^2\beta_A\beta_B, \tag{341}\end{align*}\] with the coefficients on the right evaluated at \((\lambda^2,y)\). They are smooth across \(\lambda=0\) with the stated parity. At zero the normal block is \(\kappa_h^{-2}\), the mixed block vanishes, and the tangential block is \(g|_{TS}\), so this is a nondegenerate smooth Riemannian extension. Reflection is an isometry; its fixed hypersurface is totally geodesic. The normal derivative of \(\lambda\) is \(\kappa_h\). In either case, quotient Gaussian coordinates and \(\mathrm{II}_h=0\) give \(\partial_\rho q_\rho|_0=0\). The static equation gives \(\partial_\rho^2\lambda|_0=0\), proving Equation (339).

Since \(a_0-(2\kappa_h)^{-1}\) vanishes at \(N=0\), division by \(N\) also gives Equation (338). Its remainder is closed by \(\,\mathrm d\mathcal A=0\) in the interior and smoothness at the boundary. Simple connectivity and Lemma 97 give a global \(H\) with \(\,\mathrm dH=-\mathcal A\). In either case subtract the displayed singular term, if present. The remaining differential is smooth in the original collar. Fixing values on one interior collar section and integrating the normal derivative to \(S\) gives a smooth extension; its tangential derivatives agree with the prescribed differential by continuity. This proves Equation (340). ◻

Proof of Theorem 93. Propositions 101 and 105 give \(P=M\) and its simple connectivity. Lemmas 97, 98, and 106 prove the stated closedness, equations, and boundary data.

The quotient completion has the same underlying topology as \(\Omega\): in the positive-lapse case \(N\mapsto\sqrt N\) changes the boundary smooth structure but is a homeomorphism of the closed collar, and its restriction to \(S\) is the identity. To check completeness, let a path have finite \(h\)-length. Since \(h\geq g\) in the interior, it is a \(g\)-Cauchy path and has a limit in the complete original manifold with boundary. An interior limit is an ordinary smooth quotient point because \(N>0\) there. A limit on \(S\) is included by Lemma 106; in its smooth quotient collar the same path tends to that boundary point. The unique ordinary end has infinite \(g\)-distance, and hence infinite \(h\)-distance. These exhaust all possible finite-length escapes and prove completeness.

Finally, Equation (289) gives \(N=V_0^2(1+O_2(r^{-\tau'}))\). Since spatial lowering by \(g\) preserves the stated end orders, \[N^{-1}X^\flat\otimes X^\flat=O_2(r^{-2\tau'}), \qquad \sqrt N-V_0=O_2(r^{1-\tau'}).\] These are Equation (295). In particular the attached boundary is compact, the quotient has the single designated end, and no additional completion points have been introduced. ◻

Proof. We spell out the completion because the original boundary smooth structure is needed again in the spacetime reconstruction. The boundary alternatives and first jets are supplied by [eq-causal-killing,geo-boundary-collars].

Suppose first that \(u|_S>0\). Then \(N\) is an original smooth defining function, positive into the exterior, and \[X^\flat|_S=\frac1{2\kappa}\,dN|_S.\] In coordinates \((N,y)\) on a collar, smooth division gives \[X^\flat=a(N,y)\,dN+N\beta_A(N,y)\,dy^A, \qquad a(0,y)=\frac1{2\kappa}.\] Set \(N=\lambda^2\). The coefficients of the base metric become \[ \begin{split} h_{\lambda\lambda}&=4\lambda^2 g_{NN}+4a^2,\\ h_{\lambda A}&=2\lambda(g_{NA}+a\beta_A),\\ h_{AB}&=g_{AB}+\lambda^2\beta_A\beta_B, \end{split} \tag{346}\] where the right sides are evaluated at \((\lambda^2,y)\). These coefficients extend smoothly through \(\lambda=0\), with even diagonal blocks and an odd mixed block. Reflection in \(\lambda=0\) is an isometry. At the boundary, \(h_{\lambda\lambda}=\kappa^{-2}\) and \(h_{\lambda A}=0\), proving the first three assertions in (343); its fixed hypersurface is totally geodesic.

