Boundary graph deformations for the spacetime Penrose inequality

Proof. Choose a countable dense set \{x_i\} and a number L larger than the diameter of the compact space. Set d_c(x)=\min\{L,\operatorname{dist}(x,E_c)\}, with the value L when E_c is empty. For each i, c\mapsto d_c(x_i) is a bounded nonincreasing real function, so its right discontinuities form a countable set. Avoid their countable union. Then d_{c_j}(x_i)\to d_c(x_i) for all i whenever c_j\downarrow c. Every d_c is 1-Lipschitz. A finite net chosen from the dense set therefore upgrades this convergence to uniform convergence on the compact space.

For nonempty E_c, if points y_j\in E_{c_j} stayed a fixed distance from E_c, uniform convergence evaluated at y_j would contradict d_{c_j}(y_j)=0. The opposite Hausdorff inclusion follows from E_c\subset E_{c_j}. For empty E_c, uniform convergence to L is incompatible with any nonempty E_{c_j}, since its distance function has a zero. This proves the stated convention as well.□

Proof. Use normal graphs \Sigma(v) over \Sigma and pull their expansion back to \Sigma. The map

is smooth near zero. Let \nu be the outward normal and \mathrm{II} its second fundamental form. The normal-variation formulas give the linearization L=D\Phi(0) explicitly as

The hypersurface dimension is d-1, so \Theta_{Q-cg/(d-1)}=\Theta_Q-c. This constant shift has zero derivative under hypersurface variations; hence L is also the MOTS stability operator for Q-cg/(d-1), with principal part -\Delta_\Sigma. Stability gives a principal eigenvalue \lambda\geq0 and a positive smooth eigenfunction \varphi [5, Proposition 5.1]. If \lambda>0, take v=s\varphi. Smooth dependence gives \Phi(s\varphi)=s\lambda\varphi+O(s^2) in C^0, which is positive for all sufficiently small s>0. The graph velocity is positive as well.

Suppose \lambda=0. The kernel of L is spanned by \varphi, and the adjoint kernel is spanned by a positive smooth function \varphi^*. These are the principal-eigenfunction facts for a scalar elliptic operator on a compact connected closed manifold [5, Section 4, Lemma 4.1]. For example, simplicity of the kernel also follows by writing a kernel element as v=\varphi w and applying the strong maximum principle to the resulting equation for w, whose zeroth-order term vanishes. Consider the augmented map

Its derivative at (0,0) is

This map is an isomorphism. Indeed, to solve Lv-a=f with \int v=b, the Fredholm compatibility condition uniquely fixes

The equation for v then has a solution; adding a multiple of \varphi imposes its prescribed integral uniquely. Elliptic estimates and this one-dimensional normalization give a bounded inverse between the indicated Hölder spaces.

The implicit-function theorem consequently supplies smooth v(s),a(s) with \mathcal G(v(s),a(s))=(0,s). Differentiating at zero and pairing with \varphi^* gives a'(0)=0 and

After shortening the interval, all these graphs form an outward foliation and have constant expansion c+a(s). If a(s)\leq0 at any positive s in this interval, replace only this frontier component by \Sigma(v(s)). This graph strictly enlarges D inside the chosen collar, leaves every other frontier component and designated barrier fixed, and has expansion at most c on the replaced component. This contradicts the assumed local outward-replacement maximality. Therefore a(s)>0 for every such s>0.

In either case positive graph velocity and compactness give the asserted bounds on |\nabla s|. Smooth elliptic bootstrapping of the graph equation gives smooth leaves in the zero-eigenvalue case.□

Proof. If D=\varnothing, use \operatorname{dist}(y,D)=+\infty; then D_z=\varnothing and the assertion is vacuous. Otherwise, take a support \phi to D_z at x'. Since x'\in\partial D_z, its distance from D is z. For sufficiently small radii the relevant distance neighborhood of \partial D lies in a fixed compact interior neighborhood. There is therefore a nearest point x\in D and a length-z minimizing unit-speed geodesic \gamma:[0,z]\to N from x to x'. This nearest point lies on \partial D: if it were interior to D, moving a short distance along \gamma toward x' would decrease the distance. The entire segment lies in the fixed smooth compact neighborhood and is shorter than its injectivity radius. The ball \overline B_z(x) is contained in D_z and touches its exterior support at x'. Its tangent plane there therefore agrees with that of the support and

Let P_t:T_xN\to T_{\gamma(t)}N denote parallel transport and put P=P_z.

For every X\in T_xN solve the Jacobi boundary problem

Here primes denote covariant differentiation along \gamma. The short interval and absence of conjugate points make this problem uniquely solvable. Its estimates can be seen without division by an uncontrolled quantity. In a parallel frame write j_X(t)=P_t^{-1}J_X(t) and \mathcal R(t) for the curvature coefficient. The integral equation is

It implies, by absorption for small z,

The second line follows from the same integral equation and the first-line bounds. Also \langle J_X,\dot\gamma\rangle is affine in t and has equal endpoint values. Thus \langle A_X,\dot\gamma(0)\rangle=0.

We may therefore prescribe the first jet of a smooth unit vector field V near x by

To realize it with uniform second-jet bounds, choose a smooth local orthonormal frame obtained by radial parallel transport from x. In that frame take the affine coefficient vector with the prescribed value and first derivatives, and normalize its length. The orthogonality just proved means normalization preserves its prescribed first derivatives. The frame has uniformly bounded derivatives on a fixed small coordinate ball, and Equation (3.16) therefore gives uniformly bounded second derivatives of V. This construction uses the radial direction and the ambient geometry, with no derivatives of the support \phi.

Define the local endpoint map

Its differential at x is the terminal value of the Jacobi field with initial values X,\nabla_XV(x). By Equation (3.15), it has the exact property

The last bound follows by differentiating the smooth map (y,v,z)\mapsto\exp_y(zv) twice in y, including the first and second derivatives of V. At z=0 the map is the identity and its covariant second derivative vanishes. Its z-derivative is uniformly bounded for the bounded jets just constructed, giving the displayed Cz bound. The exact middle identity, rather than a near-isometry estimate, is essential.

The map F_z takes D locally into D_z, since its defining geodesic from any y\in D has length z. Consequently \psi=\phi\circ F_z is an exterior support to D at x. Its derivative is P^*\dd\phi, so its gradient length is exactly |\nabla\phi(x')| and, by Equation (3.14), its normal is \dot\gamma(0). The covariant chain rule gives

Trace over an orthonormal basis of \dot\gamma(0)^\perp and divide by the common gradient length. The first term is exactly the corresponding normalized tangential trace at x', and the remaining term has absolute value at most Cz. Finally, the tensor trace changes by at most Cz, by the C^1 bound on Q and parallel transport. It follows that

The assumed support inequality at x proves Equation (3.13). There is no term containing z|\Hess\phi|.□

Proof. All distance neighborhoods in this proof are relative to N. Since D contains a collar of \Gamma_- and avoids \Gamma_+, their new frontiers, at sufficiently small radii, lie in a compact subset of the interior. A minimizing segment ending on a new frontier has its nearest point on the interior frontier of D; it cannot start on \Gamma_- because its covered collar has fixed positive width. Thus the proof of Lemma 3.5 applies unchanged. Choose \beta>0 so small that its conclusion holds for 0<z\leq\beta, all these frontiers avoid the prescribed boundary, and

We first construct a smooth inner enclosure without imposing any expansion bound on it. Smoothly approximate the continuous distance function d_D=\operatorname{dist}(\cdot,D) on N uniformly, with error less than \beta/32, using a finite coordinate cover, mollification, and a partition of unity. Choose a regular value in (\beta/3,\beta/2) of the resulting smooth function. Its sublevel set U has smooth interior boundary and, for example, satisfies

The boundary \Gamma_- is contained in its relative interior and \Gamma_+ is excluded. If a component of N\setminus\overline U meets no component of \Gamma_+, fill it. This only deletes components of the smooth free boundary. After this filling the second inclusion in Equation (3.20) need not hold for its interior, but every remaining free boundary component still lies in the indicated thin neighborhood, and the first inclusion still holds.

Choose \eta\in C^\infty(N), 0\leq\eta\leq1, equal to one on that free boundary and supported in \{d_D<\beta\}. Its existence uses the positive separation between this compact smooth boundary and the closed set \{d_D\geq\beta\}; it requires no smoothness of d_D. For a sufficiently large finite constant A, the tensor

makes the free boundary of U a strictly negative inner barrier:

All prescribed outer faces remain strictly positive for Q_A, because \eta=0 there and their original expansion exceeds c'. Apply the barrier part of Theorem 3.1 on each component of N\setminus\overline U. Each meets \Gamma_+ by the filling step and has nonempty inner boundary by connectedness of N and U\ne\varnothing. We obtain smooth MOTSs \Sigma bounding a domain E which contains \overline U and avoids \Gamma_+. They satisfy, with their normal out of E,

We claim \Sigma misses \{d_D<\beta\}. Otherwise set z=\min_{\Sigma}d_D<\beta. The first inclusion in Equation (3.20) gives z\geq\beta/4>0. Moreover D_z lies on the inner side of \Sigma. To see this directly, if d_D(y)<z, a shortest path from y to D of length less than z cannot cross \Sigma: such a crossing would have distance from D less than z. Since D\subset E, the whole path and y lie in E. Closure gives D_z\subset\overline E.

At a point attaining the minimum, a defining function for the smooth boundary \Sigma is therefore an exterior support to D_z, with the same outward normal as in Equation (3.22). The parallel support bound and Equation (3.19) imply

contradicting Equation (3.22). This excludes the entire modification range, so \eta=0 on \Sigma and proves Equation (3.18). The contained D_{\beta/4} gives strict enclosure of D. If desired, fill any remaining complementary components which meet no designated outer face; this deletes smooth boundary components and preserves all stated properties.□

Proof. The large-sphere comparison above confines all negative-threshold regions to a fixed compact set, independently of how close the thresholds are to zero. Choose a slightly larger compact set \mathcal K containing this range and \supp K in its interior. Let

First restrict both negative thresholds to an interval so close to zero that

while retaining Equation (3.4). Lemma 3.2 allows the required successive choices in that interval. Construct the two regions, select \Omega, and fix their disjoint collars inside \mathcal K as above. All ensuing constants may depend on these fixed choices.

Here is an explicit offset construction. On each collar choose a smooth nonincreasing cutoff \chi(s), equal to one near s=0 and zero near the collar’s far end. Choose \kappa_B>0 so that 1-\kappa_Bs>0 throughout that collar. For a common amplitude a>0, define the contribution of a black collar to be a\chi(s)(1-\kappa_Bs), and that of a white collar to be -a\chi(s)(1-\kappa_Bs). Extend by zero outside the disjoint collars and sum. This produces a smooth function on the manifold with boundary \Omega, supported in \mathcal K, with C_b=a, C_w=-a, and k_B=a\kappa_B on the respective collars. These remain valid bounds when a type is absent. Its C^1 norm tends to zero with a, even though the collar widths have already been fixed.

For h\in[b_-,b_+] and C\in[C_w,C_b],

Choose a so small that (C_b-C_w)T_0+\|\nabla C\|_\infty<m_0/8. The first two terms of Equation (3.25) are then at most m_0/4 on \mathcal K and vanish outside \mathcal K. Finally take a fixed smooth positive weight equal to r^{-4} far out and multiply it by a sufficiently small positive constant. The end lower bound for \mathfrak m and \delta<1, together with compact positivity, allow the choice \rho\leq\mathfrak m/4 everywhere. This proves Equation (3.25). All remaining assertions were proved in constructing the regions and collars.□

Proof. Apply the scalar identity of [35, Proposition 4.1] in spatial dimension three, so its surface dimension is k=2. To distinguish the two conventions temporarily, add a subscript \mathrm A to its variables. The exact dictionary is

The quantities D,d,w,a,v,H^f,F,U are identical. In particular,

and the companion metric e^{2t_{\mathrm A}}\bar g is precisely \hat g=e^{4t}\bar g. Thus the divergence term, trace prescription and metric match without rescaling their flux. The physical density conversion gives 8\pi(\mu+J(w)); the ADM energy and momentum are not changed by that conversion.

For the quadratic expression, the companion coefficient s_p=p_{\mathrm A}+(k-1)/2 is (p+1)/2, and dt_{\mathrm A}=2dt. Its transverse square is |M+(p+1)a\nabla_\perp t|^2. Combining its residual -[a^2|\nabla_\perp t|^2] with that square gives (M+pa\nabla_\perp t)\cdot(M+(p+2)a\nabla_\perp t). The remaining gradient contribution is 4(p+1)|dt|_{A_d}^2. Its normal terms become

These substitutions give Equations (4.9) and (4.10). Smoothness at zero gradient is part of the invariant identity: the temporary axis has not been used to define its coefficients. The import is pointwise and requires neither an energy condition nor a solved divergence equation.□

Proof. With x=\partial_e t and y=\nabla_\perp t, exact completion gives

Completing first in x leaves 3\chi q^2/(4+pv); completing next in F leaves 3\chi/(4+pv)-\chi^2/12=4d(9-d)/[3(4+pv)^2]. The latter coefficient is at least 32d/[3(4+N)^2]. It controls dq^2 polynomially. The F-square then controls F^2, since \chi^2q^2\le dq^2. The transverse gradient coefficient is at least three, and the shifted M-square controls M with an additional factor (1+N)^2. The squared d-norm of the shift in the x-square is \chi^2vq^2/(16d)\le dq^2/16, so it also controls dx^2. Finally |T|^2=|T^{\operatorname{tf}}|^2+(F-\chi q)^2/2. This proves (4.12). If \dd t=0, the sole cross term is -\chi Fq/2. Using |\chi Fq|/2\le\chi q^2/4+\chi F^2/4 and 0<\chi\le1 proves (4.13).□

Proof. The tangential Hessian is \Hess f|_{TB}=(\partial_\nu f)\operatorname{II}_g. Thus \tr_\perp H^f=w_\nu H and \chi q=F-P_B-w_\nu H. The normal component of (4.8), using (4.15), is

The \bar g-normal is \sqrt d\,\nu. Its mean curvature is \sqrt d\,H, because the tangential metric is unchanged and its normal first variation is multiplied by \sqrt d. The conformal mean-curvature formula gives the asserted mean curvature for \hat g.□

Proof. Fix a smooth j:\R\to[0,1] which vanishes for t\le0 and equals one for t\ge1. For 0\le\lambda\le1 put

Before the last stage use the boundary condition

Outer values remain f=Z=0 throughout.

We specify two smooth collar modifications. Let \nu_s=\nabla s/|\nabla s| be the outward unit normal to a collar leaf; the leaf coordinate s here is local and is separate from the homotopy parameter. Choose disjoint collar cutoffs equal to one at the faces. Take \eta>0 so small that b_-+3\eta<b_+. A black contribution to D_0(x,w,h) is a negative constant times a collar cutoff, times a smooth directional cutoff supported where w\cdot\nu_s<-1/2 and equal to one at -1, times a nonincreasing height cutoff equal to one below b_-+\eta and zero above b_-+2\eta. A white contribution has positive sign, its directional cutoff supported where w\cdot\nu_s>1/2 and equal to one at 1, and a nondecreasing height cutoff equal to zero below b_+-2\eta and one above b_+-\eta. Decrease \eta to separate these ranges. Their sum D_0 is smooth, bounded, supported in the collars, and nondecreasing in h. It is nonpositive for h\le b_-+\eta, nonnegative for h\ge b_+-\eta, and vanishes on black outward and white inward directions, for every length 0\le|w|\le1.

Choose the magnitudes larger than 2\sup_B|H|+1. Equation (4.1) then gives, for F=h+C+D_0 at the assigned boundary height b=b_\pm,

These are limiting directional evaluations at a=1,\chi=0. The compatible direction has equality; D_0 corrects the other one. Its derivatives on bounded height ranges are independent of N. Choose a smooth compactly supported extension b(x), constant equal to b_\pm on each smaller collar. On these collars let D_1(x,w)=A_1w\cdot\nu_s, extended by a collar cutoff, with A_1>\sup_B|H|+1. Then D_1(x,0)=0 and D_1(x,\pm\nu)=\pm A_1 on B.

The four stages are the following.

1. (i)

   For 0\le s\le1, set \lambda=1-s, keep F=h+C and h|_B=b, and use \Div V_\lambda=u\Xi and (4.20).

2. (ii)

   For 1\le s\le2, set \lambda=0, \alpha=s-1, retain h|_B=b, and replace the trace right side by F=h+C+\alpha D_0(x,w,h). Keep the same scalar equation and boundary prescription.

3. (iii)

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

   \begin{split} F={}&h+(1-\zeta)[C+D_0(x,w,h+\zeta b(x))]\\ &+\zeta[\tr_{A_\chi}K+D_1(x,w)]. \end{split} \tag{4.22}

   Keep the same scalar equation and boundary prescription. At a face, h+\zeta b=b. The value at w=\pm\nu is the convex combination of the preceding directional value and P_B+D_1(x,\pm\nu), so (4.21) persists.

4. (iv)

   For 3\le s\le4, retain the trace problem at \zeta=1 and multiply both \Xi and the right side of (4.20) by \eta_s=4-s.

The prescriptions agree at the common endpoints. Their h-derivatives are 1, 1+\alpha\partial_hD_0, or 1+(1-\zeta)\partial_hD_0, all at least one.

At \zeta=1 the trace equation is \tr_{A_\chi}H^f=h+D_1(x,w) with zero Dirichlet values. At a positive interior maximum of f, one has w=0, A_\chi=I, D_1=0, and l\Delta f=\tau f>0, a contradiction. A negative minimum is excluded in the same way. Thus f=0 and t=Z in the last stage. At s=4 the remaining equation is \Div(4L(Z)\nabla Z)=0, with zero inner conormal flux and Z=0 on S_R. Multiplication by Z and integration give Z=0. For the fixed-point construction in Section 6, the final linear scalar problem has zero source and zero inner flux for every input; consequently that last map has output identically zero.□

Proof. The collar modifications vanish at w=0. At an interior extremum the trace equation reads either \tr K+l\Delta f=h+C or \tr K+l\Delta f=h+(1-\zeta)C+\zeta\tr K. The maximum and minimum inequalities bound h by fixed bounds for |\tr K|+|C|. Boundary values range between b_-,0,b_+. This proves the height bound without using the floor. Boundedness of D_0,D_1 and |\tr_{A_\chi}K|\le3|K| proves the assertion for F.□

Proof. Take r_0 outside the supports of K,C,D_0,D_1. There F=h in the first three stages, and h=0 in the last one. For \psi=r^{-\gamma}, with axis \nabla\psi/|\nabla\psi|, the controlled end gives, uniformly for 0<\chi\le1,

after enlarging r_0. At a comparison contact, coefficients use the given Z and the common gradient of test and solution. The positive factor l/(\tau\sqrt D) preserves this sign. Height monotonicity therefore makes C(\psi-R^{-\gamma}) an upper barrier and its negative a lower barrier. Choose C to dominate C_0 on r=r_0 for every R\ge2r_0. Differentiating at the common outer value gives (4.24); tangential derivatives there vanish.

At S_R, Z=0. The monotonicity of its defining expression implies t>-\epsilon if \ell^2|\nabla f|^2<e^{8\epsilon}-1. It suffices to require C^2\ell^2\tau^{-2}R^{-2-2\gamma}<e^{8\epsilon}-1, which is a permissible lower bound depending on fixed N,\epsilon.□

Proof. Consider the graph of f in the Riemannian product with warped fiber (\Omega\times\R,g+l^2\dd b_0^2). Its induced metric is \bar g. At \dd t=0 one also has \dd l=0, and its second fundamental form, up to orientation, is H^f. Write E=l^{-1}\partial_{b_0}. The graph unit normal is \sqrt d\,E-ae. The ambient curvature formulas at this point are

with zero mixed Ricci terms. Hence the ambient contribution to half the graph scalar curvature is

The last term is nonpositive: at \dd t=0, \Hess l=lp\,\Hess t and p\ge0. The second-form contribution is [(\tr_{\bar g}H^f)^2-|H^f|_{\bar g}^2]/2. Since the inverse graph metric is A_d, its expansion is exactly (4.26). Finally the conformal contribution at this point is -4\bar\Delta t\le0. This proves (4.25).

The pointwise algebraic clause of [35, Lemma 4.3] applies under Equation (4.11). We also record its three-dimensional calculation, because the boundary homotopy below needs its precise degenerate normal coefficient. First evaluate \mathcal S on S=K+H^f, whose trace in the transverse plane is F-\chi q. Subtracting (4.26) from (4.10) at \dd t=0 gives

The normal coefficient is at least \chi, while |dFq|\le\chi q^2/2+d^2F^2/(2\chi). Because

Equation (4.13) implies \mathcal T-\mathcal S(S)\ge c\mathcal T-C(1+N)F^2 with universal c>0.

To replace S by H^f=S-K, polarize (4.26). The difference is bounded by

Indeed the only extra normal product is dq\,\tr_\perp K; all remaining products involve transverse entries, dM, or \tr_\perp S=F-\chi q. By (4.13) and d/\sqrt\chi\le C\sqrt{1+N}, this bound is at most C\sqrt{1+N}|K|\sqrt{\mathcal T}+C|K|^2. Young’s inequality absorbs a fixed small fraction of \mathcal T and costs only C(1+N)|K|^2. This proves (4.27).