If \(u|_S=0\), let \(s\) be the original inward-to-domain normal coordinate. The preceding boundary jets give \[u=s a(s,y),\qquad X=s^2Y(s,y),\qquad a(0,y)=\kappa.\] Consequently, with \(D=a^2-s^2|Y|_g^2\), \[h=g+s^2D^{-1}Y^\flat\otimes Y^\flat, \qquad \lambda=s\sqrt D, \qquad D|_S=\kappa^2.\] These are smooth in the original collar. The static Hessian equation extends continuously to \(S\) and gives \(\mathop{\mathrm{Hess}}_h\lambda=0\) there. Its tangential part is \(\kappa\mathrm{II}_h\), so the boundary is totally geodesic. In \(h\)-Gaussian distance \(\rho\) it also gives \[ h=d\rho^2+q(\rho),\qquad q'(0)=0, \qquad \lambda=\kappa\rho+O(\rho^3). \tag{347}\]

The completion has the same underlying topology as the original closed exterior. A finite-length escaping \(h\)-curve is a finite-length \(g\)-curve because \(h\geq g\). Completeness of the original metric space therefore gives a limit in the original closed exterior. Interior points are ordinary smooth base points because \(N>0\) everywhere in the interior, and points of \(S\) have just been attached by regular collars. The end has infinite \(h\)-length by its hyperbolic asymptotics. Thus there is no missing finite-length escape.

For an interior smooth enclosing cut, the quadratic-form inequality \(h\geq g\) gives its \(h\)-area at least its \(g\)-area, which is at least \(A_0\). A cut smooth in the completed base and meeting \(S\) can be approximated, in area, by cuts whose meeting portions have been moved a positive distance into a collar. To see that the full-boundary convention is preserved, apply a smooth inward collar map to the entire exterior domain defining the cut, taper it to the identity outside a fixed collar, and take its full intrinsic boundary. In the positive-lapse case perform this construction in the regular \(\lambda\) collar and keep the displacement positive before regarding the cut as an original smooth interior cut. Collar smoothness makes the area error tend to zero. Hence every completed-base cut has area at least \(A_0\), while \(S\) itself has area \(A_0\) and remains an admissible full boundary.

Choose \[\frac32<\tau_1<\min(2,\tau_{\mathrm{data}})\] slightly below the exponent in the adjoint end estimates. Those estimates give \(u-V_0,X=O_{2,\alpha_1}(r^{1-\tau_1})\). Since \(N\sim r^2\), the identity defining \(h\) gives the final estimate in (345). The identity \(\lambda-u=-|X|_g^2/(u+\lambda)\) gives the estimate for \(\lambda\); the estimate for \(h-b\) follows from that for \(g-b\).

Each metric-flux integrand is linear in the error and its first background derivative and has a static-potential factor of size \(O(r)\). The difference contributed by \(h-g\) is therefore \(O(r^{3-2\tau_1})\) after integration over a coordinate sphere, and tends to zero. The same estimate holds for all four potentials. The existing mass limits of \(g\) consequently give the stated limits for \(h\). This proves (344) and the proposition. ◻

Proof. Use (347) and set \(\vartheta=\kappa\theta\), of period \(2\pi\). On each boundary coordinate patch, write \(x=\rho\cos\vartheta\) and \(y=\rho\sin\vartheta\). The normal part is \[d\rho^2+(\lambda/\kappa)^2d\vartheta^2 =dx^2+dy^2+ \frac{(\lambda/(\kappa\rho))^2-1}{\rho^2} (x\,dy-y\,dx)^2.\] The coefficient in front of the final squared one-form is a smooth one-sided function of \(\rho\) and the bolt variables, equal to a smooth bolt function plus \(O(\rho)\). The squared one-form has quadratic Cartesian coefficients. Its \(O(\rho)\) correction is consequently \(C^{2,\alpha}\) for every \(\alpha<1\). Likewise \(q(\rho)=q(0)+\rho^2q_2+O(\rho^3)\) gives a \(C^{2,\alpha}\) tangential block. There are no mixed \(d\rho\)–bolt terms in Gaussian coordinates. Positivity follows at \(\rho=0\) from \(q(0)>0\).

The Einstein equation holds away from the bolt and, because the metric is \(C^2\), also holds at the bolt by continuity of Ricci curvature. Interior Einstein regularity in harmonic coordinates applies, for instance (DeTurck and Kazdan 1981, Theorem 5.2). This gives smooth, indeed analytic, Einstein coordinates there. The circle action and its fixed set are then smooth: isometries preserve harmonic coordinates and the local Killing field satisfies the corresponding elliptic equation. Normal exponential coordinates to the fixed set recover the usual smooth disk action. Thus this is a regular bolt, with no conical angle defect. ◻

Proof. Fix \(\gamma\) with \(3/2<\gamma<\min(2,\beta)\) and lower the Hölder exponent slightly. First cut \(G-G_0\) off toward the interior, leaving it unchanged on a sufficiently distant end. Call the resulting global metric \(G_E=G_0+E\). Because \(\gamma<\beta\), the global \(C^{2,\alpha_0}_\gamma\) norm of \(E\) tends to zero as the cutoff is moved out. The cutoff is a function of \(\eta\), so retains the stated symmetries.

Seek a harmonic map \[\Phi:(B^3\times S^1,G_E)\longrightarrow(B^3\times S^1,G_0), \qquad \Phi(x)=\exp^{G_0}_x W(x),\] with \(W\in C^{3,\alpha_0}_\gamma\). Parallel translation along the exponential segment identifies its tension field with a section of the fixed background tangent bundle. The negative linearization in \(W\) at \((E,W)=(0,0)\) is \[ J=\nabla^*\nabla-\mathop{\mathrm{Ric}}_{G_0}=\nabla^*\nabla+3. \tag{352}\] It has no \(L^2\) kernel, since its quadratic form is \(\int(|\nabla W|^2+3|W|^2)\). Theorem C(c) and Proposition E of (Lee 2006), applied to one-forms after lowering an index, give an isomorphism in physical weights \(-1<\delta<4\): the indicial radius is \(\sqrt{3^2/4+1+3}=5/2\), and the Fredholm index is zero with kernel equal to the \(L^2\) kernel. In particular, \[J:C^{3,\alpha_0}_\gamma\longrightarrow C^{1,\alpha_0}_\gamma\] is an isomorphism.

The tension expression is a smooth map between these Banach spaces. In physical unit patches its coefficients are smooth functions of \(E,G_E^{-1}=(G_0+E)^{-1}\), one derivative of \(E\), and the first two derivatives of \(W\); target curvature and its derivatives are uniformly bounded. Products of weighted errors have at least their summed weight. Consequently its nonlinear remainder satisfies, on a fixed small ball, \[\|\mathcal R(E,W)-\mathcal R(E,\widetilde W)\|_{C^{1,\alpha_0}_\gamma} \leq C\bigl(\|E\|_{C^{2,\alpha_0}_\gamma} +\|W\|_{C^{3,\alpha_0}_\gamma} +\|\widetilde W\|_{C^{3,\alpha_0}_\gamma}\bigr) \|W-\widetilde W\|_{C^{3,\alpha_0}_\gamma}.\] The inverse of \(J\) therefore gives a contraction, or equivalently the implicit function theorem gives a unique small \(W\), with \(\|W\|_{C^{3,\alpha_0}_\gamma}\leq C\|E\|_{C^{2,\alpha_0}_\gamma}\). The map is a local diffeomorphism by smallness of its first jet. Its displacement is bounded, so it is proper for the complete background metric. It is homotopic to the identity through its exponential displacements. A proper local diffeomorphism is a covering map, and this homotopy makes its induced fundamental-group map surjective; the covering therefore has one sheet. Thus \(\Phi\) is a global diffeomorphism. In compact coordinates its displacement is \(O(z_0^{1+\gamma})\), so it restricts to the identity at infinity. Uniqueness shows that it commutes with circle translations and time reflection.