If (1+p)v\le\eta_*, then d and \chi are within C\eta_* of one. At d=\chi=1 the normal block in (4.29) is 3q^2/2-Fq+F^2/2; its symmetric matrix has smallest eigenvalue (2-\sqrt2)/2>0. Choose \eta_*>0 small enough that its coefficients remain within half this spectral gap. The transverse coefficients are already bounded below. For K=0, Equation (4.13) therefore gives \mathcal T-\mathcal S(H^f)\ge c_*\mathcal T in this regime. In the remaining regime v>\eta_* /(1+p), use (4.27) with K=0 and absorb (1+N)F^2 into C(1+N)^2vF^2. This proves (4.28). All formulas extend to \sigma=0 by their invariant expressions; there d=\chi=1 and the full quadratic expressions are independent of the temporary choice of axis.□

Proof. At this contact \dd t=0 and p=N is locally constant as a function of t near its value -\epsilon. Differentiating w in an orthonormal frame gives

In the e-frame the normal vector slot is multiplied by d. Equations (4.13) and d/\sqrt\chi\le C\sqrt{1+N} therefore show

No bound for the uncontracted normal Hessian is asserted.

Each first-three-stage right side can be written as a smooth function F(x,w,h,p). Its h-derivative is at least one. On the bounded height range supplied by Lemma 4.5, its explicit x- and w-derivatives are polynomially bounded and compactly supported, apart from the derivative of the term h. This follows directly from the fixed collar cutoffs and A_\chi=I-(p+4)w\otimes w/(4+pv), whose denominator is at least four. For example its first w-derivatives on |w|\le1 are bounded by C(1+N)^2, and the other explicit modifications have bounds independent of N. Combining these bounds with (4.32) and Young’s inequality therefore produces a fixed finite power of 1+N. Since \nabla p=0 at the contact and w(h)=\tau a\sigma, the chain rule and (4.32) give, for every fixed \eta>0,

The dependence D_0(x,w,h+\zeta b(x)) only adds the fixed compact derivative of b and does not change this conclusion.

The omitted flux is (1-\lambda)uA_\chi K(w,\cdot). Let B_K=A_\chi K(w,\cdot). Its derivative is bounded by a polynomial constant times 1+\sqrt{\mathcal T} on \mathcal K, by (4.32); derivatives of p vanish. The weight term requires the contraction rather than a bound for |\nabla\log u|. At the contact,

This involves only \chi H^f_{ee} and H^f_{e\perp} and is controlled by (4.13). It follows that

Combine the scalar identity with (4.25). The difference 8\pi\mu-\Scal_g/2 equals ((\tr K)^2-|K|^2)/2 and is compactly supported; so is J. The term F\tr K is compactly bounded by Lemma 4.5. On \mathcal K, Equation (4.27) therefore contributes c_*\mathcal T-C(1+N)^A. Outside \mathcal K, K=0, F=h, and (4.28) together with Lemma 4.6 gives c_*\mathcal T-C(1+N)^Av r^{-2\gamma}. The remaining Ricci term is bounded below by -C\mathbf1_{\mathcal K}-Cv r^{-3}; enlarge \mathcal K if necessary. Both end weights are bounded by a multiple of \rho_0+\rho_1. Finally subtract the omitted flux and use (4.33)–(4.34), choosing their two \eta’s with sum at most c_*/2. Fix 0<\delta_0<\min(c_*/4,1/4). Decreasing it slightly if necessary gives (4.30).□

Proof. Choose R_{\mathrm{floor}} to include the requirement of Lemma 4.6. A floor contact cannot occur on S_R. In the first three stages, an inner contact has j(t)=0. Equation (4.20) then cancels the omitted flux on both sides and gives V_\nu/u=\mathcal B. Equation (4.16) forces H+4\partial_\nu t=-N_B, so \partial_\nu t<0. This contradicts the nonnegative derivative into \Omega_R at a boundary minimum.

Consider an interior contact. Let E_N=C(1+N)^A[\mathbf1_{\mathcal K}+v(\rho_0+\rho_1)] denote the error in (4.30). Because \rho_0 has a positive minimum on \mathcal K, \mathbf1_{\mathcal K}\le C_{\mathcal K}\rho_0. Also v\rho_0\le\rho_0, and \rho/\rho_0 is globally bounded. We can therefore choose a polynomial C_N so large that, pointwise for all 0\le v\le1,

At the floor m_0=1, so the scalar equation and the lower estimate give

a contradiction. Enlarging C_N below preserves this inequality and preserves its polynomial dependence.

In the last stage f=0, t=Z, u=L, and v=0. At an interior minimum, \mathcal T=(|K|^2+(\tr K)^2)/2. Increase C_N polynomially if necessary so that \delta_0\mathcal T+\rho-C_N\rho_0<0 at all such points. For \eta_s>0, the scalar equation at a floor minimum has left side 4L\Delta t\ge0 and strictly negative right side. At an inner floor minimum its flux is 4\partial_\nu t=\eta_s(-H-N_B)<0, also impossible. If \eta_s=0, multiplication of \Div(4L(t)\nabla t)=0 by t, with zero outer value and zero inner flux, gives t=0. Thus the last stage has no floor contact either.□

Proof. For integers j\ge1 define

These functions are lower semicontinuous and take values in (0,1). Their lower relaxed limit \liminf_{j\to\infty,\,y\to x}v_j(y) is the function I_k(x) equal to zero on D_k and one on its complement. Indeed, at x\in D_k the constant sequence gives v_j(x)\le S_j(k)\to0, while v_j\ge0 everywhere. If u(x)>k, lower semicontinuity gives a neighborhood on which u\ge k+\delta for some \delta>0, and v_j\to1 uniformly there.

Let \phi define an exterior support of D_k at x: \phi(x)=0, \dd\phi(x)\ne0, and \phi\le0 on D_k near x. Shrink to a small normal-coordinate ball compactly contained in U on which |\phi|<1/2. Then \phi is a lower test of I_k at x. Subtract a small positive multiple of the fourth power of the distance from x to make this contact strict without changing its two-jet; write the resulting function as \phi_0. Minimize v_j-\phi_0 on the closed ball. The preceding relaxed-limit description, the strict contact, and v_j(x)\to0 show that the minimizers x_j lie in its interior for large j, satisfy x_j\to x, and have contact values v_j(x_j)\to0. Thus \phi_0+a_j is a lower test of v_j at x_j, where a_j\to0.

For each fixed large j, restrict to a neighborhood of x_j on which \phi_0+a_j takes values in (0,1). The increasing function S_j^{-1} makes S_j^{-1}(\phi_0+a_j) a smooth lower test of u. Its contact height is

for all sufficiently large j. Its gradient is nonzero, since \dd\phi_0(x_j)\to\dd\phi(x)\ne0. An increasing smooth composition preserves the oriented level expansion: it preserves the unit normal, and for \theta'>0,

The hypothesis therefore gives \mathcal E_Q[\phi_0+a_j](x_j)\le q_0. Letting j\to\infty yields \mathcal E_Q[\phi](x)\le q_0, because \phi_0 and \phi have the same two-jet at x.□

Proof. We prove the lower assertion. It is enough to consider an arbitrary sequence N_i\to\infty, R_i\to\infty and solutions in either of the first two stages. The elementary height and end estimates in Lemmas 4.5 and 4.6 give

Near each white face extend h_i to the other side by the constant b_+. Extend the fixed geometric coefficients smoothly there. The extended functions are continuous. Define the lower relaxed limit

It is lower semicontinuous, bounded, and satisfies the limiting end bound. No equicontinuity of h_i is asserted.

Choose b_-<k'<\min\{0,b_-+\eta\}, where \eta is the low-height sign range of the added term D_0. Suppose that a smooth function \phi touches \underline h from below at x, with \underline h(x)\le k' and d\phi(x)\ne0. Exclude for the moment the black faces. Subtracting a small fourth-order localization term from \phi makes the contact strict without changing its two-jet. Minimization on a small closed ball then produces, after passing to a subsequence, points x_i\to x and constants c_i\to0 such that \phi+c_i touches h_i from below at x_i, and h_i(x_i)\to\underline h(x). The boundary of this ball cannot contain x_i by strictness. Nor can x_i lie on a white face or on the extended side: their values there are b_+>k'. Thus every sufficiently large i uses the interior trace equation.

At these contacts,

Here a_i=l_i|\nabla f_i|/\sqrt{D_i} is the deformation variable. Positivity of A_{\chi_i} and the Hessian inequality at a lower contact imply

In the second stage D_0\le0 at these values; in the first stage it is absent. Consequently \limsup F_i(x_i)\le k'+C_b. Passing to the limit gives

This is the expansion of the level surface of \phi, oriented toward increasing \phi. The only Hessian used in this passage is the fixed test Hessian. Removing the localization proves the assertion for non-strict contacts as well.

Fix b_-<k<k' and put E_k=\{\underline h\le k\}. Apply Lemma 5.1 with u=\underline h, tensor Q=K, and q_0=k'+C_b. The function is finite and lower semicontinuous, and Equation (5.5) supplies the required bound for every nonzero-gradient lower test below k'. We apply the lemma locally away from black faces, using the constant extensions to obtain open neighborhoods across white faces. Every exterior support of E_k there consequently has expansion at most k'+C_b, with no regularity assumption on the sublevel.

This argument also rules out a hidden layer at a white face. At every point strictly on the added side, the relaxed limit equals b_+>k. Thus E_k lies locally in the original closed exterior, and the reversed white face is an exterior support of E_k at any contact there, with expansion

For k' sufficiently close to b_-, this is strictly greater than k'+C_b. The high extension ensured that all preceding low-value tests came from the interior equation, so the support contradiction applies at the face itself. Hence E_k avoids every white face. The end estimate makes E_k compact.

Adjoin the entire closed region \mathcal B_{c_b}. At a point on its smooth black frontier, every exterior support of the union also contains the black region locally. The tangency comparison therefore bounds its expansion by c_b<k'+C_b. At all other frontier points the preceding sublevel argument applies. The resulting compact closed set contains the required original inner barrier with a collar, because the black region already does. It avoids the outer barriers. Lemma 3.6 places it in a smooth black trapped region at any threshold c''>k'+C_b sufficiently close to c_b. It is consequently contained in \mathcal B_{c''}.

Finally let Q be as in the lemma. Hausdorff right continuity at c_b gives a \delta>0 such that \mathcal B_{c_b+\delta}\cap Q=\varnothing; decrease \delta so that the enlarged threshold is still negative and all barriers remain strict. Choose k,k',c'' with

The construction implies E_k\cap Q=\varnothing for every sequence under consideration. If the uniform estimate h>k on Q failed, choose violating solutions with N_i\to\infty, R_i\to\infty and x_i\in Q, h_i(x_i)\le k. A subsequence has x_i\to x\in Q and \underline h(x)\le k, a contradiction. This gives (5.3), after decreasing its positive margin.

For the upper assertion apply the entire argument to (-h,-K,-C), interchanging colors. The reversed black faces have strictly positive white expansion -c_b. Their exclusion and compactness give positive distance from the black region before applying the white version of Lemma 3.6. Adjoining the full white region is therefore legitimate in the fixed black complement. Empty white regions and missing face types cause no change: only an actually nonempty sublevel would supply a trapped seed and hence a contradiction.□

Proof. Write s for the smooth leaf parameter in an outward collar and \nu_s=\nabla s/|ds|. All geometric coefficients of this fixed collar are bounded, with |ds| bounded above and below. At a comparison contact with h_0=b_-+\psi(s), \psi'>0, the shared gradient gives

If the slopes are in a fixed positive interval, the first bound is d\le C\ell. None of these comparisons uses an upper bound for Z.

The trace operator on this test is

where P_s is the tangential trace of K on the leaf. Thus it is the leaf expansion H_s+P_s, with an error of absolute value at most C d, plus the displayed logarithmic-slope term. The same formula for a negative slope has a replaced by -a in the Hessian terms. This is an evaluation at the contact variables; it does not differentiate the unknown t.

On a black collar H_s+P_s\ge c_b and C=C_b-k_Bs. Choose small positive slopes so that \psi(0)=0 and \psi(s)\le k_Bs/2. At the outer end of a sufficiently short fixed collar, Lemma 5.2 lets this test lie below the solution. In the compatible direction the add-on D_0 vanishes. The residual of the trace equation on the test is therefore at least

Choose a smooth nonnegative \beta_N(s) equal to AN for 0\le s\le1/N, supported in 0\le s\le2/N, and satisfying \int\beta_N\le2A. Define

Choose the fixed A large enough that the final term of (5.9) dominates Cd on s\le1/N, using (5.7) and a\ge1/2 there. On s\ge1/N, the first term dominates Cd\le C\ell for large N; the taper contributes with the good sign. Since the slope multiplier is at most e^{2A}, a fixed sufficiently small m>0 enforces all the previous small-slope requirements. This constructs a strict lower barrier with slopes bounded above and below independently of N.

An upper barrier from b_- is constructed in the same collar with

The smooth leaf expansion differs from its boundary value by O(s). Choose the fixed M so large that \psi(s)\ge C_2s dominates this variation, the offset variation, and the height bound at the outer collar end. The logarithmic-slope term is now negative. On s\le1/N it dominates Cd; on the rest of the collar the negative margin proportional to s does so. The upper barrier is strict at every possible positive contact. Its slopes lie in [Me^{-2A},M], an interval independent of N.

To check the comparison signs, at a negative interior minimum of h-h_0 the Hessian of h dominates that of h_0, the gradients and Z agree, and hence so do t,l,A_\chi. The right side is strictly increasing in the height. Thus a strict positive residual for a lower barrier contradicts the equation. At a positive maximum the inequalities reverse, proving the upper-barrier assertion. Taking the one-sided derivative at s=0 proves the black part of (5.6). Replacing (h,K,C) by (-h,-K,-C) proves the white part.

In the third stage the boundary height is the assigned interpolated constant, say b_\zeta. Construct tests b_\zeta+\psi(s) and b_\zeta-\psi(s) with large positive \psi' and (\log\psi')'=-\beta_N. At s=0, the two directional inequalities of Proposition 4.4 say precisely that the positive-slope test has nonpositive limiting residual and the negative-slope test nonnegative limiting residual. At every comparison contact, the fixed positive barrier slope and Equation (5.7) give d\le C\ell, so a\ge1/2 for large N. On 1/2\le|w|\le1, the first w-derivatives of A_\chi in Equation (4.5) are bounded independently of 0\le p\le N. The other collar terms have fixed smooth coefficients on the bounded height range. Thus, at fixed boundary height the coefficient and geometric errors are O(s+d+\chi), since |w-(\pm\nu_s)|=1-a\le d and the radial comparison segment stays in this range. Changing the test height supplies, by height monotonicity, a residual of magnitude at least \psi(s) with the required sign. Large fixed slopes dominate the O(s) error and the outer-end height bound. The logarithmic-slope term has the required sign for both tests, as follows from (5.8); it dominates the O(d) error on the initial 1/N interval. The remaining margin dominates it beyond that interval. This proves the two-sided upper derivative bound, without using sublevel exclusion in the third stage.

Finally, on each own-color collar the lower barrier just constructed gives the required one-sided height bound. On the rest of a fixed compact region use Lemma 5.2; the opposite height bound on an own-color collar follows from that lemma as well, since the closed collar is disjoint from the other region. A finite cover proves b_-\le h\le b_+ on the compact regions in the statement.□

Proof. For \sigma>0, write e_f=\nabla f/\sigma, the axis denoted by e in Section 4. Set

Its norm in \bar g is a\le1. For any covector \xi,

Direct differentiation, using uW=Lw, gives

For example, \Div w=\tr_{A_d}H^f+dp\,a\partial_{e_f}t, while differentiating L adds p_La\partial_{e_f}t to the first formula. Coercivity in Lemma 4.2 bounds both right sides by \Pi_N(1+\sqrt{\mathcal T}). More explicitly, writing q=(K+H^f)_{e_fe_f}, the normal Hessian term has coefficient d^{3/2}, so it is bounded by \sqrt d|q|+C. The other Hessian terms are transverse traces, and the gradient term is at most (2N+2)\sqrt d|\partial_{e_f}t|. All are controlled by the stated coercivity estimate. This also establishes the formulas and bounds at zero gradient by their smooth direction-free extensions.

Let \varepsilon_B=1 on a black collar and -1 on a white collar. The signed slopes give w_\nu=\varepsilon_B a on B and a\ge1/2 for large N. Thus

The outward integration normal is -\nu. Apply the divergence theorem to -\varepsilon_B uW\varphi, with a fixed collar cutoff. Its boundary flux is La\varphi. Equations (5.12) and (5.13), and the fixed cutoff derivative, give (5.10). Replace u by \sqrt D and use (5.14) for (5.11). No factor \ell^{-1} was used.□

Proof. Extend the fixed base patch smoothly and embed a compact enlargement isometrically in Euclidean space by Nash’s embedding theorem [32, Theorem 2, p. 59]. Its product with the vertical line gives an isometric Euclidean immersion of the graph. The second fundamental form of the base embedding is bounded, so its contribution to the graph mean curvature vector is bounded. The scalar mean curvature in the product is

The prefactor is bounded by C(N), and d_0/\sqrt d=\sqrt{1+l^2\sigma^2}/(1+\sigma^2)\le e. Coercivity therefore gives |\boldsymbol H_\Gamma|\le C(N)(1+\sqrt{\mathcal T}). This estimate involves controlled quadratic energy, not a pointwise bound for the Hessian.

The ordinary Michael–Simon inequality [31], in the compact-support form stated in [39, Chapter 4, Section 5, Theorem 5.7, p. 98], gives after the boundary extension described below, for nonnegative \varphi,

One can obtain the boundary form directly from the compact-support version by smoothly extending each smooth graph across its boundary and cutting off on a collar tending to zero: the cutoff derivative integral converges to the boundary integral, and the added curvature integral tends to zero. This argument is applied to each smooth graph before the uniform estimate is taken. The only boundary in the support is an inner face, where f is constant and \dd A=\dd A_g. Equation (5.11) and the fixed-N graph comparisons bound its contribution by

Apply the resulting inequality to \varphi=|\omega|^4. Cauchy–Schwarz bounds its right side by

The left side is \|\omega\|_6^4; division, with the zero case separate, and squaring prove the assertion.□

Proof. It remains to extract the bounds for the conformal variable and the gradient. Since Z=t+\log D/8 and D\ge1, the floor gives -\epsilon\le t\le Z. The upper bound for Z gives

This proves the asserted fixed-N gradient bound. The height bounds are Lemma 4.5. The finite collection of sublevel and collar estimates and the polynomial absorption choose N_0 once. Enlarge R_{\min}(N) to include the outer estimates and all fixed compact patches used above. The next section derives Hessian and continuity estimates from these bounds and uses them to prove existence.□

Proof. We first obtain an L^p Hessian estimate for some p>2. A divergence identity then shows that a gradient nearly attaining its upper bound forces the solution to be nearly affine. The equation with a fixed gradient direction improves that affine approximation. The same steps survive reflection at a constant Dirichlet face, and iteration gives the stated estimate.

A Cordes estimate above exponent two. For a Euclidean unit vector e,

For a compactly supported smooth function v, Plancherel’s theorem gives \|D^2v\|_2=\|\Delta v\|_2. The Calderón–Zygmund operator taking \Delta v to the full array D^2v is bounded on L^p, and interpolation makes its norm arbitrarily close to one as p\to2. Choose once and for all

Rescale a metric-normalized patch until the coordinate principal matrix differs from its Euclidean axial matrix by less than \chi_0/2 in Frobenius norm. Absorbing this difference in the Laplacian estimate, applying a cutoff, and using \|Dv\|_p\le\eta\|D^2v\|_p+C_\eta\|v\|_p gives

The Christoffel terms are bounded first-order terms in this estimate. The same argument applies when v=z-L_0 and L_0 is coordinate affine: its extra right side is bounded by C\|Dg\|_\infty|DL_0|. The estimate uses bounded measurable principal coefficients only; the assumed C^3 regularity allows all of its applications without an approximation issue. The linear estimates just used are the ordinary Laplacian estimates; see [18, Chapters 7 and 9].

A nearly maximal gradient forces small Hessian energy. The preceding estimate does not yet give continuity of Dz. To gain continuity we use the fact that the axial direction is \nabla z itself. Put H=\Hess z and P=\nabla z. Equation (6.1) gives the pointwise vector identity

Indeed HP=|P|H(e,\cdot) and G=\Delta z-(1-\chi)H(e,e), so the two axial terms cancel. The identity is also valid at P=0. Taking the divergence of its right side, rather than differentiating \chi, yields

Choose a_0>0 so small that (1+a_0)(1-\chi_0)^2<1. Young’s inequality and the equation show

where c_0>0 depends only on \chi_0.

Consider solutions normalized by |\nabla z|\le1 on B_1. After making the scale small, let the forcing and the rescaled geometric errors have size at most \delta. The nonnegative function s=1-|\nabla z|^2 then satisfies

in a weak sense. In the Euclidean case the last term can be replaced by C\delta^2. We allow C\delta to include the geometric errors.

For the compactness step, suppose the normalized errors tend to zero in a sequence and \inf_{B_{1/2}}s\to0. The inhomogeneous weak Harnack inequality, applied to Equation (6.10) after dropping its favorable Hessian term, gives s\to0 in measure on B_{1/2}. The boundedness 0\le s\le1 also gives convergence in every finite L^p norm there. To recover the Hessian information, take a cutoff \eta supported inside B_{1/2} and use \eta^2(b-s)_+ in the weak supersolution inequality. Ellipticity and Young’s inequality give

for each fixed b>0 as the errors vanish. The error fields in Equation (6.10) are bounded and tend to zero, so their products with Ds are absorbed in deriving this inequality. Equation (6.6) supplies a uniform L^2 bound for Ds. Its integral over \{s\ge b\} tends to zero in L^1 by Cauchy’s inequality and convergence in measure. On \{s<b\} use Equation (6.11), then let b\downarrow0. Thus Ds\to0 locally in L^1. Testing Equation (6.8) with a nonnegative compact cutoff and using Equations (6.9) and (6.7) now gives

Uniform Lipschitz compactness implies that, after subtracting a constant and taking a subsequence, z tends uniformly on a smaller ball to an affine function. Its slope has length one because |\nabla z|\to1 in measure and the Hessians tend to zero.