Set \(\widehat G=(\Phi^{-1})^*G_E\) and \(k=\widehat G-G_0\). The identity map from \(\widehat G\) to \(G_0\) is harmonic, and \(k\in C^{2,\alpha_0}_\gamma\). On a smaller end \(\widehat G\) is Einstein. The Ricci equation with this harmonic-map gauge has linearization \[ P=\frac12(\Delta_L+6) =\frac12\bigl(\nabla^*\nabla-2\mathring R_{G_0}\bigr) \tag{353}\] at \(G_0\); see (Lee 2006, Equations (1.2)–(1.3)). In each fixed-size physical patch the gauged Ricci expression is uniformly elliptic. Differentiating its difference from the background equation and applying interior Schauder estimates on nested patches gives successively \[\|k\|_{C^{j+2,\alpha_0}(B_1)} \leq C_j\bigl(\|k\|_{C^{j+1,\alpha_0}(B_2)} +\|k\|_{C^{2,\alpha_0}(B_2)}\bigr).\] The constants are uniform along the end because the background has bounded geometry and \(k\) is uniformly small in \(C^{2,\alpha_0}\). The defining function is comparable on each such patch. Induction therefore preserves the factor \(z_0^\gamma\) and gives \(k\in C^{j,\alpha_0}_\gamma\) for every \(j\).

Taylor expansion of the gauged equation now has the exact form \[ Pk=\mathcal Q(k,\nabla k,\nabla^2k),\qquad \mathcal Q\in C^{j,\alpha_0}_{2\gamma}\quad\hbox{for every }j. \tag{354}\] Indeed its constant term vanishes at \(G_0\), its first-order term is \(P\), and every remaining term contains at least two error factors, counting derivatives as factors in physical norms.

We verify injectivity for the next use of the weighted theorem. On the tube, \(Z=\partial_\eta\) has \[\operatorname{div}_{G_0}Z=2\coth\eta+\tanh\eta\geq3.\] For a compactly supported scalar \(f\), integration by parts and Cauchy–Schwarz give \[3\int f^2\leq-2\int f\,Zf \leq2\|f\|_2\,\|df\|_2, \qquad \int|df|^2\geq\frac94\int f^2.\] The integration can first omit \(\eta<\varepsilon\); its additional boundary term tends to zero because the tubular area is \(O(\varepsilon^2)\). Kato’s Inequality gives the same lower bound for \(\int|\nabla T|^2\) in terms of \(\int|T|^2\). For a tracefree symmetric two-tensor on curvature \(-1\), \(\mathring R T=T\), so the operator inside the parentheses in (353) has quadratic form at least \(\frac14\|T\|_2^2\). On a pure trace tensor, \(\mathring R(fG_0)=-3fG_0\), so its zeroth-order term is favorable. The trace and tracefree bundles are parallel. Thus \(P\) has no \(L^2\) kernel.

Theorem C(c) and Proposition D of (Lee 2006) apply to this smooth, complete conformally compact background. For \(\Delta_L+6\) in dimension four the indicial radius is \(3/2\), so \(P\) is an isomorphism at every physical weight \(0<\delta<3\). Cut \(k\) off once more toward the interior. Its equation on the whole tube is (354) plus a smooth compactly supported commutator. Since \(2\gamma>3\), that source belongs to \(C^{j,\alpha_0}_\delta\) for every \(\delta<3\). Solve it at any \(\delta\) with \(2<\delta<3\) and compare with the cut-off \(k\) at weight \(\gamma\). Their difference is in the kernel at the smaller weight, hence vanishes. Higher weighted regularity follows by the same equation. Smaller positive weights follow by inclusion.

Finally, for an ordinary compact coordinate component of \(z_0^2k\), two coordinate derivatives have size \(O(z_0^{\delta-2})\). The physical Hölder estimate, compared on patches of compact diameter comparable to \(z_0\), gives an ordinary \(C^{2,\alpha_2}\) extension whenever \(0<\alpha_2<\min(\alpha_0,\delta-2)\). Equivalently this is the weighted-to-ordinary inclusion of (Lee 2006, Lemma 3.7(b)), with the tensor weight included. This proves the claimed preliminary conformal compactness. ◻

Proof. Equations (345) and (348) give \(G-G_0\in C^{2,\alpha_1}_{\tau_1}\) in physical four-dimensional tensor norm: the relative circle coefficient is \((\lambda^2-V_0^2)/V_0^2=O(r^{-\tau_1})\). Apply 109. The resulting metric is smooth in the interior, Einstein with \(\mathop{\mathrm{Ric}}=-3G\), has an ordinary \(C^2\) conformal compactification, and has the smooth conformal representative \(d\theta^2+\sigma\).

These are the hypotheses of (Chruściel et al. 2005, Theorem A). That theorem is local to a collar of a compact conformal boundary and does not require a global bolt filling. It supplies a boundary-identity collar diffeomorphism placing the metric in geodesic form. Its compactification is smooth in even total dimension, hence in the present dimension four. The defining function is chosen for the specified representative. Both its defining eikonal equation and the inward normal flow are invariant under circle translation and reflection; uniqueness with that boundary normalization makes the collar map equivariant. Equivariance under translations permits at most a spatially dependent circle shift, and reflection together with boundary identity forces that shift to vanish. Thus the geodesic gauge is a spatial one and has the block form in (355).

For completeness we compute the coefficients needed later. Write \[G=z^{-2}(dz^2+\gamma(z)),\qquad \gamma(0)=\gamma_0=d\theta^2+\sigma.\] The tangential Einstein equation and its mixed constraint are \[\begin{align*} 0={}&z\gamma''-z\gamma'\gamma^{-1}\gamma' +\frac z2\operatorname{tr}(\gamma^{-1}\gamma')\gamma' -2\gamma'-\operatorname{tr}(\gamma^{-1}\gamma')\gamma -2z\mathop{\mathrm{Ric}}_{\gamma}, \tag{357}\\ 0={}&\operatorname{div}_{\gamma} (\gamma^{-1}\gamma') -d\operatorname{tr}(\gamma^{-1}\gamma'). \tag{358}\end{align*}\] They follow by using the second form \(\gamma'/2\) of the compact metric \(dz^2+\gamma\) and the conformal Ricci formula. Expanding the first equation at order zero gives \(\gamma'(0)=0\). Writing \(\gamma=\gamma_0+z^2\gamma_2+z^3\gamma_3+\cdots\), its order-\(z\) equation gives \[\gamma_2+\operatorname{tr}_{\gamma_0}(\gamma_2)\gamma_0 =-\mathop{\mathrm{Ric}}_{\gamma_0}.\] Here \(\mathop{\mathrm{Ric}}_{\gamma_0}=\sigma\) and \(R_{\gamma_0}=2\), so \[\gamma_2=\frac12d\theta^2-\frac12\sigma.\] The order-\(z^2\) equation gives \(\operatorname{tr}_{\gamma_0}\gamma_3=0\), and the same order in (358) gives \(\operatorname{div}_{\gamma_0}\gamma_3=0\). Since the circle coefficient of \(\gamma\) is \((z\lambda)^2\), these statements are exactly (355)–(356).

We verify the spatial mass comparison from the two asymptotic representations of the same metric \(h\). In the original spatial chart, (345) gives \(h-b=O_2(r_{\mathrm o}^{-\tau_1})\) with \(\tau_1>3/2\). In the new geodesic chart put \(r=z^{-1}-z/4\). The exact identities \[\frac{dr^2}{1+r^2}=z^{-2}dz^2,\qquad r^2\sigma=z^{-2}(1-z^2/4)^2\sigma\] and the smooth expansion (355) give \(h-b=O_j(r^{-3})\) for every fixed differentiated order \(j\). Also \(R_h=-6\) by the static equations. In each chart, therefore, the weighted quadratic integrability of (Chruściel and Herzlich 2003, Equation (2.8a)) follows from \(\int^\infty r^{2-2q}\,dr<\infty\) with \(q>3/2\); its scalar-curvature condition (2.8b) is automatic. The frame decay is \(o(r^{-3/2})\), as required by (2.13), and uniform metric equivalence holds sufficiently far out. Theorem 2.3 of that paper identifies the metric mass functionals up to a hyperbolic background isometry. The transition estimates are conclusions of that theorem, so differentiated estimates for the geodesic coordinate map are not an additional input here. Both the harmonic-map identification and the geodesic collar identification induce the identity on the spatial conformal boundary. The resulting hyperbolic isometry acts trivially on \(\mathbb S^2\) at infinity and is therefore the identity. This proves the asserted mass comparison.