Consequently, for any prescribed sufficiently small b_*>0 there are fixed r_*\in(0,1/4), k_*\in(r_*,1) and \delta_*>0 such that every normalized solution with errors at most \delta_* has one of the following properties:

If this conclusion failed, solutions whose gradient suprema approach one and whose errors approach zero would contradict Equation (6.12). A gradient-drop constant can always be weakened toward one to ensure k_*>r_*.

Regularity when the direction is fixed. The second alternative in (6.13) supplies a nonzero affine slope. The comparison equation for that slope has a fixed direction, although its axial coefficient remains merely measurable. Let e_0 be a Euclidean unit vector and suppose

The derivative v=\partial_{e_0}Y_0 belongs to W^{1,2} and, in distributions, satisfies

This follows by differentiating the distribution \chi\partial_{e_0}v as a whole. It does not assume a derivative of \chi exists as a function. De Giorgi’s estimate for this uniformly elliptic divergence equation, followed by Caccioppoli applied to v-v(x_0), gives an exponent \alpha_1\in(0,1) and

for balls in a smaller fixed interior region. These are the scalar divergence estimates with ellipticity \chi_0; see [18, Chapter 8].

All components of DY_0, not only v, are controlled by this estimate. In fact,

The last inequality follows from Cauchy’s inequality and Equation (6.16). Take the Newton potential of f_0 restricted to a slightly larger interior ball; its difference from Y_0 is harmonic in that ball. For two points at distance h, the contribution to the difference of gradient potentials from annuli of radius r\le2h is bounded by C\sum r^{\alpha_1}\le Ch^{\alpha_1}. On the remaining annuli the difference of gradient kernels is at most Ch r^{-3}, so their contribution is

The harmonic remainder has its usual interior derivative estimates. Therefore, decreasing the interior ball if needed,

Improving the affine approximation. We now compare the original equation with the fixed-direction equation. Assume that, for a coordinate-affine function L_0,

Let the forcing and the first-derivative metric errors be at most b\delta. For Y=(z-L_0)/b, Equation (6.6) on fixed nested balls gives

In particular, this estimate does not require a bound for DY in the supremum norm. Since Dz=DL_0+bDY, the variable axis converges in measure to e_0=DL_0/|DL_0| as b\to0. More quantitatively, the coefficient error E caused by replacing this axis by e_0 is bounded and satisfies

Indeed the unit-direction map is Lipschitz where |bDY|<|DL_0|/2, and elsewhere its difference is bounded; the L^{p_0} bound in Equation (6.21) and interpolation give Equation (6.22). The same estimate includes the metric principal-part error. Hölder’s inequality, with 1/2=1/q+1/p_0, shows that

Solve for a zero-Dirichlet correction w on B_{3/4} with the left side of Equation (6.23) as its source. This does not require a nondivergence Green function. The convex Dirichlet Hessian inequality \|D^2w\|_2\le\|\Delta w\|_2 and Equation (6.4) make the Laplace solution operator a contraction on W^{2,2}\cap W^{1,2}_0. Hence

The last implication uses W^{2,2}\hookrightarrow C^{0,1/2} in dimension three. The convex Hessian inequality is the Miranda–Talenti estimate [40, Section 2, Theorem 3, p. 297]. Now Y_0=Y-w satisfies Equation (6.14) with a uniform W^{2,2} bound.

Choose 0<\alpha_2<\alpha_1 and then a fixed \lambda_0\in(0,1/4) such that C\lambda_0^{1+\alpha_1}\le\lambda_0^{1+\alpha_2}/4 in Equation (6.19). By first taking \delta small and then b\le b_* small, Equation (6.24) has supremum norm at most \lambda_0^{1+\alpha_2}/4. Taylor expansion of Y_0 at the center therefore gives an affine function L_1 with

All constants are uniform in the slope range in Equation (6.20).

Reflection at a constant Dirichlet face. Subtract the boundary value and use normal coordinates with the face given by x_3=0. Reflect the metric across this plane and reflect z oddly. Tangential derivatives of z vanish on the face, so the extension is C^1. There is no distributional Hessian atom. The doubled metric is Lipschitz; the coefficient and lower-order estimates in Equation (6.6) hold with its essential first-derivative bound.

The flux in Equation (6.7) may have a normal jump. On either side of the face its normal trace is

Because the boundary value is constant, \tr_T\Hess z is its mean-curvature coefficient times \partial_\nu z. The jump is therefore bounded by C|\mathrm{II}|\,|\nabla z|^2. At scale r it is a bounded plane density of size O(r). Write the plane density and flux after multiplication by the reflected coordinate volume density, so that the following divergence is Euclidean. Such a signed density a(x')\,\delta_{\{x_3=0\}} is the divergence of the bounded field a(x')\mathbf 1_{\{x_3>0\}}\partial_3. Thus its negative part is another small divergence error in Equation (6.10). Smooth-side curvature errors are O(r^2). Weak Harnack and the low-truncation argument remain valid, and the flatness comparison uses the same Lipschitz metric estimates. This proves all the preceding alternatives for balls meeting the reflection plane as well.

Iteration and the required Hessian bound. There are now two ways to improve the solution on a smaller ball: its gradient bound decreases, or its approximation by a nonconstant affine function improves. We choose the constants once so that either route can be repeated. Keep \alpha_2,\lambda_0 from the affine improvement fixed. Choose b_* sufficiently small that Cb_*/(1-\lambda_0^{\alpha_2})<1/4, with C the slope-increment constant in Equation (6.25). Then choose r_*,k_*,\delta_* in Equation (6.13) for this fixed b_*. Normalize initially by a fixed multiple of L+Q and choose a uniformly small initial radius so that the metric and forcing errors satisfy all the preceding smallness conditions. If the first case of Equation (6.13) holds, rescale the ball by r_* and the gradient bound by k_*. The normalized forcing decreases by r_*/k_*<1. Repeating this step either continues forever or reaches the second case. In the former case the gradient decays with exponent \log k_*/\log r_*. In the latter case apply Equation (6.25) successively, retaining exponent \alpha_2. At each affine step, rescale z by dividing by the spatial scale, subtracting only an additive constant; this preserves its gradient bound. Its slope changes have total size at most Cb_*/(1-\lambda_0^{\alpha_2}), so the slopes remain in [1/4,4]. At the jth step the error size is b_*\lambda_0^{j\alpha_2}, while the normalized forcing and metric first-derivative errors have size O(\lambda_0^j). Since \alpha_2<1, their ratio to the error size only decreases. This justifies every subsequent application of the improvement lemma. To combine the two branches, now choose

At every center we have consequently obtained affine functions L_{x,r} satisfying

The slope differences at consecutive radii are bounded by C(L+Q)r^\alpha; comparing the affine approximations on overlapping balls gives Equation (6.2). This also identifies their limiting slope with Dz, since z is C^1. The reflected version gives the estimate up to a boundary.

Finally apply the L^2 version of Equation (6.6) to z-L_{x,2r} after rescaling. The affine error contributes C(L+Q)r^{1/2+\alpha} to the Hessian L^2 norm; the bounded forcing and affine Christoffel term contribute C(L+Q)r^{3/2}. Squaring and using \alpha<1 proves Equation (6.3).□

Proof. The proof has two steps. An exponential change absorbs the quadratic term in DZ; a small potential correction then permits weak Harnack to decrease the essential oscillation. Both steps work in the two dimensions stated in the lemma. Restricting all radii to at most one, we may replace \beta by \min\{\beta,1\} in the measure bound. We retain the symbol \beta for this positive reduced exponent; this also makes the small-scale mollification bound below immediate.

Write \mathcal L=-\operatorname{div}(aD) and, for s\in\{-1,1\}, set V_s=e^{skZ}. The assumed product rule is

The term V_se uses the fixed bounded representative. For a continuous piecewise smooth function the identity follows by integration on the two sides; its common trace multiplies the flux jump. Choose k\ge\max\{1,4C_0/\lambda\} and use B|DZ|\le(\lambda/2)|DZ|^2+B^2/(2\lambda). The gradient-square term from e is absorbed by the negative quadratic term for both signs. Since e^{-kM}\le V_s\le e^{kM}, this gives

Thus no small-ball estimate for |DZ|^2 is needed.

On a ball B_R in the smaller patch solve the zero-Dirichlet problem \mathcal L w_s=\operatorname{div}F_s+\nu. For the bounded vector source, the scalar bounded-solution estimate with any fixed p>n gives

this is the scaled divergence-source estimate of [29, Theorem 2.6, p. 50]. For the positive measure source, extend a elliptically to B_{2R}. The scalar Green comparison theorem [29, Theorem 7.1 and Remark 2, p. 66], and domain monotonicity, give

Using B_{2R} keeps both points in a fixed compact interior region of the comparison ball after rescaling. Dyadic annuli and (6.30) imply

All balls used in this estimate lie in the chosen larger patch.

This potential also gives an energy solution. Mollify the restriction of \nu to B_R and restrict the resulting positive density again to B_R. Its ball-growth bounds are uniform: above the smoothing scale use an enlarged ball, and below it use the mollified density bound. The variational solutions have the preceding supremum bound and obey

Weak compactness in W^{1,2}_0 and distributional convergence produce the desired measure-source solution. Adding the vector-source solution, and taking R\le1, yields

All extrema in the rest of this proof are essential extrema. Put M_R=\operatorname*{ess\,sup}_{B_R}Z and m_R=\operatorname*{ess\,inf}_{B_R}Z. The functions

are nonnegative almost everywhere and satisfy \mathcal L H_\pm\ge0. At least half of B_{R/2} satisfies either Z\le(M_R+m_R)/2 or Z\ge(M_R+m_R)/2. On the first set the positive exponential deficit is at least c(M_R-m_R); on the second set the negative exponential deficit has that lower bound. Their derivatives are bounded above and away from zero on [-M,M]. Weak Harnack for the corresponding H_\pm therefore either lowers the essential supremum or raises the essential infimum on B_{R/4} by at least c'(M_R-m_R)-C\varepsilon_R. Hence

Choose a positive exponent smaller than \min\{1,\beta\} and than -\log(1-c')/\log4. Iteration gives an essential oscillation bound Cr^\gamma at every center in a smaller patch. The averages of Z on balls centered at x are consequently Cauchy as their radii decrease to zero. Their limit defines Z^*(x) at every center; comparison on overlapping balls gives |Z^*(x)-Z^*(y)|\le C|x-y|^\gamma. Lebesgue differentiation identifies Z^*=Z almost everywhere. This proves the representative assertion. For continuous Z the representatives agree everywhere. No conclusion about arbitrary values of the initially chosen Borel representative on a null set is required.

For the boundary application, flatten a face and set S=\operatorname{diag}(1,\ldots,1,-1). An even extension uses a_-=Sa_+S and b_-=Sb_+; its plane source is twice the prescribed total normal flux. An odd extension after subtracting a constant Dirichlet value uses a_-=Sa_+S, b_-=-Sb_+ and the sign-reflected bulk source; its normal flux is continuous. The coordinate volume density is included in these coefficients. For the smooth reflected solutions used here the product rule holds, and a bounded plane density has ball mass O(r^{n-1}), giving \beta=1 for that part of the measure. This verifies the boundary clause whenever the remaining reflected measure has the stated growth.□

Proof. First, control Df and the Hessian measure. Proposition 5.6 bounds f,Z,t and \sigma=|\nabla f|. At fixed parameters this bounds u,l,D above and away from zero and gives \chi\ge\chi_0>0. After dividing the scalar trace equation by l/\sqrt D, its right side is

It is bounded throughout the homotopy, including its added collar terms. Lemma 6.1 therefore gives a uniform C^{1,\alpha} estimate for f and

on the uniform patches, including boundary patches where f is constant.

Second, control the oscillation of Z. After multiplication by the Riemannian volume density, the second equation has coordinate principal coefficients a^{ij}=4\sqrt{\det g}\,uA_\chi^{ij}, a bounded flux vector b, and a source bounded in absolute value by

Indeed t is a smooth function of x,Z,Df on the bounded variable range. Thus Dt is a bounded linear combination of DZ,D^2f and fixed smooth terms; the expression \mathcal T is quadratic in these quantities. All other source terms have lower order. The same volume density is included in b,e; it is smooth and bounded above and away from zero on the chosen patches, so it preserves these bounds and uniform ellipticity. The prescribed conormal flux at the inner boundary is bounded. At the outer face Z=0. Apply Lemma 6.2, with \mu=C|D^2f|^2\dd x and the bounded reflected plane density. Equation (6.38) supplies its measure growth hypothesis. We obtain a uniform positive Hölder exponent for Z.

Third, gain two derivatives of f. The coefficients and source of the scalar equation are smooth functions of x,Z,Df on the bounded range. They are therefore Hölder now that Z and Df are Hölder. Interior and constant-Dirichlet boundary Schauder estimates give f\in C^{2,\theta} for a uniform \theta>0. Expand the Z-equation in nondivergence form. Its principal coefficients are uniformly elliptic and C^{0,\theta}, while its right side is bounded by C(1+|DZ|^2). On an inner face Df is normal. Therefore A_\chi has zero normal–tangential components there and normal eigenvalue \chi. Its boundary equation solves for

where \mathcal H_s is smooth on a fixed bounded variable range. Its derivatives with respect to x and its displayed scalar arguments are bounded at fixed parameters. The outer condition remains constant Dirichlet. These are different, disjoint smooth boundary components; there is no change of boundary condition along an edge or a corner.

Fourth, absorb the quadratic growth and control DZ. Continuity of Z supplies the small factor needed for this step. Fix p>3. The linear local W^{2,p} estimate for the leading nondivergence operator with Equation (6.40) or outer Dirichlet data has the form, on fixed nested patches U_x\Subset V_x,

Here the boundary norm is controlled by

using trace and composition after extending its smooth expression through the collar. The derivatives of f,Df entering this extension are already bounded. The leading matrix has a fixed Hölder modulus, so its small oscillation on sufficiently small patches permits the usual freezing and absorption in the linear estimate. The normal boundary operator is uniformly oblique and satisfies the complementing condition. The general boundary theory is developed in the two papers of Agmon, Douglis and Nirenberg [1, 2]. The scalar estimate used here is [1, Theorem 15.2, estimate (15.5), p. 704].

The outer patches U_x,V_x,V_x^+ and the constant in (6.41) are fixed before the next radius h is chosen; only the interpolation is localized further. Let [Z]_{C^{0,\theta}}\le H_0. On balls or half-balls of radius h, the scaled Gagliardo–Nirenberg inequality, applied after subtracting a constant, gives

One can use a bounded extension operator on a half-ball; its constants are uniform in the boundary charts. Here the interpolation exponents are spatial dimension 3, derivative orders 2 and 1, top-derivative exponent p>3, height exponent \infty, and interpolation parameter 1/2; the resulting gradient exponent is 2p. This is [33, Lecture II, Equation (2.2), p. 125, and Remark 5, p. 126], including the lower-order term on a bounded patch. Subtracting a constant gives the oscillation, rather than the absolute height, as the coefficient of \|D^2Z\|_p. Cover V_x by radius-h patches of bounded overlap and take the \ell^p sum of Equation (6.43). It follows that

Also, for any \eta>0, \|Z\|_{W^{1,p}(V_x)} \le\eta\|D^2Z\|_{L^p(V_x^+)}+C_\eta. Each enlarged patch V_x^+ is covered by a bounded number of the patches U_y, independently of R. Set S=\sup_y\|Z\|_{W^{2,p}(U_y)}, finite for each smooth finite-domain solution. Equations (6.41) and (6.44) give

Choose h and then \eta so the first coefficient is less than 1/2. We obtain a uniform W^{2,p} bound and hence a C^{1,1-3/p} bound for Z.

Finally, obtain all remaining derivatives. The bounds f\in C^{2,\theta} and Z\in C^{1,1-3/p} make the expanded Z source Hölder and its normal datum \mathcal H_s(x,Z,f,Df) of class C^{1,\theta'}, after reducing the positive exponent. Its principal coefficients are Hölder as well, so both the interior and normal-boundary Schauder estimates apply. First the nondivergence Z-equation has Hölder source, giving Z\in C^{2,\theta'}; the scalar f-equation then gains another derivative. Repetition gives every finite classical derivative supported by the smooth data. The same procedure holds on the uniform unit patches of the asymptotic end and on the large outer spheres, whose normal-coordinate constants are uniform. This proves the asserted independence of R. The prepared background has end symbol bounds of every order by Definition 2.2; no regularity constant here requires additional derivatives of the original weak-decay data.□

Proof. For the scalar continuation below, use the affine space of C^{3,b} functions with the assigned Dirichlet values and the equation space C^{1,b}. Its individual solutions meet the C^3 hypothesis of Lemma 6.1; no uniform third-derivative bound is assumed before that lemma is applied. The monotonicity \partial Z/\partial t=1+pv/4>0 defines t=t(x,Z,Df) smoothly for every finite input, including Df=0. The principal matrix has no dependence on the value of f. The right side is increasing in h=\tau f, with derivative at least one. At a height extremum the collar terms vanish and Df=0, so the equation gives fixed constant upper and lower barriers. Including all prescribed boundary heights, we obtain

where the constant can be chosen independently of Z and the homotopy parameter. We include the fixed boundary values in its choice.

Large slopes. Only |Z|\le M is assumed here. As \sigma\to\infty, the implicit equation forces t\to-\infty uniformly in Z. There p=N and l=e^{Nt}, so its exact form is

Consequently

The constants are uniform for |Z|\le M at fixed N. Since N\ge4, the exponent 8/(N+4) is at most one. Combining the large-slope estimates with compactness of the remaining variable range yields

whenever Equation (6.47) holds.

Boundary slopes. The estimate \chi\ge c/(1+\sigma) suggests a concave logarithmic barrier: its axial second derivative can dominate terms of linear growth in its slope. Let \delta be smaller than a fixed collar radius, half the distance between distinct boundary components, and a normal injectivity radius of a smooth extension of the finite domain. On a boundary distance collar use

The upper and lower barriers are the assigned face value plus or minus \psi. On these tests the axis is normal to the distance leaves, so their principal trace is \chi\psi''+\psi'\Delta r. At slopes at least one, Equation (6.50) gives

Choose k large enough to dominate all the bounded geometry terms and the linear growth in Equation (6.50). Next shrink \delta so that 1/(2k\delta) exceeds any fixed large-slope threshold used here. Finally choose B large enough that B\delta\ge1 and \psi(\delta)>4C_0/\tau. The latter exceeds twice the bound for \operatorname{osc}f from (6.47). Then \psi'\ge1/(2k\delta) on this collar and the two barriers compare by the height monotonicity. This gives a uniform boundary gradient bound.

Interior slopes. Use the same modulus for differences at two points. After increasing B if needed, consider

The distance is taken in the fixed smooth extension. The boundary barriers exclude a positive maximum with one endpoint on a face: the other point lies in that face’s collar, its distance to the face is at most the distance between the pair, and \psi is increasing. The separation choice prevents endpoints on distinct faces. The inequality \psi(\delta)>2\operatorname{osc}f excludes the other boundary of the two-point domain. At a positive interior maximum let r=\operatorname{dist}(x,y) and q=\psi'(r). The two gradients have length q and are parallel transports along the connecting geodesic. Vary both endpoints simultaneously in parallel transverse directions. The second variation formula for the short geodesic yields

Varying each endpoint separately along that geodesic gives f_{;e_xe_x}(x)\le\psi''(r) and f_{;e_ye_y}(y)\ge-\psi''(r). Subtracting the two scalar equations therefore gives

By Equations (6.50) and (6.51), the last term is at most -ckq at the large comparison slopes. Our choice of k gives a contradiction. Thus Equation (6.53) is nonpositive. Reversing x,y proves the same bound for the absolute difference and, on letting y\to x, a global Lipschitz bound for f. The endpoint variations are interior variations in the smooth extension; the connecting short geodesic need not stay inside \Omega_R. This argument used no continuity modulus for \chi_x-\chi_y; only the common transverse eigenvalues and the radial lower bound were needed.

Continuation and uniqueness. The height and gradient estimates supply compactness for a scalar continuation. Interpolate the normalized equation with \Delta f=\tau f, retaining its prescribed boundary values. Explicitly, for a\in[0,1] take principal matrix (1-a)I+aA_\chi and right side (1-a)\tau f+aG_f. The height sign, the linear gradient growth, and Equation (6.50) persist uniformly in a. The height and logarithmic gradient bounds just proved are therefore uniform. After these bounds the interpolating equations are uniformly elliptic. Their matrices still have the axial form, with axial eigenvalue (1-a)+a\chi, so Lemma 6.1 applies. Schauder estimates then give a uniform C^{2,\theta} bound at fixed input Z. The coefficients and right side are now Lipschitz in position on bounded input sets, which permits their C^{0,b} estimates and an upgrade to C^{2,b}. Differentiating once, or using the next Schauder estimate, yields f\in C^{3,b} because Z\in C^{1,b}.

The scalar linearization has the form a^{ij}D_{ij}\varphi+b^iD_i\varphi-c\varphi with c\ge c_1\tau>0 on the bounded variable range. Indeed the principal matrix is independent of the height, while \partial_fG_f\ge\tau\sqrt D/l>0. The Dirichlet maximum principle gives uniqueness and a zero kernel; the linear elliptic Dirichlet theory gives an inverse from C^{1,b} to C^{3,b}_0, where the subscript denotes zero data on the entire boundary. The implicit-function theorem gives openness of the interpolation parameter set. The uniform classical bounds and compactness give closedness, starting from the invertible Laplace endpoint. This proves existence. The same maximum principle applied to the difference of two solutions proves uniqueness of the nonlinear solution. The implicit-function theorem, or compactness and uniqueness, gives continuous dependence on Z and on the homotopy parameter. At the finitely many concatenation points the scalar equations agree, so the same continuity holds there.□

Proof. The fixed points in \overline{\mathcal U} form a compact set \mathfrak F. Indeed, boundedness of \mathcal U, compactness of I and of the images of T give a convergent subsequence of any sequence of such fixed points; continuity preserves the fixed-point equation. The hypothesis places \mathfrak F inside \mathcal U. Also \{s\}\times\partial\mathcal U_s lies in the relative boundary of \mathcal U, so each section degree is defined.

Fix s_0\in I. Cover its compact fixed-point slice by finitely many open balls in X whose closed balls remain in \mathcal U_s for all s sufficiently close to s_0. Relative openness and the finite cover permit this common parameter interval. Let U be their union. All fixed points in the closed sections for these nearby parameters lie in U after shortening the interval: otherwise compactness of \mathfrak F would give an omitted fixed point in the slice at s_0. Excision identifies the degree on each \mathcal U_s with that on U, and fixed-domain homotopy invariance makes the latter constant. If the slice is empty, compactness instead gives no nearby fixed points in the closed sections, so those degrees are all zero. Thus the section degree is locally constant, and connectedness of I makes it constant. These are the excision and homotopy properties of Leray–Schauder degree [27, Sections 12–14, pp. 59–61]; see also [13]. At the stated terminal map, normalization gives degree one; the existence property of nonzero degree proves the last assertion.□

Proof. Choose the parameters before defining degree. Fix the prepared data, thresholds, collars and \epsilon. Choose an integer N_0 so that

and so that the floor test and all estimates of Proposition 5.6 apply for N\ge N_0. Let R_{\mathrm{ap}}(N) dominate every lower-radius requirement in the a priori estimates and in the floor exclusion; let R_{\mathrm{geo}} contain all compact boundary faces, collars and supports of the construction. The end barrier of that proposition gives a constant C_{\mathrm{out}}, independent of N,R,s, such that every admissible or floor-contact homotopy solution has

The scalar last stage has f=0 and satisfies the same bound. Choose, enlarging to a radius of a smooth truncation if necessary,

At Z=0, the value of its implicit expression at t=-\epsilon is -\epsilon+\frac18\log(1+\ell^2\sigma^2). Equations (6.57) and (6.58) make this strictly negative. Strict monotonicity of that expression in t therefore excludes an outer floor contact. The term N in Equation (6.58) ensures R_{\min}(N)\to\infty.

The sublevel and collar estimates apply to arbitrary smooth admissible or floor-contact solution sequences with N\to\infty and expanding outer cutoffs, uniformly over the compact ranges of the homotopy parameters. If the needed conclusion failed for arbitrarily large N with R\ge R_{\min}(N), one could select such a violating sequence. The chosen radius convention would make it precisely an expanding sequence excluded by those proofs. Thus a single N_0, followed by the radius choices above, suffices for every homotopy parameter. No upper bound for Z is inserted in the earlier contact estimates: there \sigma\asymp\tau^{-1} and l\ge\ell alone give d\lesssim(\tau/\ell)^2.

Define the compact map on all bounded input sets. For the rest of the degree argument fix N\ge N_0 and R\ge R_{\min}(N). Work in the Banach space

For an input Z\in X_b, first solve f=\mathcal S_s(Z) by Lemma 6.4. Compute t,u,A_\chi,\mathcal T and the full homotopy source from this input pair. Write B_s for the frozen drift summand in the flux, E_s for the full right side of its divergence equation, and q_s for the prescribed inner flux. Thus before the final scaling B_s=\lambda_suA_\chi K(w,\cdot) and E_s=u\Xi; on the final interval E_s and q_s have the simultaneous scalar factor prescribed in Proposition 4.4. Define \mathcal K_s(Z)=\widetilde Z by the linear mixed problem

All coefficients in this problem are computed from the input (\mathcal S_s(Z),Z); the only unknown is \widetilde Z. The normal \nu on the inner faces points into the domain. For a test function \varphi vanishing on S_R, the weak form is

The sign of the boundary term comes from using -\nu as the integration normal. The scalar estimate bounds the leading coefficient above and away from zero on each bounded input set. The nonempty outer Dirichlet face supplies Poincaré’s inequality, so the bilinear form is coercive and has a unique solution.

On bounded input sets in C^{1,b}, Lemma 6.4 bounds f in C^{3,b}. Hence the leading coefficient and drift in Equation (6.60) are C^{1,b}, the bulk source is C^{0,b}, and the prescribed boundary data are C^{1,b}. The derivatives of t(x,Z,Df) involve only DZ and D^2f, so this assertion includes the quadratic expression \mathcal T. Linear regularity gives a bounded output set in C^{2,b}. The inclusion C^{2,b}\hookrightarrow C^{1,b} is compact on the finite domain. Continuous dependence of the scalar solve and the linear problem makes (s,Z)\mapsto\mathcal K_s(Z) a continuous compact homotopy on bounded sets. These assertions hold also at the concatenation parameters because the defining equations coincide there. A fixed point solves the coupled system; repeated Schauder estimates make it smooth.

Exclude fixed points from the boundary of admissibility. Write t_s[Z]=t(x,Z,D\mathcal S_s(Z)). The scalar solution must be computed before this admissibility test. Define

It is relatively open because the scalar solve and the implicit function defining t depend continuously on (s,Z). Propositions 5.6 and 6.3 give a uniform bound in C^{1,b} for all fixed points satisfying t\ge-\epsilon. If initially needed, use their higher classical bounds to reach the chosen exponent b. Choose M_1 larger than this bound. No fixed point lies on the norm boundary of \mathcal O. No fixed point lies on its floor boundary either: outer contacts were excluded above and all remaining contacts are excluded by Proposition 4.9.

Apply degree on the varying admissible sections. The sections \mathcal O_s need not be constant in s. Lemma 6.5 applies with I=[0,4], X=X_b, T_s=\mathcal K_s and \mathcal U=\mathcal O. We have proved continuity and compactness on bounded input sets; the norm condition makes \mathcal O bounded and the scalar solve makes it relatively open. Its relative boundary can contain only norm contact or floor contact, and neither can be a fixed point by the preceding estimates. Thus its section degree is constant.

At s=4, the scalar equation has the unique solution f=0. The drift vanishes, and both E_4 and q_4 vanish for every input. Equation (6.60) then has output identically zero. Moreover, t_4[0]=0>-\epsilon, so 0\in\mathcal O_4 and

Homotopy invariance gives the same degree at s=0. There is therefore a fixed point in \mathcal O_0, which is the desired smooth solution with t>-\epsilon.

Remove only the outer truncation. Keep N and all other deformation parameters fixed. Proposition 6.3 gives estimates independent of R on an exhaustion by compact sets, using boundary charts at the fixed inner faces. A diagonal subsequence converges smoothly on compact subsets of \overline\Omega. Both equations and their inner boundary conditions pass to the limit. The strict finite-domain inequality gives the non-strict limit inequality t\ge-\epsilon. The following section establishes end normalization, decay and mass flux; those conclusions do not follow from compact exhaustion alone.□

Proof. The compact convergence, including at the fixed inner boundary, follows from Theorem 6.6 and its uniform fixed-N classical estimates. All equations and inner boundary conditions pass to this limit. To preserve the end normalization, we prove the decay estimates on the truncations before passing to infinity.

Outside a fixed large sphere, K=C=0, and the first equation is

At fixed N the preceding estimates make the left side uniformly elliptic, and its positive height coefficient is bounded below by a positive constant depending on N. It is bounded above as well. For sufficiently small c_N>0, the functions \exp[-c_N(r-r_0)] are positive supersolutions of the corresponding linear equation outside a sufficiently large r_0: their radial second derivative has size c_N^2, while their tangential second derivatives have size c_N/r, and the fixed positive height term dominates both. Multiplying by a constant bounds the solution on S_{r_0}; the zero outer Dirichlet value is also bounded. The maximum principle applied to both signs of f gives the uniform exponential estimate. The fixed-N unit-patch estimates, followed by the local homogeneous linear estimates for this equation, give the same decay for derivatives. The corresponding boundary estimates apply on the large outer spheres. Iterating the smooth equations gives each required finite derivative order.

The relation Z=t+\tfrac18\log(1+l^2|\nabla f|^2) now shows that t-Z is exponentially small, with derivatives. The graph errors have the same property. On the end the second equation becomes

Indeed A_\chi=I and u=L(Z) up to exponentially small errors, whereas \mathcal T=4(p_L(Z)-1)|dZ|^2 up to such errors. The \rho_0 penalty weight is O(r^{-4}); the term with \rho_1 is multiplied by the exponentially small v. All coefficients in this statement are bounded on the fixed-N solution range. Differentiably controlled exponential errors can absorb any fixed polynomial power of r.

Define an increasing smooth function on that range by

Its derivative is bounded above and below by positive fixed-N constants. The chain rule in (7.2) gives

The function is bounded and vanishes at the outer cutoff. To obtain a bound independent of that cutoff, choose 0<\eta<1. For large r, a positive multiple of r^{-1}-r^{-1-\eta} satisfies

Here the Euclidean leading term is \eta(1+\eta)r^{-3-\eta}, and the metric error is O(r^{-4}). It dominates the right side after increasing the multiple and the fixed inner radius. Comparison for both signs yields |\Phi_N(Z)|\le C'(N) r^{-1}, where C'(N) is a fixed-N constant with no polynomial claim. Consequently Z=O(r^{-1}).

At sufficiently large radius this estimate and the exponential graph error give t>-\epsilon+1/N; hence the floor penalty vanishes there. Rescale each annulus of radii comparable to r to unit size. The equation for \Phi_N(Z) has a bounded rescaled source and uniformly controlled metric coefficients. Local W^{2,p} estimates with p>3 first give the gradient decay. Its source is then a smooth weight times a smooth function of the bounded variables, plus the already controlled exponential errors. Schauder estimates, or a further Sobolev bootstrap followed by them, give Z=O_2(r^{-1}). Inverting \Phi_N preserves these bounds, and the same holds for t.

Since L(0)=1, we have u=1+O(r^{-1}) and A_\chi=I+O(e^{-c_N r}). With K=0 on the end, the definition of V gives (7.1). Moreover \Div V=u\Xi is integrable there: \mathcal T=O(r^{-4}), \rho=O(r^{-4}), the penalty has vanished, and \tau a\sigma decays exponentially. The divergence theorem on annuli proves the existence of the outward flux limit.□

Proof. At the physical endpoint the curvature identity gives

The height bounds on the compact supports of K and C give h\in[b_-,b_+]. Since |w|\le1, the terms |F\tr K|+|\nabla C|+\rho are dominated by 8\pi(\mu-|J|), by the choices in Proposition 3.7 and the choice of \rho. Outside those supports the same conclusion uses 8\pi(\mu-|J|)-\rho\ge0. All remaining terms are nonnegative. This proves R_{\hat g}\ge0.

The boundary identity and the prescribed flux give H+4\partial_\nu t=-N_B<0. Since f is constant on each face, this is exactly the mean-curvature sign for \hat g. Also \hat g\ge e^{-4\epsilon}g, so any escaping curve has infinite length; the compact boundary is included. Smoothness and completeness therefore follow.

Lemma 7.1 gives the stated metric decay. The prepared-data hypothesis supplies R_g=O(r^{-3-\delta}) on the end. Since graph errors are exponentially small, the conformal formula gives

Equation (7.2), the vanishing far-end penalty, and the gradient estimate give \Delta_gt=O(r^{-4}). This proves the asserted scalar-curvature decay, including the pointwise falloff required by the AF convention of [8, Definition 21 and Theorem 19].

Exponential graph errors contribute no ADM flux. The conformal change and t=O_2(r^{-1}) give

Replacing the metric normal and area form by their Euclidean versions costs o(1), and the O(r^{-3}) error in (7.1) has vanishing sphere integral. This proves (7.3).□

Proof. The normal \nu on B points into \Omega, so the divergence theorem has the sign

Each integral is finite for a fixed N, by Lemma 7.1 and compact smoothness.

At a black face, F-P_B=H and w_\nu=a. At a white face, F-P_B=-H and w_\nu=-a. Thus at either type of face,

The signed boundary gradient bounds give \sigma\ge c/\tau, so

Use the polynomial collar trace estimate from Section 5 with the constant test function:

where \mathcal C is a fixed compact union of collars. Its polynomial constant is independent of the upper Z bound. Since \rho has a fixed positive minimum on \mathcal C,

The factor C(1+N)^C\tau/\ell =C(1+N)^C\ell^{1/2} tends to zero. For large N, the entire negative boundary term in (7.4) is therefore at most one quarter of \int_\Omega u(\delta_0\mathcal T+\rho)\dd V_g.

It remains to estimate the floor penalty. On its support, -\epsilon\le t\le-\epsilon+1/N, so l and L are comparable to \ell, with constants independent of N. Directly from their definitions,

Consequently

Young’s inequality, with the square term chosen to be a quarter of \delta_0\tau ua\sigma, yields

Squaring the constant only changes the polynomial C_N. The integrals of \rho_0, \rho_0^2, and \rho_1^2 are finite: on the end their orders are respectively r^{-4}, r^{-8}, and r^{-4\gamma}, with 4\gamma>3. Integrating leaves at most C(1+N)^C(\ell+\ell^2/\tau), after absorption.

Substitute both bounds in (7.4). Positive fractions of the three bulk terms \delta_0\mathcal T,\rho,\delta_0\tau a\sigma remain. Dropping them proves the claim. Finally \ell^2/\tau=\ell^{1/2}, so the error tends to zero exponentially against every polynomial loss.□

Proof of Theorem 2.3. The fixed subexterior and preservation of every full cut are supplied by Proposition 3.7. Fix \epsilon and take the global solutions of Lemma 7.1, obtained by first exhausting the end at fixed N. Lemma 7.2 gives completeness, the two curvature signs, and the end decay. The pointwise inequality \widehat g\ge e^{-4\epsilon}g gives the two-dimensional area inequality on every full cut, including every component and every coincident portion. The energy formula and Lemma 7.3 give the asserted upper error with \eta_{\epsilon,N}=\varepsilon_N/(8\pi). This completes the geometric construction before any Riemannian Penrose comparison.□

Proof. For each fixed sufficiently large N, minimize \hat g-area among filled cuts enclosing B, using a large outer sphere as an auxiliary barrier. The independent enclosure results of [34, Lemmas 2.1–2.3] apply to this smooth AF metric; we check their hypotheses and the strict detachment here. A fixed enclosing competitor bounds the area, and local perimeter density estimates prevent minimizers from escaping to infinity. The positive mean curvature of sufficiently large outer spheres excludes contact there, as in that enclosure theorem. For the inner barrier, let U_s be a short outward normal collar of the filled obstacle and Y its outward unit foliation field. Strict negativity gives \Div_{\hat g}Y<0 throughout the collar. If a perimeter minimizer E omitted a positive-volume portion D=U_s\setminus E, Gauss–Green and |Y|_{\hat g}\le1 would give

Thus E contains a whole inner collar; obstacle regularity upgrades this inclusion to its regular representative. Its free boundary is disjoint from the obstacle. The resulting cut \Gamma_N is compact, smooth, and minimal in dimension three; it may be disconnected. It is outer area-minimizing by its minimizing property. Equivalently one may first use the ordinary trapped-region theorem for (\hat g,0) and then take the outer minimizing enclosure. The boundary version of the Riemannian Penrose theorem applies to the exterior of \Gamma_N, by Lemma 7.2 and [8, Theorem 19].

For every two-plane the restriction of \bar g=g+l^2df^2 dominates that of g. The floor therefore gives

The last inequality uses that B, and hence \Gamma_N, cuts off the original filled obstacle. Lemmas 7.2 and 7.3 now give

Let N\to\infty with \epsilon fixed, and then \epsilon\downarrow0. This proves the reduced inequality.

Finally apply the weak three-dimensional rest-preparation theorem of [35, Proposition 7.13] to an exterior satisfying the hypotheses of Theorem 2.4. Write (g_j,K_j) for the resulting data. Their tensor fields have compact support, their momenta vanish, and their metrics have all-order r^{-1} decay and the stipulated scalar falloff. Discard a finite initial segment so that E_j>0. Strict trapping and strict DEC hold on the compact core. The positive minimum there of 8\pi(\mu_j-|J_j|_{g_j})\varrho^{3+\delta}, together with the end margin, gives the global weighted bound after decreasing c_j. The full enclosing area is positive by the geometric enclosure lemma. Thus each fixed j satisfies Definition 2.2; no constant uniform in j has been used in its deformation solve. The reduced energies converge to its invariant ADM mass and the full enclosing infima converge to the original enclosing infimum. Applying the reduced bound to each member of that sequence and passing to the limit gives Equation (2.10).□

Proof. Fix \eps>0 and a sufficiently large N, and abbreviate \widehat g=\widehat g_{\eps,N}. Proposition 8.3 applies to (\Omega,\widehat g): Theorem 8.2 gives smoothness, completeness, one end with exponent q=2>1, and the strictly negative boundary mean curvature. Let \Sigma be its detached minimal frontier. Its exterior is connected, complete with boundary, and has the same end and energy. Outer area-minimization allows all components of every enclosing competitor. Since \Sigma is itself a full cut and the metric comparison holds on its tangent three-planes,

The numerical part of the four-dimensional Riemannian Penrose inequality of Bray–Lee [11, Theorem 1.4] now applies. Its input is a smooth connected exterior, complete including its minimal outer-area-minimizing compact boundary, with R\ge0, metric decay O_2(r^{-p}), p>1, and scalar decay O(r^{-s}), s>4. Here the exponents are p=2 and s=4+\delta_1; the scalar curvature is integrable and the energy is finite. That numerical theorem permits disconnected boundary and imposes no spin hypothesis. Its additional spin premise concerns an equality conclusion, which is not used. Its mass normalization is 1/(6\omega_3), the normalization in Equation (8.1). Consequently

First let N\to\infty at fixed prepared data and \eps, then let \eps\downarrow0. Only this numerical inequality is passed to the limit.□

Proof. Replace Q by Q_c=Q-(c/3)g. Since the hypersurfaces have dimension three,

for every oriented hypersurface. It suffices to prove the result for c=0, and we use Q_c throughout this proof.

The global input is the smooth-frontier part of [4, Theorem 4.6]: for smooth complete asymptotically flat data in spatial dimensions 2\le n\le7, a nonempty total trapped region has smooth embedded outermost stable MOTS frontier. No dominant energy condition is required for this conclusion. The higher-dimensional construction uses the Perron and almost-minimizing method developed by Eichmair [14], who also explains its adaptation to MOTSs. We now reduce the compact problem to the boundaryless asymptotically flat setting of the cited total-region theorem, preserving exactly its trapped competitors.

First attach an outward cobordism from \partial X to one three-sphere. Each component of \partial X is a closed orientable three-manifold. Relative one-handles join these components, and the smooth surgery theorem for orientable three-manifolds gives a framed link surgery from the resulting connected three-manifold to S^3 [42, Theorem 6 and its proof, pp. 521–522]. Its trace consists of relative two-handles: reversing a four-dimensional two-handle trace again uses two-handles. Thus the cobordism has a handle decomposition relative to \partial X with indices at most two. Attach S^3\times[0,\infty) at its other end, obtaining a boundaryless manifold \widetilde X. The handle Morse function provides a proper smooth height q on the extension, with q=0 on \partial X, increasing in an outward collar, with only critical points of index at most two, and equal to a radial coordinate sufficiently far out. Extend q a short distance into X as a collar coordinate with negative values.

Extend g smoothly across \partial X. We may choose its values near every critical point of q so that

at that critical point, and hence throughout a smaller neighborhood. Indeed, at an index-two point the eigenvalues of the Hessian can be arranged as -a,-b,A,B, with a,b>0 and \min(A,B)>a+b; the sum of the three smallest is positive. At indices zero or one the same choice is easier. This is achieved by rescaling the positive and negative coordinate directions in the metric in a Morse chart. The neighborhoods are disjoint and away from the prescribed original data, so these local metrics extend to a smooth positive metric on the whole extension. Choose that metric Euclidean on the remote end.

At a regular point of a level of q, its mean curvature toward increasing q is

It is positive near the critical points by (9.4). Choose a smooth preliminary tensor extension agreeing with Q_c in the prescribed boundary collar. Near each critical point there is a>0 such that \tr_V\Hess q\ge a on every unit three-plane. After shrinking its neighborhood, H\ge a/|\dd q| dominates the bounded tangential trace of the preliminary tensor. On the remaining compact regular part the level mean curvature is bounded below. Retain the tensor near the original boundary, and add a sufficiently large nonnegative multiple of the metric on the remaining compact part of the extension. The tangential trace of this addition is three times its coefficient. We can therefore ensure strictly positive expansion on every regular level of q. The given strict boundary inequality permits the matching in a collar of \partial X, including a collar on the original side. On the Euclidean remote end take the tensor to be zero; its round levels already have positive mean curvature. The transition to zero can be made inside that Euclidean region with a nonnegative metric multiple. We have obtained smooth complete one-ended data without boundary, exactly Euclidean with zero tensor sufficiently far out.

Every compact smooth weakly trapped region in this extension stays a positive distance inside X. To see this, extend q to a smooth function on the interior core with values below those of a fixed smaller collar. If a region enters that collar or the attached extension, the maximum of q on it occurs at a point in that part. A regular maximum cannot lie in the interior of the region. At a regular boundary maximum the region is on the inner side of a positively expanding level, with the same outward normal at contact. Tangential Hessian comparison gives \Theta_{Q_c}(\partial E)\ge\Theta_{Q_c}(\{q=q_{\max}\})>0, a contradiction. At a critical maximum in the interior, a direction of positive Hessian contradicts maximality. At a critical boundary maximum, the inward open half-space of a smooth domain contains a vector v with \Hess q(v,v)>0: choose the sign of a positive Hessian direction to point inward, and perturb it inward if it is tangent. A curve entering the region with this initial velocity has q(\gamma(t))=q(\gamma(0))+t^2\Hess q(v,v)/2+o(t^2), again a contradiction. Thus one fixed original-side collar is avoided by every such region.

The global total region consequently coincides with (9.2); all its candidate regions lie in the same compact subset of \operatorname{int}X. The cited theorem gives its smooth stable frontier. With the closed representative used here it is a smooth compact region, its frontier has \Theta_{Q_c}=0, and it contains every candidate. This proves the assertion at threshold c by (9.3). If a complementary component has closure in \operatorname{int}X, filling it deletes whole smooth boundary components and changes no remaining outward expansion. The resulting larger region would still be a candidate in (9.2), contradicting maximality.□

Proof. Inclusion of the defining classes in (9.2) proves that T_c(X,Q) increases with c. These are closed subsets of one fixed compact metric space (X,\operatorname{dist}_g). Apply Lemma 3.2 to this family. Its empty-set convention gives precisely the asserted eventual emptiness. Since the exceptional set is countable, every nonempty open subinterval of I contains a right-continuity threshold.□

Proof. Lemma 9.1 supplies a compact smooth frontier with \Theta_Q=c, stable for the outward normal. Its component \Sigma is closed, connected, and two-sided. Replacing only \Sigma by a sufficiently small positive outward graph with expansion at most c would produce a larger admissible region in (9.2), contradicting that lemma’s maximality. The graph collar can be chosen inside \operatorname{int}X and disjoint from the other frontier components.

Apply Lemma 3.3 with ambient dimension d=4 and tensor Q. Here the shifted tensor is Q-(c/3)g; (9.3) holds for every nearby graph, so its linearization is the same operator L used in that lemma. In particular, when the principal eigenvalue is zero, the normalized linear map is

for 0<\alpha<1. This is the common zero-mode isomorphism (3.11), whose inverse and resulting foliation are proved there. The lemma gives \Theta_Q=c at s=0 and \Theta_Q>c for s>0, hence (9.5), with a smooth leaf coordinate and positive normal speed. Apply it to the finitely many components and shorten the collars to be disjoint.□

Proof. Apply Lemma 3.5 with ambient dimension d=4, tensor Q, and the compact closed set D. Choose its fixed smooth compact neighborhood inside the interior in the present hypotheses, and shorten z_0 to keep the distance neighborhoods there. That lemma includes the empty case and gives exactly the required estimate, uniformly in the support curvature. We record its contact identities in the present notation.

Let a support \{\psi=0\} touch D_z at x', choose a nearest point x\in\partial D, and let \gamma:[0,z]\to X be the unit-speed minimizing segment from x to x'. Put u_0=\dot\gamma(0) and let P_t be parallel transport along \gamma. The support normal at x' is P_zu_0. The common proof constructs, for each X\in T_xX, the Jacobi field

Writing J_X(t)=P_t(X+W_X(t)), its short-interval estimates give

For A_zX=D_tJ_X(0) the first-jet compatibility is

In normal coordinates y^i at x, let E_v(y) denote radial parallel transport of v\in T_xX. The unit field of the common construction is

Its prescribed first jet and uniform second-jet bounds are those established in that proof. The endpoint map is

The exact endpoint identity and its covariant second-derivative bound specialize to

The pulled-back level \psi\circ F_z is an exterior support to D at x, with normal u_0 and the same gradient length as \psi at x'. For an orthonormal basis e_1,e_2,e_3 of u_0^\perp, the Hessian identity from the common proof reads

Its normalized tangential trace is the four-dimensional formula

There is no error involving \Hess\psi. The tensor estimate is

These identities specialize the common proof’s comparison of the two support expansions; the assumed inequality at x gives the bound k+Cz at x'.□

Proof. Assume D\ne\varnothing. Choose \beta>0 small enough that D_\beta lies in \operatorname{int}X, Lemma 9.4 applies for 0<z\le\beta, and

Smooth approximation of the distance function, followed by a regular-value choice, gives a smooth compact region E with

For example, approximate the distance uniformly to accuracy \beta/16 and choose a regular level between 7\beta/16 and 9\beta/16.

Choose a smooth function a\ge0, supported where \operatorname{dist}(\cdot,D)<\beta, that is a sufficiently large constant near \partial E. For Q'=Q-ag we then have \Theta_{Q'}(\partial E)=\Theta_Q(\partial E)-3a<c'. The boundary barriers of X are unchanged. Lemma 9.1 applied to Q' produces a nonempty smooth total region T' containing E. Its frontier is nonempty because T' is compactly contained in the interior of the connected manifold X with nonempty boundary. That outward frontier satisfies

Since D has an open neighborhood in E\subset T', the number z=\operatorname{dist}(D,\partial T') is positive and attained.

Suppose z<\beta. Then D_z\subset T'. Indeed, a minimizing segment of length at most z from D to a point outside T' would first cross \partial T' at distance strictly less than z. Such segments stay in X by our choice of \beta. At a pair realizing the minimum distance, \partial T' is therefore an exterior support to D_z. Its expansion is at most k+Cz<c' by Lemma 9.4 and (9.11), contradicting (9.12). Consequently z\ge\beta.

The frontier of T' is outside the support of a, so its expansion for the original tensor is exactly c'. Thus T' itself is an admissible region in (9.2) for (X,Q,c'). Since it contains D, the defining maximality gives D\subset T'\subset T_{c'}(X,Q).□

Proof. Choose a distant sphere S_{R_0} beyond \supp K, sufficiently large that it and all farther coordinate spheres have positive mean curvature. Their expansions for both K and -K are positive. Let X_0 be the original compact core between S and S_{R_0}. It is connected: paths using the remote end can be replaced at its connected spherical cross section. Attach a smooth compact filling F behind the entire original boundary. Such a filling can be constructed by taking an oppositely oriented copy of X_0 and capping its spherical boundary by a four-ball. Its remaining boundary identifies with S. Extend g and K smoothly across S into this filling, keeping the metric positive. Extension of the collar jets followed by a partition of unity does this; no constraint condition is required in F. The resulting compact manifold X has only the connected outer boundary S_{R_0}.

For later parameter choices, fix now

Choose \gamma>0 so small that

and the original boundary has expansion less than -2\gamma. By strictness and continuity, F together with a fixed short exterior collar of S is a smooth compact region whose outward boundary has expansion less than -\gamma. For every c\in(-\gamma,0) form

Here K denotes its smooth extension into F. Lemma 9.1 applies and \cD_c contains that entire filled neighborhood. Its smooth frontier therefore lies strictly in the original data. Its complement in X is connected. Indeed the collar of the connected outer sphere belongs to a unique complementary component; every other component would be an interior hole and could be filled, contrary to maximality.

Choose c_b\in(-\gamma,0) at a right-continuity threshold by Lemma 9.2. Write Y for the closed complement of \operatorname{int}\cD_{c_b} in X. This is a smooth compact connected manifold with the outer sphere and the black frontier as its boundary. For the white tensor -K, all these boundary components have strictly positive outward expansion. On a black component the outward normal of Y is the reverse of the outward normal of \cD_{c_b}, so the expansion is

This changes both the normal and the tensor in an auxiliary problem; the original boundary still has its prescribed future sign.

For c\in(-\gamma,0) define the white family

It may be empty. Lemmas 9.1 and 9.2 give a right-continuity threshold c_w\in(-\gamma,0). Every nonempty white region lies strictly inside Y, so \cD_{c_b} and \cW_{c_w} are disjoint closed sets, a positive distance apart. Notice that the black threshold and manifold Y are fixed before making this second threshold choice.

In the filled one-ended manifold take the component of the open complement of \cD_{c_b}\cup\cW_{c_w} that reaches the end, and let \Omega be its closure. Its boundary is the union of those entire frontier components adjoining this exterior. To justify the word “entire,” a smooth connected frontier component has a connected thin collar on each side; the component of its exterior side is therefore constant along the frontier. Both frontiers are compact and smooth with finitely many components, and they are disjoint. Thus \Omega is smooth, closed, and connected, and its boundary B is a union of complete smooth components. It has one end, the original end, and lies in the original manifold because the black region contains the filling and a collar of S. Its boundary is nonempty: the component reaching infinity is separated from that nonempty filled region. The inherited metric is complete up to B; boundary charts give local completeness at finite limit points, and completeness of the original end prevents escape in finite distance.

Call its retained \cD_{c_b} faces black and its retained \cW_{c_w} faces white. The normals into \Omega are the outward normals of these regions. Their threshold equations give (9.13). Figure 1 illustrates these orientations; \Omega_R there denotes the part of \Omega inside a distant coordinate sphere S_R.

Apply Lemma 9.3 to their respective total regions and shorten the collars so they lie on the \Omega side and are mutually disjoint. This gives all the asserted leaf inequalities. The support enclosure Lemma 9.5 is available for the full families \cD_c and \cW_c on X and Y, respectively; in the latter case the reversed black barriers remain fixed.

All these collars lie in the fixed original core X_0, so the lower bound \sigma_* was fixed before either threshold or any collar width. For each collar choose a smooth profile q_B(s)\in[0,1], supported in that collar, with q_B(s)=1-\alpha_Bs near its face and \alpha_B>0. Such a profile is obtained by multiplying this linear function by a nonnegative cutoff supported before the linear function can become negative. With disjoint supports, define

There is a finite constant M_*, now depending on the chosen collars and profiles, such that |C|\le a_0 and |\dd C|\le a_0M_*. Formula (9.14) holds with \kappa_B=a_0\alpha_B>0. Only at this point choose a_0>0 small enough that

For b_-\le h\le b_+ we have |h+C|\le\gamma+2a_0, and hence on X_0\cap\Omega

At points outside \supp K the trace term vanishes, so this estimate requires no restriction there on the value of h.

Finally choose a constant \rho_*>0 so small that

On X_0\cap\Omega, (9.20) plus this choice bounds the left side of (9.16) by \sigma_*/2. Outside X_0, K=C=0, and (9.1) gives the required inequality directly. This proves (9.16). This order of choices uses no lower bound on collar widths as c_b,c_w approach zero: the thresholds are chosen first, the collars next, and their amplitude last. Ordinary collar cutoffs also give the stated compactly supported smooth extension of the constants b_- and b_+.

If D_e is an admissible closed connected outer domain for a cut in \Omega, it is also closed in M, because \Omega is closed in M. It is a smooth codimension-zero submanifold of M, its interior lies in \operatorname{int}M, and it contains the distant original end. Its entire intrinsic boundary, including every part coincident with B, is therefore an admissible original enclosing cut. Its area is at least A_*. This proves the final assertion without assuming the existence of a minimizing cut.□

Proof. We derive the identity in a stationary Lorentzian metric, keeping its normal second form distinct from the prescribed tensor K. On the product with coordinate s' use

Its constant-s' slices have metric g. Their lapse, shift, and normal second form, with the convention that the second form is half the normal metric rate, are

Here \operatorname{sym} includes the factor 1/2. Put \theta=\tr_gk, and let N_0 be the future unit normal. The contracted Gauss and normal expansion identities are

Stationarity gives UN_0\theta=-\beta(\theta) and \divg\beta=-U\theta. If

the preceding two formulas consequently yield

In the static coordinate s'-f one instead has

Direct differentiation of U=l\sqrt D gives

Moreover, using J=\divg P_K, symmetry of P_K, and \operatorname{sym}\nabla\beta=-Uk gives

Substitute these identities into (10.8), use \mu=(\Scal_g-\mathsf B(K,K))/2, and put Q=K-k. The result is

For example, the divergence arising directly from (10.8) is U^{-1}\divg(UP_kw); adding J(w) combines it with -U^{-1}\divg(UP_Kw) to give the last term in (10.9). This also fixes its sign.

The four-dimensional conformal formula is

Since \Delta_{\gbar}t=U^{-1}\divg(U\Ad\grad t)-p|\dd t|_{\Ad}^2 and u=e^{2t}U, changing the divergence weight from U to u gives

The following differential identities follow from (10.2):

They imply

Adding uFw inside the divergence in (10.10) therefore leaves w(F)+F u^{-1}\divg(uw). Its trace contribution is exactly -F\tau, while the remaining quadratic expression is

This is smooth even at w=0, since (1-\chi)j=(3+p)S_f(w,w)/(3+pv). For the block expansion, Q has blocks T, M+pay, j+2pax, and \tr T=F-\chi j. Expanding (10.13) gives (10.6). At w=0 it reduces to

which is independent of the auxiliary axis. This proves all assertions.□

Proof. Completing squares in (10.6) gives the exact identity

For the final coefficient, the identity used in this completion is

The coefficient of |y|^2 is at least two, and the last coefficient is at least 45d/[8(3+N)^2]. Thus d j^2\le C(3+N)^2\cT. The inequalities \chi^2\le d^2\le d then recover F^2 from its shifted square and d x^2 from its shifted square, at polynomial cost. Recovering M costs at most C(1+N)^2|y|^2, and recovering T uses \tr T=F-\chi j. This proves (10.14); these observations, for instance, allow one common bound C(1+N)^2 in that inequality.

When \dd t=0, complete only the (F,j) terms to obtain

Since 5/8\le1-3\chi/8\le1 and \chi^2\le\chi, this gives (10.15) with absolute constants.□

Proof. With \mathrm{II}_B(X,Y)=g(\nabla_X\nu,Y) for tangent vectors, one has tangentially \Hess f=(\partial_\nu f)\mathrm{II}_B, so \tr T=P_B+w_\nu H and \chi j=F-P_B-w_\nu H. Use this relation in the first identity of (10.12); as w is normal on the face, it gives

which is the first assertion because w_\nu^2=1-d. The induced metric of \gbar on the face equals g|_B, and its normal in base coordinates is \sqrt d\,\nu. In the tangential first variation, l^2\dd f^2 contributes zero since the tangential derivatives of f vanish there. Hence H_{\gbar}=\sqrt d\,H. Applying the boundary conformal formula with \nu_{\gbar}=\sqrt d\,\nu proves the second assertion. Both assertions also hold when \dd f=0.□

Proof. At an interior extremum of h, w=0 and both auxiliary terms vanish. The trace equation becomes

Since l>0, comparison at a maximum and a minimum bounds |h| by \max(\sup_B|b|,\|C-\tau\|_\infty). Stage 4 has h=0. All prescribed trace right sides then bound |F| by a fixed constant, because D_0,D_1 are bounded and \|\Achi\|\le1.

Choose r_0 beyond the supports of K,C,b,D_0,D_1. At a comparison with a radial function, the axis of \Achi is \grad_g r/|\dd r|_g. Uniformly for 0<\chi\le1, the prepared end expansion gives

after increasing r_0. The leading coefficient is at most -3/4, so the choice is independent of N. Take C_* large enough that C_*(r_0^{-3/2}-R^{-3/2}) dominates the global height bound for every R\ge2r_0. On this annulus the scalar equation is

The functions \pm C_*(r^{-3/2}-R^{-3/2}) are respectively upper and lower barriers by (10.29). At a putative first comparison point the gradients coincide, so the same positive trace matrix and the same positive prefactor occur in both expressions. The Hessian order and the proper height sign give a contradiction. This proves the end estimate. Differentiating the comparison at the outer boundary, where h=0 tangentially, gives (10.27), absorbing the uniform end normal comparison into C_*.

Finally let G(t)=t+\frac16\log(1+l(t)^2\sigma^2) at an outer point. It is strictly increasing and G(t)=Z=0. Thus t>-\eps if

By (10.27), it suffices to choose R_{\min}(N) so large that C_*^2\ell^2\eta_N^{-2}R_{\min}(N)^{-5}<e^{6\eps}-1. This uses only fixed-N enlargement of the outer radius.□

Proof. Use the Riemannian warped product g+l^2(\dd b')^2 and its graph b'=f. At \dd t=0 one has \dd l=0, so the graph second form is H^f, up to its immaterial overall sign. Its inverse metric is \Ad. The contracted second-form contribution is

as is seen by writing \Ad=\operatorname{diag}(1,1,1,d). The ambient scalar curvature and Ricci blocks are

for the scalar, horizontal block, and unit vertical direction; the mixed block is zero. The graph unit normal has horizontal part -w and vertical component \sqrt d. Therefore the ambient scalar minus twice its normal Ricci, divided by two, is

At a stationary point \Hess l/l=p\Hess t, and the conformal term is -3\tr_{\Ad}\Hess t. Thus we have, more precisely, the equality

At a minimum the last two terms are nonpositive, proving (10.31).

For (10.32), first subtract \mathcal S(S_f). An expansion using \tr T=F-\chi j gives exactly

At d=\chi=1 the (F,j) quadratic is F^2/3-2Fj/3+4j^2/3, whose matrix has determinant 1/3 and positive trace. It is uniformly positive in a fixed neighborhood of this point. Since

there is an absolute \kappa_0>0 such that this neighborhood contains all inputs with (1+p)v\le\kappa_0. In that region (10.34) controls an absolute fraction of |T^0|^2+|M|^2+\chi j^2+F^2.

In the remaining region, use \chi(1+d-2\chi/3)\ge\chi and |\chi/3-d|\le d to absorb the mixed term into, for example, \chi j^2/4 and a multiple of (d^2/\chi)F^2. The only ratio required is

Keeping also a fixed fraction of F^2 at the cost of another multiple of F^2, this proves a lower bound by an absolute fraction of the stationary norm minus C(1+N)^2vF^2. Equation (10.15) converts the norm to \cT.

It remains to replace S_f by S_f-K. Polarizing (10.30) bounds the difference by

The stationary norm controls |T| and |M|, while d|j|\le(d^2/\chi)^{1/2}(\chi j^2)^{1/2}. By (10.35) and Young’s inequality, for every fixed \alpha>0 this is at most \alpha\cT+C_\alpha(1+N)|K|^2. Choose \alpha to retain half the already obtained positive coefficient of \cT. This proves (10.32) with absolute c_2>0.□

Proof. At \dd t=0, differentiation gives

These formulas and (10.15) show

For clarity, the axial entry in \nabla w is dH^f_{ee}, and its control uses d^2/\chi\le(3+N)/3. In |\dd\log u|_{\Achi} the axial entry has the factor \sqrt\chi. Differentiating \Achi=\Id-(3+p)w\otimes w/(3+pv) uses only the preceding \nabla w and coefficients bounded by powers of 1+N, since 3+pv\ge3. This proves (10.37) without dividing by l or \eta_N.

Write the relevant prescribed trace as \Phi(x',w,t,h). Its height derivative is at least one, so the chain rule contains the favorable term

All other terms are supported in \mathcal K. Lemma 10.4 bounds their height arguments in a fixed range; derivatives of D_0,D_1 and b are consequently bounded. Derivatives of \tr_{\Achi}K use \nabla K and (10.37). The t derivative has zero contribution at the point in question. Thus the remaining absolute error is bounded by \Pi_N(1+\sqrt{\cT})\mathbf1_{\mathcal K}. Young’s inequality proves the first assertion.

Expanding the divergence in the second assertion differentiates \Achi, K, and w, and adds \inner{K(w,\cdot)}{\dd\log u}_{\Achi}. The latter is bounded by C|K|\,|\dd\log u|_{\Achi}. Equation (10.37), compact support, and Young’s inequality give the second assertion with the same allowed dependencies.□

Proof. Suppose first that a floor minimum is interior and the parameter is in one of stages 1–3. At that point \dd t=0 and \Hess t\ge0. Subtract (10.31) from (10.5) and use V_\lambda=V-(1-\lambda)u\Achi K(w,\cdot). This gives

Choose \delta_2>0 in Lemma 10.6 so small that the two losses \delta_2\cT leave at least c_2\cT/2 in (10.32). The remaining tensor expressions \mu-\Scal_g/2=(\tau^2-|K|^2)/2, J, and F\tau are compactly supported and bounded by fixed constants, using Lemma 10.4. On the complement of \mathcal K, F=h and F^2\le C r^{-3}; also |\Ric_g|\le C r^{-4}. On \mathcal K the positive weight \rho_0 has a fixed positive minimum. Hence

Combining these bounds yields an absolute c_3>0 and a polynomial Q_N\ge0 such that, at the proposed minimum,

Fix 0<\delta_0<\min(c_3/2,1/4). Since t=-\eps gives m_0(t)=1, choose C_N to exceed Q_N+1+\sup_\Omega(\rho/\rho_0). The right side of (10.38) then strictly exceeds \delta_0\cT+\rho+\delta_0\eta_N a\sigma-C_N(\rho_0+v\rho_1) =\Xi, contradicting (10.21). This choice is polynomial in N.

At an interior minimum in stage 4, (10.25) gives f=0, t=Z, and v=0. If \gamma>0, its equation at the stationary point is

Enlarge C_N, still polynomially, so that C_N>1+\sup_\Omega(\delta_0\cT_K+\rho)/\rho_0. The right side is strictly negative, contradicting \Delta t\ge0. This additional bound is finite because K is compactly supported.

Consider next an inner boundary minimum in stages 1–3. There j_0(-\eps)=0, so the same drift correction appears on both sides of (10.21). Cancelling it gives V_\nu/u=\cB. Equation (10.18) and d>0 imply

This contradicts the nonnegative derivative into \Omega_R at a boundary minimum. In stage 4 with \gamma>0, the corresponding exact formula from (10.25) is 3\partial_\nu t=\gamma(-H-N_B)<0, giving the same contradiction. The outer boundary is excluded by (10.28).

Finally, at \gamma=0, f=0 and the remaining problem is \divg(3u\grad t)=0 with homogeneous inner conormal data and t=0 on S_R. Testing by t gives \int_{\Omega_R}3u|\dd t|^2=0. Connectedness and the nonempty Dirichlet boundary imply t=0. Thus no stage admits floor equality.□

Proof. We give the lower-height argument first. Consider any sequence N_i\to\infty of the indicated solutions; its radii tend to infinity. Extend h_i by the constant b_+ into a small collar across each white face. The functions on these enlarged neighborhoods are continuous. Define their lower relaxed limit by

It is lower semicontinuous and locally bounded by Lemma 10.4. Fix levels

where \xi_1 is the height-cutoff constant in D_0. We initially work away from black faces.

Suppose a smooth \phi touches \underline h from below at x, with \underline h(x)<k' and \dd\phi(x)\ne0. Subtracting a small fourth-order function of distance makes the contact strict on a small closed neighborhood without changing its second jet. The definition of the relaxed limit and minimization on this neighborhood give, after passing to a subsequence, points x_i\to x and constants c_i such that \phi+c_i touches h_i from below at x_i, and h_i(x_i)\to\underline h(x). These contacts are interior contacts of the original exterior: their heights are eventually strictly below k'<b_+, so they cannot occur on a white face or in its constant extension. This remains true when x itself is on a white face.

At contact, \dd h_i=\dd\phi and \Hess h_i\ge\Hess\phi. The positive trace matrix therefore gives

Here \alpha_i=0 in the first stage. The cutoff construction gives D_0\le0 at these low heights, everywhere: its white contribution vanishes there, and its black contribution is nonpositive. Moreover

Consequently |w_i|\to1, d_i\to0, \chi_i\to0, and A_{\chi_i} tends to the orthogonal projection onto (\grad\phi)^\perp. Passing to the limit in (11.2) proves

The normal here points toward increasing \phi.

We next transfer this test inequality to the closed set D_k=\{\underline h\le k\} by Lemma 5.1, with ambient dimension four, u=\underline h, tensor Q=K, and q_0=k'+a_0. The coarse height bound makes u finite, its defining relaxed limit makes it lower semicontinuous, and Equation (11.3) gives the required bound for every nonzero-gradient lower test at a height below k'. Away from black faces these hypotheses hold on open neighborhoods; at white faces use the constant extensions already constructed. The lemma therefore bounds every exterior support of D_k there by k'+a_0, including when the sublevel has a plateau.

This argument also rules out a boundary layer at a white face. The constant extension need not make \underline h continuous there. Nevertheless all approximating low contacts used above are interior, so the support inequality holds at a limiting white-face contact. At every point strictly on the added side the relaxed limit equals b_+>k, so D_k lies locally in the original closed exterior. The reversed white face would therefore be an exterior support of D_k with expansion

contradicting k'+a_0<0. Hence D_k misses every white face.

Adjoin the full black total region and consider D_k\cup\cD_{c_b} in the filled black barrier manifold. At a frontier point belonging to \partial\cD_{c_b}, an exterior support of this union also supports the smooth region \cD_{c_b}, so smooth tangency bounds its expansion by c_b. At other frontier points the bound is k'+a_0>c_b. The coarse end-height decay makes the additional sublevel compact. It lies strictly inside the distant barrier sphere: otherwise a sphere at its greatest radius is an exterior support with positive expansion, a contradiction. There are no remaining contacts with the boundary of the filled barrier manifold. Lemma 9.5 therefore gives

in the chosen threshold range.

If A is disjoint from \cD_{c_b}, its distance from that closed region is positive. Right continuity first permits c'>c_b close enough that A\cap\cD_{c'}=\varnothing. We then choose k,k' as in (11.1), close enough to b_- that k'+a_0<c'. It follows that A\cap D_k=\varnothing. If a uniform eventual lower bound h\ge k on A failed, a sequence of violating points in the compact set A would give a point of A\cap D_k in its lower relaxed limit. This contradiction proves the first claim, with, for example, \xi_A=k-b_-.

For the second claim apply the same proof to (-h,-K,-C). The low-height modification is now -D_0 and again has the required nonpositive sign. Extend across black faces by the high constant -b_-. At a limiting contact, the reversed black face has expansion -c_b>0 for -K, and is excluded before using the white barrier manifold. Adjoining \cW_{c_w} and applying Lemma 9.5 now places the sublevel in \cW_{c'}. Right continuity at c_w finishes the proof. This also covers an empty chosen white region: the distance-cap formulation of right continuity makes the nearby regions empty if necessary, and a nonempty enclosed sublevel is then impossible.□

Proof. Write \nu_s=\grad s/|\dd s| and let P_s,H_s be the tangential trace of K and the mean curvature of the collar leaf, with normal \nu_s. For a test height h_*=b_*+\psi(s) of positive slope, comparison with a solution uses the test gradient and the solution’s value of Z. With the resulting implicit t, direct contraction gives

Indeed f_*=h_*/\eta_N and \Hess f_*=(\psi'\Hess s+\psi''\dd s^2)/\eta_N. The factor l\psi'/(\eta_N\sqrt D) equals a/|\dd s|, and \tr_{\nu_s^\perp}\Hess s=|\dd s|H_s. For a test b_*-\psi(s) with \psi'>0, all terms with prefactor a in (11.7) change sign.

If the slope is between two fixed positive constants, the floor and the fixed collar geometry imply

and a\ge1/2 for large N. Choose a smooth nonnegative cutoff \vartheta_2, equal to one on [0,1] and zero on [2,\infty), and set

Thus slopes whose logarithmic derivative is \pm b_N remain within a fixed multiplicative factor e^{2A_2} of their initial slopes.

On a black collar a lower barrier is b_-+\psi_-(s), with \psi_-(0)=0 and \psi_-''/\psi_-'=b_N. First choose A_2 large, depending only on bounded collar geometry. Then choose the slope multiplier sufficiently small that \psi_-'\le\kappa_B/2 throughout the collar and that its value at s_0 is below the solution by the strict outer-leaf gap. The direction of its gradient is compatible with the face, so D_0=0 at a contact. Since P_s+H_s\ge c_b and C=a_0-\kappa_Bs, the left side of the trace equation minus its prescribed right side at this test height is at least

The constant multiplying d depends on collar geometry, not on the size of the chosen slope multiplier. On s\le1/N, the last term dominates Cd by (11.8) and the choice of A_2. On s\ge1/N, the first term dominates the exponentially small Cd for sufficiently large N. The residual is therefore strictly positive.

For an upper barrier use b_-+\psi_+(s) with \psi_+''/\psi_+'=-b_N. Choose its minimum slope sufficiently large to dominate the coarse height at s_0 and all fixed linear spatial variations. Smoothness and P_0+H_0=c_b give |P_s+H_s-c_b|\le Cs. The residual is consequently bounded above by

for a fixed c>0. It is strictly negative by the same two ranges of s. In particular, the favorable linear term in these comparisons comes from the height and the offset; we have not assumed a positive derivative of the leaf expansion at s=0.

At any hypothetical crossing extremum, the test and the solution have the same gradient and the same Z, hence the same t and positive trace matrix. The prescribed right side is increasing in h with derivative at least one. The strict residual signs, Hessian comparison, and the endpoint inequalities exclude such a crossing. Differentiating the resulting barriers at s=0, and using upper and lower positive bounds for |\dd s|, proves (11.5) on black faces. Sign reversal of h,K,C proves the white assertion. The compatible-direction modification again vanishes there.

In stage three let b_*=(1-\zeta)b be the assigned face height. Use b_*+\psi(s) and b_*-\psi(s), with \psi'>0, \psi''/\psi'=-b_N, and with a large fixed minimum slope. For the limiting coefficients a=1, \chi=0 at s=0 and height b_*, the directional inequalities in the homotopy construction are

Their residuals have precisely the signs needed for the upper and lower tests, respectively. At positive s and finite d, smooth spatial variation and 1-a\le d, \chi\le d cost at most C(s+d). Increasing or decreasing the height by \psi(s) improves the respective residual by at least \psi(s). Choose the minimum slope to dominate the Cs error and the coarse height at the outer collar leaf. The logarithmic curvature contribution in (11.7) has the favorable sign for both tests and dominates Cd on s\le1/N; on s\ge1/N the linear height term dominates it. The same comparison proves (11.6). All constants are uniform in \zeta\in[0,1] by smoothness of the prescribed convex combination. The assertion for stage four is the scalar maximum-principle conclusion already proved in the homotopy construction.□

Proof. Set W=\sqrt d\,w. For every covector \xi,

The first assertion follows along the axis from |\xi(W)|^2=dv\xi(e)^2\le d\xi(e)^2 and is immediate transversely.

For completeness, write Q_f=\tr_{\Ad}H^f. Differentiating w and D gives

The two weighted divergences are exactly

Indeed the second formula follows from uW=L_0w and \dd\log L_0=(p+2)\dd t. Since \sqrt d\,|w(t)|\le|\dd t|_{\Ad} and 0\le d-\chi\le d, coercivity (10.14) bounds both right sides by \Pi_N(1+\sqrt{\cT}) on the fixed collars. No factor l^{-1} or \eta_N^{-1} occurs.

On a face \varepsilon_Bw_\nu=a\ge1/2 by (11.5). Integrate the divergence of \varepsilon_B\zeta uW\varphi separately on each collar, where \zeta=1 at B and \zeta=0 near its outer leaf. The outward normal of \Omega_R at B is -\nu, so the boundary term is -\int_B L_0a\varphi. Taking absolute values of the remaining terms and using (11.15)–(11.17) proves (11.12). Replacing u by \sqrt D proves (11.13). The derivatives of \zeta cost only a fixed constant.

On the boundary the compatible slope makes D_0=0, and the prescribed heights and offsets give exactly F-P_B=\varepsilon_BH. Moreover w_\nu=\varepsilon_Ba and (\Achi K(w,\cdot))_\nu=\varepsilon_Ba\chi K(\nu,\nu). Writing q_\lambda=1-(1-\lambda)j_0(t)\in[0,1], the flux formula (10.21) is therefore

This is bounded in absolute value by Cd since \chi\le d. Finally ud=L_0\sqrt d and \sqrt d\le C\eta_N/\ell at B by (11.5). This proves (11.14).□

Proof. The test k_0(Z) vanishes on S_R. Integration by parts gives

The last integral is absorbed into part of the gradient term plus C(N). Indeed it is supported in a fixed compact set and in the strip Z_0\le Z\le Z_0+1, where

Thus the residual strip integral has bounded background volume and a bounded weight, without any preliminary gradient estimate.

By Lemma 11.3, the absolute boundary contribution is at most

The fixed collars have \min_{\mathcal C}\rho>0. Consequently 1+\sqrt{\cT}\le C(\rho+\delta_0\cT) there. Choose N large using (11.18) to absorb the nondifferentiated term into the positive source in (11.19). Young’s inequality absorbs the differentiated term into another part of \int u k_0'|\dd Z|_{\Achi}^2, with a bounded remainder by (11.21). This proves (11.20).□

Proof. Extend the compact base neighborhood smoothly to a compact Riemannian manifold and fix a smooth Euclidean isometric embedding by Nash’s theorem [32, Theorem 2, p. 59]. Its product with a line embeds the product graph isometrically. The scalar mean curvature of the product graph is, up to its normal sign,

The prefactor is at most \ell^{-1} and E_f^{-1}\le e^2d. Coercivity therefore bounds its magnitude by C(N)(1+\sqrt{\cT}). The additional mean-curvature vector of the fixed embedding has magnitude at most four times the bound for its second fundamental form. Thus the Euclidean mean-curvature vector has the same bound, independently of the graph slope.

The four-dimensional Michael–Simon L^1 Sobolev inequality [31], in the compact-support form of [39, Chapter 4, Section 5, Theorem 5.7, p. 98], applied to |\varphi|^3, gives

We have retained the boundary term. Its form follows, for each individual smooth graph, by applying the boundaryless inequality to cutoffs tending to one up to the face: the integral of their normal derivatives tends to the face integral. Artificial patch edges do not contribute because \varphi vanishes there. Along B, f is constant on each face, so the graph face measure is exactly \dd A_g. Apply (11.13) to |\varphi|^3 and use (11.22). This bounds the face contribution by the same two interior expressions. If I=\int_{\Gamma_f}|\varphi|^4, Cauchy–Schwarz now gives

For I>0 divide by I^{1/2} and square. The case I=0 is immediate, proving (11.24).□

Proof. We first treat stage one, where the compactly supported drift can prevent a direct maximum argument. Let \psi be a smooth base cutoff supported in a fixed compact patch, permitted to meet B. For k\ge k_*(N)\ge1, test the divergence equation with \psi^2e^{2kZ}/L_0. We claim

where C(N) is independent of k.

Here are the error estimates establishing the claim. Let \dd\mu_D=\sqrt D\,\dd V_g, and write \kappa=K(w,\cdot) in this computation. Since \dd\log L_0=(p+2)\dd t, the exact tested identity is

The global lower bound in (11.19) supplies \delta_0\cT and costs only C(N)\psi^2 on the right. Coercivity bounds the final integrand, by Young’s inequality, by

Choose k_*(N) sufficiently large to absorb this fixed multiple of the gradient square. The drift term carrying 2k is bounded by k\psi^2|\dd Z|_{\Achi}^2+C(N)k\psi^2. The cutoff terms cost another fixed fraction of the k-weighted gradient square plus C(N)k(\psi^2+|\dd\psi|_g^2).

For the boundary term, (11.14) gives |(V_\lambda)_\nu|/L_0\le C\sqrt d\le C. Apply (11.13) to \psi^2e^{2kZ}. The resulting integrand, apart from its common factor e^{2kZ}, is bounded by

For any fixed coefficient b=C(N) the estimates

show that the boundary cost is absorbed with the claimed remainder linear in k. Leaving fixed positive fractions of the coercive terms and using (11.22) proves (11.26).

Set U=e^Z. Applying (11.24) to \psi U^k and using (11.26) yields

Choose nested base patches Q_r\Subset Q_s in a fixed coordinate neighborhood, or corresponding half patches at B, with 0<s-r\le1. Cutoffs have |\dd\psi|\le C/(s-r). Iterate (11.27) with k_j=2^jk_* and geometric gaps. The factor at the jth step is at most

Its product is finite, since \sum_j(j+1)/k_j<\infty. Tracking the total gap exponent gives

All comparisons here use the same graph measure of the individual solution; its constants are uniform by the preceding lemmas.

The starting exponent 2k_* can exceed the exponent in (11.23). A second, elementary nested-patch argument closes this point. Write M(s)=\sup_{Q_s}U. Interpolation gives

The last integral is uniformly bounded by (11.23). If k_*=1 this already suffices. Otherwise Young’s inequality applied to (11.28) gives, for any q\in(0,1),

Choose increasing radii r_j tending to a fixed larger radius, with r_{j+1}-r_j proportional to 2^{-j}, and take q<1/4. Iteration sums a geometric series with ratio 4q<1. The remaining q^jM(r_j) tends to zero: for each individual smooth solution all these patches lie in one compact neighborhood with finite supremum. Thus M(r_0)\le C(N) uniformly. A finite cover gives a uniform Z bound on a compact set containing B and the support of K.

Outside this set K=0. At an interior maximum with Z>Z_0(N), \dd Z=0 and

whereas u\Xi>0 by (11.19). The outer value is Z=0, so this maximum argument extends the compact bound through the end. This proves the stage-one Z estimate.

In stages two and three \lambda=0, so the same maximum argument works everywhere in the interior. On B the already established face-gradient bounds show that Z can be high only if t\ge1: if t<1, then

For t\ge1 the boundary switch satisfies j_0=1 and the prescribed conormal flux in (10.21) is zero. A high boundary maximum would contradict the boundary point principle applied locally on the high-level set to \divg(3u\Achi\grad Z)=u\Xi>0. For this application the individual smooth solution gives a smooth positive elliptic matrix; no uniform ellipticity constant is being assumed before the bound. Equivalently, the outward derivative at a nonconstant boundary maximum is strictly positive, whereas the zero conormal flux forces it to vanish. This bounds Z in these two stages.

In stage four f=0, hence t=Z. At a positive scale multiplying the source and boundary data, the same high-level interior and boundary arguments apply. At zero scale the homogeneous mixed problem gives Z=0. The resulting bound is uniform in the scale.

Finally Z-t=\frac16\log D\ge0, so the floor and the upper bound give D\le e^{6(C(N)+\eps)}. Since l\ge\ell>0, |\dd f|\le\ell^{-1}\sqrt{D-1}\le C(N). The coarse bound for h=\eta_Nf gives |f|\le C(N). This proves all of (11.25).□

Proof. We prove the assertion on uniformly small full balls, using a reflection at the flat face. Rescaling back and a finite local cover give the stated form. The argument first establishes a local Hessian estimate and a dichotomy: on a smaller ball, either the gradient bound decreases or the solution is close to a nonconstant affine function. A fixed-axis estimate then improves the affine approximation. Iterating these alternatives gives Hölder gradient control, from which the local Hessian estimate yields the Morrey bound.

Doubling and a quantitative Hessian estimate. Subtract the constant boundary value. With R_4=\operatorname{diag}(1,1,1,-1), extend the tangential metric block evenly and set \widetilde z(x)=-z(R_4x) below the plane. Tangential derivatives of z vanish on the face and the two normal derivatives agree. Thus \widetilde z is C^1; its distributional coordinate Hessian has no surface measure. The doubled metric is Lipschitz and has the prescribed smooth bounds on each closed side. On the reflected side the gradient and covariant Hessian are -R_4\grad z and -R_4(\Hess_g z)R_4, respectively. Extend \widetilde\chi evenly and G oddly. Equation (12.1) then holds almost everywhere on the double. Each individual doubled function is locally W^{2,p} for every finite p, although these norms are not yet uniformly bounded.

Make a constant linear coordinate change so that the metric at the center of a ball is the identity. A plane in the double remains a plane under this change. If the ball is sufficiently small, the coordinate principal matrix a=(a^{ij}) in (12.1) satisfies

The reason for the strict gap is the exact identity |\Id-A_z|_F=1-\widetilde\chi in an orthonormal frame; the small metric oscillation consumes at most half the gap.

Let T=D^2\Delta^{-1} on \R^4, with its values measured in the Frobenius norm. Its L^2 operator norm is exactly one, since its Fourier multiplier satisfies

The Calderón–Zygmund bound gives a finite L^4 operator norm C_4. Interpolation between 2 and 4 therefore gives \|T\|_{p}\le C_4^{\vartheta(p)}, where 1/p=(1-\vartheta(p))/2+\vartheta(p)/4 and \vartheta(p)\downarrow0 as p\downarrow2. Choose once and for all 2<p_0<4 such that C_4^{\vartheta(p_0)}\kappa<1. For a compactly supported y, write \Delta y=a:D^2y+(\Id-a):D^2y and absorb to obtain

This proves the needed Cordes estimate without differentiating a.

For later use its local, scaled form is

Here is the localization. A cutoff across a gap h gives the additional terms Ch^{-1}\|Dy\|_p+Ch^{-2}\|y\|_p. On a slightly larger ball, derivative interpolation bounds the first by \varepsilon\|D^2y\|_p+C_\varepsilon h^{-2}\|y\|_p. Use concentric gaps decreasing geometrically and choose \varepsilon so that the resulting geometric series absorbs the larger-ball Hessian norms. For each individual function these norms are finite, so the terminal term tends to zero. This proves (12.4); the same argument absorbs bounded first order coefficients. In particular the Christoffel terms in the covariant equation cause no difficulty. The analytic inputs here are the scalar Calderón–Zygmund and interpolation estimates [18].

A divergence inequality for the gradient deficit. Divide by M_1+N_1 unless this number is zero, in which case the claim is immediate. Subtract a constant and rescale space by r and height by r. The new gradient has norm at most one and the new forcing has norm at most r. Metric first derivatives and the second fundamental form of a reflecting plane are O(r); curvature on the smooth sides is O(r^2). After normalizing the metric at the center, all of these errors can be made uniformly small. In this paragraph write P=\grad z, H_z=\Hess_g z, and s=1-|P|_g^2\ge0.

There is an exact cancellation of the measurable axial coefficient:

Indeed \grad(|P|_g^2/2)=H_zP and \Delta_gz=G+(1-\widetilde\chi)H_z(e_z,e_z). The axial parts in the first formula cancel. In the second formula, \divg H_z=\grad\Delta_gz+\Ric_g(P,\cdot) cancels the third derivatives. Both identities hold also at P=0, because the first has both sides zero there. At no point is a derivative of G or \widetilde\chi taken. Choose \varepsilon_0>0 with (1+\varepsilon_0)(1-\chi_0)^2<1 when \chi_0<1. Young’s inequality, or directly \Delta_gz=G if \chi_0=1, gives

We compute the only surface error in this identity. In boundary normal coordinates on the original side, P=z_4\partial_4 at the plane and, for a,b<4,

Writing H_B=\tfrac12g^{ab}\partial_4g_{ab} for the mean curvature toward increasing x_4, this gives

Thus the jump of the volume-weighted normal flux on the double is bounded by C|\operatorname{II}_B||P|^2. In particular it contains no uncontrolled second derivative of z. A bounded plane density j(x') is the distributional divergence of j(x')\mathbf1_{\{x_4>0\}}\partial_4; only a normal derivative is taken in this assertion. After a linear coordinate change use the corresponding constant normal direction. Subtracting this bounded step field handles the signed jump, without imposing any tangential regularity on j.

More explicitly, put J_g=\sqrt{\det g} and let

be the coordinate divergence matrix. Let j=[J_gQ^4] be the upper trace minus the lower trace. Reflection gives Q^-=R_4Q^+, so j=-2J_gH_Bz_4^2. Then

Since a_1Ds+2J_gGP=-2J_gQ, the error field 2J_gGP+2j(x')\mathbf1_{\{x_4>0\}}\partial_4 cancels the plane contribution exactly.

Consequently (12.5)–(12.7), with volume factors included, imply on a full normalized ball

as the normalized source and geometry errors tend to zero. The matrix a_1 is bounded, symmetric and uniformly elliptic, with constants depending only on \chi_0 and the fixed comparison bounds. For example, away from the plane one may take the volume-weighted matrix of A_z, E_1=2\sqrt{\det g}\,GP, and e_1\le C(|G|^2+|\Ric_g|); the step field just described supplies the remaining term in E_1.

The gradient drop or affine alternative. Fix r_*=1/32. We claim that, for every b_0>0, there are r_*<k_*<1 and \delta_{\rm alt}>0, depending only on the fixed parameters and b_0, such that a normalized solution on B_1, with gradient at most one and errors at most \delta_{\rm alt}, satisfies one of

where L is a coordinate affine function.

To prove the claim, consider a sequence with errors tending to zero and \inf_{B_{r_*}}s\to0. Weak Harnack for nonnegative scalar divergence supersolutions, applied to (12.8) with its bounded vector and scalar errors, yields for some q>0

One obtains these fixed radii from the usual concentric weak Harnack inequality by a finite chain of overlapping balls. Thus s\to0 in measure. The scalar weak Harnack and energy estimates used here are valid for bounded measurable divergence matrices [18].

There is also Ds\to0 in L^1 on smaller balls. Indeed, for a fixed cutoff \zeta and b>0, test the weak inequality by \zeta^2(b-s)_+. Discarding its favorable Hessian term and using Young’s inequality gives

The full local L^2 norm of Ds is bounded by (12.4), the gradient bound and the geometry bounds. On \{s\ge b\}, Cauchy–Schwarz and convergence in measure give a vanishing L^1 norm. On \{s<b\} the displayed estimate bounds the limiting L^1 norm by Cb. Letting b\downarrow0 proves the claim. Testing (12.8) with a fixed nonnegative cutoff equal to one on B_{1/4} now gives

The coordinate Hessian also tends to zero in L^2, because the connection coefficients tend to zero and Dz is bounded. After subtracting z(0), the functions are equi-Lipschitz. Poincaré gives convergence of their gradients in L^2 to constants, and compactness gives uniform convergence to an affine function on smaller balls. Since s\to0 in measure and the metrics converge to the identity, its slope has length one. If (12.9) failed for every k_*\uparrow1 and every tolerance tending to zero, this conclusion would contradict failure of its affine alternative. This proves the claim.

All derivatives for a fixed exceptional axis. For a constant Euclidean unit vector e_0, let

If Y_0\in W^{2,2}(B_{3/4}) and \mathcal A_0Y_0=0 almost everywhere, we claim that, for some \alpha_1\in(0,1),

Set v_1=\partial_{e_0}Y_0\in W^{1,2}. Differentiating the equation in distributions, with no differentiation of its coefficient as a separate function, gives

This is a scalar uniformly elliptic divergence equation. Its local boundedness and Hölder estimate bound v_1 in C^{\alpha_1}. Caccioppoli, subtracting the value at the center of a ball of radius h, then gives

at all centers in a fixed smaller ball. The power follows from h^{-2} for the cutoff, h^{2\alpha_1} for the squared oscillation, and volume h^4.

Normalize the norm on the right to one. Since \Delta Y_0=(1-\widetilde\chi)\partial_{e_0}v_1, it follows that

Let \mathcal N be the Newtonian potential of \zeta\Delta Y_0, where \zeta is supported inside B_{1/2} and equals one on a slightly smaller ball. The remainder Y_0-\mathcal N is harmonic there, with bounded local L^2 norm; the potential has such a bound by local integrability of |x|^{-2} and the L^2 bound for \Delta Y_0. For the gradient of \mathcal N, the kernel and its derivative are bounded by C|x|^{-3} and C|x|^{-4}. If two points are separated by h, the annuli of radii s\le4h give, by (12.13), C\sum s^{\alpha_1}\le Ch^{\alpha_1}. On annuli s>4h the kernel difference instead gives Ch\sum s^{\alpha_1-1}\le Ch^{\alpha_1}. The sums run over dyadic scales; both converge because 0<\alpha_1<1. The harmonic remainder has bounded smooth derivatives on the final smaller ball. Thus every component of DY_0, and not only \partial_{e_0}Y_0, satisfies (12.11).

Improvement of a nonzero affine approximation. Suppose on B_1 that

and the source, metric oscillation from the identity and metric first derivatives are at most b\delta_{\mathrm{aff}}. Put Y=(z-L)/b. Its coordinate equation has uniformly bounded right hand side on B_1: the covariant Hessian of L is a Christoffel symbol times its bounded slope, and division by b leaves a bound C\delta_{\mathrm{aff}}. Consequently (12.4) gives a uniform W^{2,p_0}(B_{7/8}) bound for Y.

Let e_0=DL/|DL| and let a_0=\Id-(1-\widetilde\chi)e_0\otimes e_0. Since Dz=DL+bDY, the set E_b=\{|bDY|>\sqrt b\}\cap B_{3/4} has measure at most Cb^{p_0/2}. On its complement the actual gradient is bounded away from zero and the coordinate principal matrix differs from a_0 by at most C(\sqrt b+b\delta_{\mathrm{aff}}). On E_b this difference is bounded. Hölder’s inequality, with 1/2=1/p_0+1/q, shows explicitly that

Zeros of Dz can occur only in the exceptional set when b_0 is small. The arbitrary permitted choice of their axis is therefore covered by the same estimate.

Remove this residual by solving a_0:D^2y_1=a_0:D^2Y on B_{3/4} with zero Dirichlet data. To justify this solve for measurable \widetilde\chi, recall the Miranda–Talenti boundary identity on a Euclidean ball for smooth zero-trace y [40, Section 2, Theorem 3, p. 297]:

Here mean curvature is the outward trace convention, equal to minus three times Talenti’s signed normalized mean curvature. The identity extends by density to W^{2,2}\cap W^{1,2}_0. Hence the Dirichlet inverse of \Delta has Hessian norm at most one. On this space the map

is a contraction in the Hessian L^2 norm with factor at most 1-\chi_0. Poincaré and the Dirichlet Laplace estimate identify this as a complete norm and give

The function Y_0=Y-y_1 has uniformly bounded W^{2,2} norm and satisfies the homogeneous fixed-axis equation. Therefore (12.11) applies to it.

The passage to a small supremum norm must use a separate estimate in dimension four. Choose any 0<\eta<\min\{1,2-4/p_0\}. The W^{2,p_0} bound gives a uniform C^{0,\eta} modulus for Y on smaller balls. The fixed-axis estimate and the local L^2 norm give a uniform Lipschitz bound for Y_0. Thus y_1 has a common C^{0,\eta} modulus on B_{1/4}. If m=\|y_1\|_{L^\infty(B_{1/8})}, a ball of radius c m^{1/\eta} around a point with absolute value at least m/2 has absolute value at least m/4, provided m is small. Hence

If m is not small, the same reasoning on a fixed smaller ball first excludes that case when \varepsilon_1 tends to zero. This argument uses no embedding of W^{2,2}(\R^4) into C^0.

Choose 0<\alpha_2<\alpha_1, then 0<\lambda_0<1/8 so that the Taylor remainder constant in (12.11) satisfies C\lambda_0^{1+\alpha_1} \le\tfrac12\lambda_0^{1+\alpha_2}. Next choose \delta_{\mathrm{aff}},b_0>0 so that the correction supremum above is at most \tfrac12\lambda_0^{1+\alpha_2}. Taylor approximation of Y_0 at the center gives an affine L_1 such that

Reduce b_0 further so that

These estimates concern the almost everywhere coordinate equation on full balls and therefore apply equally to doubled patches.

The complete scaling iteration. The choices are ordered as follows: first p_0,\alpha_1 and their constants; then \alpha_2,\lambda_0,\delta_{\mathrm{aff}},b_0; then k_*,\delta_{\rm alt} from (12.9) for that b_0. Finally shrink the initial spatial scale so that its normalized source and geometry errors are bounded by \delta, where

Here the fixed c>0 incorporates all comparison constants in the geometry rescalings. This scale depends only on the theorem’s parameters, since division by M_1+N_1 makes the initial source at most one. Write the resulting function as u and center all following balls at zero.

After m successive gradient drops the actual rescaled solution is

The metric remains the identity at the center; its oscillation and first derivative errors are at most C\delta r_*^m, with its curvature error shrinking at least as fast. A reflecting plane has the correspondingly scaled second fundamental form. Since r_*<k_*, the next application of the alternative is legitimate. If the affine alternative first occurs after these m drops, rescale its ball by setting v_0(x)=u_m(r_*x)/r_*. Then v_0 has affine error at most b_0, slope in [1/2,2], bounded gradient, and source at most

Its geometry satisfies the same tolerance by (12.19).

In all subsequent affine steps use

The derivative Dv_i(x)=Dv_0(\lambda_0^i x) retains the actual gradient bound. The source and first order geometry errors are at most b_0\delta_{\mathrm{aff}}\lambda_0^i\le b_i\delta_{\mathrm{aff}}, because \alpha_2<1. The affine function used at stage i is rescaled in the same way, including its constant term. Thus (12.17) applies inductively. Its changes of slope sum to less than 1/4 by (12.18), so all slopes remain in [1/4,4]. Only the auxiliary residual in the proof of an improvement is divided by b_i; that residual is never asserted to solve the original nonlinear axial equation.

Choose

If the drops never end, the gradient on B_{r_*^m} is at most k_*^m\le r_*^{m\alpha}, and subtraction of the central value gives a constant affine approximation with error Cr_*^{m(1+\alpha)}. If the drops end after m steps, the affine approximation at R_i=r_*^{m+1}\lambda_0^i has error at most

The previous drop scales have the same type of estimate. Adjacent radii have ratio r_* or \lambda_0, both fixed, so every intermediate radius has an affine approximation with error Cr^{1+\alpha}. This conclusion holds at every center of the smaller full patches, including centers whose balls cross the plane.

For clarity, the usual comparison of affine approximations gives the gradient conclusion directly. If L_r,L_{r/2} are two such approximations, then \|L_r-L_{r/2}\|_{L^\infty(B_{r/2})}\le Cr^{1+\alpha}, whence |DL_r-DL_{r/2}|\le Cr^\alpha. Their slopes converge, and the limit is Dz since z is C^1. At points separated by h, compare the two approximations on overlapping balls of radius 4h; the same affine norm estimate gives a slope difference Ch^\alpha. Their limits differ from those slopes by Ch^\alpha as well. This proves the first estimate in (12.2).

Finally take an affine approximation L on B_r with error Cr^{1+\alpha}. Its L^2 error is at most Cr^{3+\alpha}. The coordinate equation for z-L has right hand side equal to G plus a bounded Christoffel coefficient times Dz, whose L^2 norm is at most Cr^2 before these iterative rescalings. Equation (12.4) with p=2 therefore gives

The difference \Hess_gz-D^2z has L^2 norm at most Cr^2. Since \alpha<1, squaring and restoring M_1+N_1 proves the second estimate in (12.2).□

Proof. We use the uniform family of interior and boundary patches described in the statement, shrinking them by fixed factors when necessary.

The first Hölder estimates and the reflected equation. The bounded range (11.25) keeps f,Z,t,\grad f bounded and l,d,\chi,u above positive constants depending on N. Divide the trace equation by l/\sqrt D to obtain

All prescribed trace expressions have bounded values on this range. Every boundary value of f is constant on its face. Thus Theorem 12.1 gives a Hölder bound for Df and

The coordinate and covariant Hessians differ by a bounded Christoffel term times Df, so they have the same bound at these radii. The bound also holds for the odd double at a face.

In coordinates set

using \lambda=0 in the later stages. The coordinate Z equation is (12.20), with these bounded coefficients and the appropriate scalar homotopy factor in E. Its matrix is symmetric and uniformly elliptic at fixed N. Differentiating the implicit relation for t gives the exact covector identity

Since 3+pv\ge3 and all its coefficients are bounded on the range, |D_xt|\le C(N)(1+|D_xZ|+|D_x^2f|) in coordinates. The quadratic formula (10.6) and the bounded lower order terms in the source therefore give

in the interior. No derivative of the principal matrix has been used to obtain this bound.

At a face take the domain to be x_4>0 and write Q=\mathcal A DZ+B_1. For x_4<0 define, with s=1 or -1,

The reflected flux is \widetilde Q(x)=sR_4Q(R_4x). Integration by parts on the two sides shows that its divergence has the reflected bulk source sE(R_4x) and the plane term

Indeed the upper normal component is Q_4^+ and the lower one is -sQ_4^+, so their difference is (1+s)Q_4^+. For an inner face use even reflection, s=1. The total conormal flux is prescribed by (10.21); its coordinate density Q_4^+ is bounded on the established range. For an outer face use odd reflection, s=-1. The zero Dirichlet condition makes Z continuous across the face, and the plane term vanishes. Even reflection at the inner face is also continuous. Each individual reflected function belongs to W^{1,2} and is smooth on the two closed sides, which verifies the weak chain rule required in Lemma 6.2.

In the full patch, (12.25) and (12.26) give a signed measure dominated by

where the Hessian density is reflected and the plane term is omitted for an odd face or an interior patch. By (12.23), this positive measure satisfies, at every center,

For balls crossing the plane this follows either from the doubled estimate or by enlarging a ball centered on the plane by a fixed factor. Lemma 6.2 now supplies a uniform Hölder modulus for Z, including at both boundary types.

Schauder for f and the ordinary normal condition for Z. The actual coefficients in (12.22) are smooth functions of (x,Z,Df) on the bounded range. This remains true at Df=0, as is manifest from

The source is smooth in these variables and f, uniformly over the four parameter ranges. The Hölder bounds just proved therefore make the coefficients and source uniformly C^{0,\theta} for some \theta>0. The scalar Dirichlet Schauder estimate, with the constant face data, gives

on every smaller interior or boundary patch [18]. This application has Hölder principal coefficients, a Hölder right hand side, smooth boundary, and uniform ellipticity; it does not apply a system estimate.

Now expand the divergence equation for Z. Derivatives of its coefficients, which depend smoothly on (x,Z,Df), are sums of bounded terms and bounded multiples of DZ,D^2f. The latter Hessian is bounded by (12.27). Thus

with bounded C^{0,\theta} uniformly elliptic principal coefficients. The source contains no second derivative of Z.

On an inner face Df is normal and hence \Achi\nu=\chi\nu. Dividing the prescribed flux by 3u\chi>0 gives

The function b_2 is smooth on the bounded range, separately on each homotopy stage, with uniform bounds through their matching endpoints. Extend its displayed expression smoothly in the spatial variable into the collar. Equation (12.27) then gives

Consequently the Sobolev trace theorem bounds the W^{1-1/p,p} norm of this normal datum by C(N)(1+\|Z\|_{W^{1,p}}) on a larger patch, for each finite p>1. This derivative count costs DZ and D^2f, never D^2Z.

Linear estimates and absorption of the quadratic term. Fix p>4. On a ball or smooth half-patch of size h, the local linear W^{2,p} estimate for (12.28), with zero outer Dirichlet data or (12.29), has the form

Here C_* is independent of sufficiently small h; C_h may contain inverse powers of h. The constants are uniform over the locations and the truncations in question.

We specify the linear estimate underlying this assertion. Freeze the continuous principal matrix at the patch center. The interior constant-coefficient Hessian estimate follows from the Euclidean one by a fixed linear change of coordinates. At the boundary, normal coordinates give no mixed normal–tangential entries in the frozen matrix, because its exceptional axis is normal there. A tangential linear transformation and a normal dilation reduce the frozen principal part to the Laplacian and preserve the plane. Subtract a Sobolev extension of the normal datum, or of the Dirichlet datum, to obtain homogeneous boundary data. Even reflection for homogeneous normal data and odd reflection for homogeneous Dirichlet data give the constant-coefficient half-space estimate. The extension costs precisely the W^{1-1/p,p} norm for the normal datum. Coefficient oscillation contributes \sup|a_2-a_2(x_0)|\,\|D^2Z\|_p and is absorbed on small patches, uniformly by the established Hölder bound. Cutoffs and interpolation handle the first order terms. Scaling shows that the coefficient of the interior L^p source norm is the constant C_*, independent of h; the lower order and trace terms account for C_h. This proves (12.30) by the scalar local estimates [18]. The construction requires no meeting of different boundary types, and the inner and outer faces here are disjoint.

For use in its nonlinear right hand side, the scaled scalar interpolation inequality on Q_h is

One can obtain it on the unit patch by subtracting a constant, using a bounded Sobolev extension on a smooth ball or half-patch, and applying the bounded-domain Gagliardo–Nirenberg inequality with derivative orders 1,2, exponents 2p,p,\infty and parameter 1/2. A convenient modern formulation is [28, Theorem 1.3]. To see the interpolation mechanism directly for a compactly supported function v, integration by parts gives

Division gives the required product estimate. Extension and cutoff produce the additional \|v\|_\infty^2 term on a bounded unit patch. This lower-order term is essential: the bounded-domain Gagliardo–Nirenberg estimate includes an additive multiple of \|v\|_\infty, so one may not drop its square after applying the estimate to v=Z-c. Taking c between the maximum and minimum replaces this norm by the oscillation used here. Finally rescaling x=hy multiplies a squared L^{2p} gradient norm and the Hessian product by h^{4/p-2}, which proves precisely the last power in (12.31).

Here is a cover argument that absorbs larger-patch norms without assuming their uniform boundedness in advance. Use uniformly comparable patches of radius h covering the finite truncation, and their fixed-factor enlargements. Each enlargement is covered by at most M_{\mathrm{cov}} smaller patches, where M_{\mathrm{cov}} is independent of h, R and the total number of patches. Let

For each individual smooth solution on the finite truncation, S is finite. The Hölder estimate for Z gives \operatorname{osc}_{Q_{2h}^j}Z\le C(N)h^\gamma. Apply (12.31) in the enlarged patches and then cover them by smaller ones. Its contribution to (12.30) is at most C_*C(N)h^\gamma M_{\mathrm{cov}}^{1/p}S+C(N,h). Choose h small enough, also satisfying the linear freezing condition, that this coefficient is at most 1/4. For the remaining first order term, local interpolation gives, for any \varepsilon>0,

Choose \varepsilon so that C_h\varepsilon M_{\mathrm{cov}}^{1/p}\le1/4. The bounded range controls the last term. Taking the supremum in (12.30) now gives S\le C(N,h)+S/2, and hence a uniform bound for S. First order interpolation supplies the corresponding local W^{2,p} bounds for Z. All constants may depend on the now fixed h and on N, but not on the total number of patches or R.

Alternating Schauder estimates and uniform geometry. Since p>4, Sobolev embedding gives Z\in C^{1,\theta'} for some \theta'>0. In conjunction with (12.27), the trace equation now has C^{1,\theta''} coefficients and source for 0<\theta''\le\min\{\theta,\theta'\}. One differentiation and the scalar Dirichlet Schauder estimate give f\in C^{3,\theta''}. The equation for Z then has C^{0,\theta''} source, and its normal datum (12.29) is C^{1,\theta''}. The scalar normal-derivative or Dirichlet Schauder estimate gives Z\in C^{2,\theta''}. The boundary estimate is justified by the same frozen half-space problem used above: its normal derivative does not annihilate a nonzero decaying solution of the homogeneous principal equation. Smooth normal coordinates and smooth normal field therefore satisfy its obliqueness condition.

Repeating in the order f then Z increases the number of derivatives by one each time. The trace equation uses Z without derivatives; the expanded second equation uses only D^2f and DZ in its lower terms; and its boundary expression uses only Df. These are exactly the derivative counts needed for the iteration. The smooth fixed coefficients give all finite orders.

The compact inner faces have uniform smooth collars. On the prepared end, the metric and all fixed coefficient fields have uniform bounds on unit patches. Large coordinate spheres have boundary normal charts with uniform bounds of each finite order, and are separated from the inner faces. Thus every radius, ellipticity bound, chart constant and local covering number used above is independent of the sufficiently large truncation radius. Finally a uniform local C^2 bound controls any prescribed C^{1,\alpha_0} norm, \alpha_0<1, on a fixed finite truncation. This proves the proposition.□

Proof. We use the scalar method of continuity in the affine space of C^{3,\alpha} functions with the assigned boundary values, with equation space C^{1,\alpha}. Thus every solution estimated along this continuation already has the differentiability required by Theorem 12.1; the estimates below are uniform bounds, not an assumption of uniform third derivatives.

The normalized equation is

The implicit relation defining t is smoothly solvable for every (Z,\dd f). Neither t, the matrix, nor the positive prefactor \sqrt D/l depends on the value of f. The prescriptions satisfy

with the other arguments held fixed. For a second parameter \beta\in[0,1], use the scalar interpolation

with the same Dirichlet values. At a positive interior maximum above a fixed height threshold, both source summands are positive, whereas the left side is nonpositive. The negative minimum argument is identical with signs reversed. Including the assigned boundary values gives |h|\le C, uniformly in both parameters and the bounded trial set.

We next check the degeneracy at large gradient, before invoking any regularity theorem. Put \sigma=|\dd f|_g. Uniformly for bounded Z, \sigma\to\infty forces t\to-\infty: a lower bound on t would bound e^{6t}l(t)^2 away from zero, contradicting e^{6Z}=e^{6t}(1+l(t)^2\sigma^2). Thus eventually t<0 and l(t)=e^{Nt}, giving the exact equation

Writing \gamma_N=6/(N+3), its consequences, uniformly for Z in a fixed bounded interval, are

For example, the first line follows by taking logarithms of e^{(2N+6)t}\sigma^2=e^{6Z}(1-d); the last follows directly from \sqrt D/l=(\sigma^2+l^{-2})^{1/2}. Since N\ge4, one has \gamma_N<1. Compactness of the remaining bounded-gradient range therefore gives constants c,C>0 such that

The estimate for the source uses the height bound just proved. Notice also the exact structural identity

In particular, all three transverse eigenvalues remain exactly one throughout the scalar interpolation.

Here is a gradient estimate using only (13.5). Extend g smoothly across the boundary to a compact enlargement. All distances below are distances of this extension, at scales below its injectivity radius. If H_0 is the uniform bound for |f|, set

First choose k large relative to the constants in (13.5) and the bounded geometry. Next choose h_0<1/(4k) below all collar, injectivity, and distinct-boundary-component separation scales. Finally choose B' so large that

These requirements are compatible: q_1(h_0) tends to 1/(kh_0)>4 and \psi(h_0) tends to infinity as B'\to\infty.

On the inward collar of a boundary face with constant value f_0, let r be its distance function. The functions f_0\pm\psi(r) are upper and lower barriers. To check the upper barrier at a hypothetical positive maximum of f-f_0-\psi(r), the common gradient is q_1\grad r. The tangential Hessian of the barrier contributes at most Cq_1, and its axial contribution is at most -ckq_1^2/(1+q_1). By enlarging k this is strictly less than -C(1+q_1), the lower bound for the equation’s source at the solution height. Ellipticity and the second derivative test contradict the equation. For the lower barrier both inequalities reverse. At the outer edge of the collar the comparisons follow from (13.6). Consequently

For completeness, the same modulus controls interior pairs. Suppose that

has a positive maximum on pairs of distance at most h_0. Neither the diagonal nor the distance-h_0 set contains such a maximum. If an endpoint lies on a face, the other endpoint lies in that face’s collar; (13.7) and monotonicity of \psi exclude this as well. Thus both endpoints are interior. Let r>0 be their distance and orient their unique short geodesic from y to x. The first derivative test says that their gradients are q_1(r) times the corresponding parallel unit tangent vectors.

Vary both endpoints in each of three parallel transverse directions. The index-form estimate for the second variation of distance, using the parallel field along the geodesic, is bounded above by Cr. The second derivative test for the two-point maximum therefore gives

Separate axial variations at the two endpoints give

Because the transverse coefficients are exactly one at both endpoints, while the axial coefficients satisfy (13.5), the difference of the two left sides of (13.2) is at most

Their source difference is at least -2C(1+q_1). Since q_1\ge1, these inequalities contradict each other for the already sufficiently large k. No comparison of the two axial coefficients was used. Only variations of the interior endpoints were required, so the short geodesic need not remain inside \Omega_R. We have proved the two-point modulus; letting y\to x gives |\dd f|\le\psi'(0)=B'/k everywhere, including the boundary by (13.7).

The scalar interpolation is now uniformly elliptic. On the given smooth interior and constant-Dirichlet boundary patches, its equation has the exact form required by Theorem 12.1: the gradient and source are bounded, and 0<\chi_0\le\chi_\beta\le1. That theorem first gives a uniform Hölder bound for \dd f, at some positive exponent. The coefficients and source in (13.2) are smooth compositions of x,Z,f,\dd f, so the scalar Dirichlet Schauder estimate gives C^{2,\alpha'} bounds for some \alpha'>0. These bounds make f,\dd f Lipschitz. Together with Z\in C^{1,\alpha}, this yields C^{0,\alpha} coefficients and source, hence C^{2,\alpha} bounds. The compositions are now C^{1,\alpha} and the next Dirichlet estimate gives a C^{3,\alpha} bound.

Subtract a smooth extension of the assigned boundary values and work in the affine space with model Y_R=C^{3,\alpha}_0(\overline\Omega_R), where the zero trace is on the entire boundary. The equation defines a continuously differentiable map into C^{1,\alpha}(\overline\Omega_R). Its derivative in f is

Derivatives of A_\beta and t with respect to \dd f contribute only first derivative terms in \varphi. Its coefficients are C^{1,\alpha} and the ellipticity and positivity are uniform on the bounded range. The maximum principle removes the homogeneous Dirichlet kernel. Linear Dirichlet solvability and Schauder estimates therefore make L:Y_R\to C^{1,\alpha} an isomorphism [18]. The implicit function theorem proves openness in \beta. The bounds above prove closedness: subsequences converge in lower Hölder norms, their limits solve the equation, and the uniform C^{3,\alpha} estimates retain that regularity. At \beta=0 the equation is \Delta_gf=\eta_Nf, with ordinary Dirichlet solvability. Thus the scalar equation is solvable at \beta=1.

For uniqueness, at a positive maximum of the difference of two solutions their gradients agree. Their matrices and positive prefactors therefore agree, whereas strict height monotonicity makes the source difference positive, contradicting the Hessian comparison. The reverse comparison completes uniqueness. Applying the same implicit function theorem with parameters (s,Z) proves continuous dependence into C^{3,\alpha}. At stage junctions use one-sided parameter neighborhoods and the agreement of the prescriptions. The estimates were uniform on bounded trial sets, which proves the last assertion.□

Proof. Choose N large enough for Proposition 11.1, the collar estimates, and the polynomial absorption conditions used in Proposition 11.6. Fix an allowed R. By (11.25) and Proposition 12.2, choose M_0 strictly larger than every C^{1,\alpha} norm of an above-floor fixed point on this truncation. Define

Lemma 13.1 and smooth implicit dependence of t make their union relatively open in [0,4]\times X_R.

Apply Lemma 6.5 to I=[0,4], X=X_R, T_s from (13.8), and \mathcal U=\{(s,Z):Z\in\mathcal O_s\}. The norm cutoff makes \mathcal U bounded; the scalar solution map makes it relatively open; and (13.9) supplies continuity and compactness. Every fixed point is smooth. A fixed point on its relative boundary would satisfy t\ge-\eps and meet either the norm cutoff or the floor. The first is excluded by the choice of M_0, and the second by Proposition 10.7. The section degree is therefore constant. This checks the varying-domain hypotheses for this boundary problem without imposing the floor on trial inputs.

At s=4, the scalar prescription is \tr_{\Achi}H^f=h+D_1(x,w) with zero boundary heights. At an interior extremum w=0 and D_1=0, so the maximum principle gives f=0 for every trial Z, and then t=Z. The terminal scale sets both frozen right sides to zero and the drift is absent. Testing the frozen equation with Y gives Y=0; thus T_4 is the zero map, even though its positive coefficients may depend on the input. Its sole fixed point, Z=0, belongs to \mathcal O_4 and has degree one. The degree at s=0 is consequently one, proving existence there.

Now keep N fixed. The preceding choices of sufficiently large N used only the already proved uniform height and collar results and polynomial losses such as \Pi_N\eta_N/\ell\to0. The arbitrary fixed-N constants in the scalar construction impose no subsequent condition on N. The bounds in Proposition 12.2 are uniform in allowed R on each compact set, including its portions on B. A diagonal Arzelà–Ascoli argument along R\to\infty gives smooth convergence on all these sets. The equation and inner data pass to the limit, as do t\ge-\eps, the height interval, and the collar estimates. Smooth convergence on the fixed collars passes the weighted trace and flux inequalities as well. Strictness of the floor in this exhaustion limit is unnecessary below.□

Proof. Choose a fixed sphere beyond the supports of K,C, and all collar terms. Along each finite-truncation solution the trace equation is the homogeneous linear equation

By (11.25) the coefficients are uniformly elliptic; by Proposition 12.2 all their derivatives are bounded on uniform unit patches, independently of R. On the end,

Choose a=a_N>0 so small that the first term is less than c_N/4, and then enlarge the fixed sphere so the second term has magnitude less than c_N/4. Thus e^{-a_Nr} is a strict supersolution. A fixed multiple dominates both signs of f on the initial sphere and dominates the zero Dirichlet data on S_R. Linear comparison gives |f|\le C(N)e^{-a_Nr} uniformly in R. Local estimates for (13.13), including the zero-Dirichlet estimates on the uniform outer-sphere charts, give the same exponential decay for every derivative, after harmless reduction of a_N if necessary. The identity

and the uniform smooth local bounds then give (13.11). Every graph contribution to the metric and to the equations is exponentially small with unit-patch derivatives.

At zero graph and K=0, the exact formulas give

Hence the actual divergence equation, divided by u, becomes

where \mathcal E_N and its derivatives on unit patches are O_j(e^{-a_Nr}). Define the strictly increasing function

Its derivative is bounded above and away from zero on the fixed bounded range of Z. The chain rule cancels the quadratic term:

Put \beta_1=\delta_1/2 and W(r)=r^{-2}(1-r^{-\beta_1}). In dimension four,

Since 0<\beta_1<\delta_1<1, a sufficiently large multiple of W is positive and has negative Laplacian dominating the absolute source in (13.16) on a sufficiently distant end. Enlarge the multiple to dominate |\Phi(Z)| on its initial sphere. Its value is positive on S_R, whereas \Phi(Z)=0 there. Applying the maximum principle to both \Phi(Z)-CW and -\Phi(Z)-CW gives

uniformly on truncations and then in the exhaustion limit.

Here are the differentiated estimates, which do not follow from the pointwise barrier alone. On an annulus of radius r_1, rescale by x=r_1y and set U(y)=r_1^2\Phi(Z(r_1y)). Its supremum and its Poisson source in the rescaled equation are bounded: the source is r_1^4 times the right side of (13.16), of size O_N(r_1^{-\delta_1}). The rescaled background coefficients have uniform smooth bounds. Interior W^{2,p} estimates, for a fixed p>4, give bounded C^{1,\theta} norms of U on a smaller annulus. The weights have their differentiated power decay, the compositions with Z now have bounded scaled Hölder norms, and the exponential errors remain bounded with all scaled derivatives. The right side therefore has a bounded C^{0,\theta} norm. Schauder estimates give a bounded C^{2,\theta} norm of U. The inverse of \Phi has bounded smooth derivatives on the relevant range. Rescaling back gives Z=O_2(r^{-2}), and (13.11) gives the same statement for t. Finally |\dd Z|^2=O(r^{-6}) in (13.14); since \delta_1<1, this is lower order than r^{-4-\delta_1}. Equation (13.11) now gives the asserted bound for \Delta_gt.□

Proof. We first establish the exact sign and normalization at fixed N. Lemma 13.3 gives u=1+O_N(r^{-2}) and

Also u\Xi\in L^1(\Omega,\dd V_g): on the end the quadratic term is O_N(r^{-6}), the graph and height-gradient terms are exponentially small, and the remaining weights are O_N(r^{-4-\delta_1}). The divergence theorem on a large truncation consequently gives a finite limit

The plus sign before the inner flux arises because the domain-outward normal on B is -\nu.

To compute the metric flux, write \ghat=e^{2t}g up to exponential errors. The leading perturbation relative to g is 2t\delta_{ij}, so in four dimensions the change of its energy numerator is

All omitted derivatives are O_N(r^{-5}) or exponentially small. Their sphere integrals tend to zero. Since \dd t=O_N(r^{-3}) and g-\delta=O_2(r^{-2}), changing the normal and measure in its flux from Euclidean to background ones also has vanishing error. Equation (13.22) thus proves both existence of the new energy and the exact relation

We now use polynomial constants only. Apply Lemma 11.3 with \varphi=1 and use (11.14). On the fixed compact collars \mathcal C one has \min_{\mathcal C}\rho>0, and therefore

The polynomial absorbs the fixed comparison constant in the second line. Because \eta_N/\ell=e^{-\eps N/2}, for sufficiently large N this is at most one half of \int_\Omega u(\rho+\delta_0\cT).

It remains to estimate the penalty. Its support in an above-floor solution lies in -\eps\le t\le-\eps+1/N. For N>1/\eps this interval is negative, and uniformly there

For example uv=L_0l^2\sigma^2/\sqrt{1+l^2\sigma^2} \le L_0l\sigma. Thus on that band

Young’s inequality applied to \sqrt{\eta_N}\ell\sigma and \Pi_N\ell(\rho_0+\rho_1)/\sqrt{\eta_N}, together with (13.25), yields everywhere

The enlarged \Pi_N is still polynomial. The three residual weights are integrable against \dd V_g in dimension four: \rho_0=r^{-4-\delta_1}, \rho_0^2=r^{-8-2\delta_1}, and \rho_1^2=r^{-6} on the end, and they are bounded on the compact part. Combining (13.23), the boundary absorption, and (13.26) gives

This and (13.24) prove (13.21). Finally \ell=e^{-\eps N} and \ell^2/\eta_N=e^{-\eps N/2}; both dominate every polynomial loss. No constant from the fixed-N decay or Schauder estimates appears in this bound.□

Proof of Theorem 8.2. The geometric preparation in Proposition 9.6 chooses a closed connected one-ended exterior \Omega inside the original data. It has nonempty compact smooth boundary and does not change the asymptotic energy. The last assertion of that Proposition compares every full cut in \Omega with the original infimum A_*. Fix 0<\eps<1. For each sufficiently large N, Proposition 13.2 gives the desired smooth solution after exhausting the outer radius at that fixed N. Equation (13.17) gives nonnegative scalar curvature; Equation (13.18) gives strictly negative inner mean curvature on every face. Equations (13.19) and (13.20) establish the required asymptotics, integrability and completeness, as well as the lower metric bound. Proposition 13.4 gives finite ADM energy with the stated error. Its polynomial bound uses only the fixed geometric data and \eps. Since \ell=e^{-\eps N} and \ell^2/\eta_N=e^{-\eps N/2}, this error tends to zero. No estimates uniform over a further sequence of prepared data, and no convergence of the constructed metrics as N\to\infty, are needed.□

Proof. Set m=\sqrt{E^2-|P|^2}. The rest-end replacement [35, Proposition 8.8] applies with exactly these one-ended weak-decay hypotheses. It produces data (g_R,K_R) on the same exterior, satisfying the dominant energy condition and the weak future-trapping inequality, with compactly supported K_R and an exact static Schwarzschild–Tangherlini tail. Their energy is m_R\to m, their momentum is zero, and their full enclosing infima satisfy

This is the lower bound required below; no equality or convergence of these infima is used.

Fix R first. The strictification of [35, Lemma 8.9] gives (g_{R,s},K_{R,s})=(e^{2s\phi_R}g_R,e^{s\phi_R}K_R), for all sufficiently small s>0. It preserves completeness and compact tensor support, makes the boundary strictly future trapped, and gives, for a fixed 0<\delta_1<1,

The densities are integrable and the ADM energy is finite. Thus every hypothesis of Theorem 8.2 is met, with all data and their constants now fixed. Its complete boundary solve and Corollary 8.4 give

The latter corollary already exhausts the domain, takes N\to\infty, and removes the conformal floor for this fixed pair (R,s).

Still at fixed R, strictification supplies E_{g_{R,s}}\to m_R and A_*(g_{R,s})\to A_*(g_R) as s\downarrow0. Taking this limit and then using (14.1) gives

Finally take R\to\infty. The mass and density conventions coincide with those of both preparation results, so no normalization factor is introduced. The proof has used their preparations and the boundary construction, without importing a numerical spacetime theorem.□

Proof. The coefficients in (15.3) are uniformly bounded and elliptic at fixed N. For sufficiently small c>0, substitution of e^{-cr} into its left side gives an upper bound

outside one fixed sphere. A multiple of this positive supersolution dominates |f| on that sphere; the zero outer value is already dominated. Comparison for f and -f gives exponential decay, uniformly in the truncation radius. The assumed unit-patch estimates, followed by the smooth linear interior and zero-Dirichlet boundary estimates for (15.3), give the asserted differentiated decay. These are the smooth-coefficient estimates of [18, Chs. 6–9] and [1]. The implicit formula for t then shows that t-Z and the graph metric error have the same type of bounds.

It remains to identify what survives when those graph errors are removed. Set the graph height and its first two derivatives equal to zero in the coefficients, retaining the exponentially small difference as an error. Then H^f=0, F=0, w=0, D=d=\chi=1, t=Z and u=L_0(Z). The smooth scalar quadratic form at this zero graph is

One can also verify this directly from its block formula. Choose any unit vector e, write j=K(e,e), M=K(e,\cdot)|_{e^\perp} and T=K|_{e^\perp}, and use \tr T=-j. The terms involving K become \tfrac12|T^0|^2+|M|^2+\tfrac23j^2. Since |T|^2=|T^0|^2+j^2/3, this is exactly |K|^2/2. The terms linear in \dd Z contain w and vanish, leaving the second term in (15.6). In particular setting the graph to zero does not set the tensor to zero.

Now (\log L_0)'=p+2. Divide (15.4) by L_0, insert (15.6), and move the quadratic gradient term to the left. Smooth dependence on the bounded solution variables absorbs the graph errors into \mathcal E_N and gives (15.5).

Define \Phi(0)=0 and

On the common bounded range of Z, both \Phi' and its reciprocal are bounded at fixed N. The chain rule gives

The right side is O_N(r^{-4-\delta}), because |K|^2=O(r^{-2-2q}) and 2+2q>4+\delta. With \alpha=\delta/2, set b_0=r^{-2}-r^{-2-\alpha}. For sufficiently large r, this is positive and

A fixed multiple of b_0 dominates both the absolute source and |\Phi(Z)| on the fixed inner sphere. Since \Phi(Z)=0 at the outer sphere, comparison for both signs gives |\Phi(Z)|\le C(N)r^{-2}, and hence |Z|+|t|\le C(N)r^{-2}. These estimates hold before exhaustion, so the local limit has the asserted normalization at infinity.

Finally rescale annuli R<r<2R to unit size. The metric has uniformly controlled scaled C^{1,\beta} coefficients for each fixed 0<\beta<1. Interior W^{2,p_1} estimates with p_1>4 first give |\dd Z|=O_N(r^{-3}). The first derivative bound on K, the differentiated weights and the exponential error then give a controlled scaled Hölder norm for the source. Schauder estimates yield \nabla^2Z=O_N(r^{-4}). The same follows for t from the graph error.□

Proof. Choose a positive smooth function b\ge|K|_g which equals a constant multiple of r^{-1-q} sufficiently far out. We first solve

where \nu points into the exterior. In particular, the normal derivative in this equation is the negative of the domain’s outward normal derivative at its inner boundary.

Here are the existence and decay details. On the compact truncation M_R, impose \phi=0 on the outer coordinate sphere and replace b on the right by \lambda b, 0\le\lambda\le1. The linearization, with homogeneous boundary data, is

Its drift is bounded by b. Its kernel is zero by the strong maximum principle and the boundary point lemma, since the outer Dirichlet face is nonempty and the inner face has homogeneous normal derivative. The mixed operator has Fredholm index zero, by deformation to the mixed Laplacian, and hence is invertible. The two boundary components are disjoint, smooth, and compact, so ordinary scalar elliptic estimates apply without a junction between boundary types; see [18, Chapters 3, 6, and 9] and [1, Theorems 7.3 and 15.2–15.3].

For fixed R, the global mixed W^{2,p} estimate, p>4, and interpolation give

These constants are uniform in \lambda. If the suprema were unbounded, divide by their sup norms and take a subsequence. Compact embedding of W^{2,p} in C^{1,\alpha}, with a smaller \alpha>0, gives a nonzero limit v satisfying

This is a homogeneous linear equation with a bounded measurable drift: set the drift to \lambda_*b\grad v/|\dd v| where \dd v\ne0, and to zero otherwise. The maximum principle and boundary point lemma again exclude a nonzero solution. Thus (17.4) is bounded, and Schauder estimates and bootstrapping give closedness in the continuity argument. The \lambda=0 mixed Poisson problem is solvable, so the implicit function theorem and compactness give a smooth solution for \lambda=1. It is nonnegative: a negative minimum is excluded in the interior by -\Delta\phi>0, on the outer face by its zero value, and on the inner face by \partial_\nu\phi=-1.

We next bound these solutions independently of R. On the end consider

Choose a fixed A>0, then a large fixed r_0, so that T>0 and

Indeed, -\Delta_\R T=A\delta(2+\delta)r^{-4-\delta}, while both the metric error and b|\dd T| are O(r^{-4-q}). Also

Comparison on \{r_0\le r\le R\}, including the zero outer value, therefore yields

If m_R were unbounded along expanding truncations, divide by m_R. Local versions of (17.4), including the inner boundary estimates, give a nonnegative limit attaining value one in the fixed core. It solves -\Delta v=b|\dd v|, has homogeneous inner normal derivative, and is O(r^{-2}) by (17.6). Its positive maximum is therefore attained at a finite point. The strong maximum principle or the boundary point lemma excludes it. We have a uniform core bound, and diagonal compactness gives a smooth solution of (17.3).

Rescale annuli by their radii. The bound \phi=O(r^{-2}), the scaled metric bounds furnished by g-\delta=O_2(r^{-q}), and the linear gradient growth in (17.3) first give scaled W^{2,p} estimates by interpolation. They give Hölder control of the gradient; the right side then has the scaled Hölder control needed for Schauder estimates. Hence

The source in (17.3) is O(r^{-4-q})+O(r^{-4-\delta}), and is integrable. The divergence theorem shows that