The Kerr–Newman Penrose Inequality for Axisymmetric Electrovacuum Exteriors

Proof. Truncate the given end at a sufficiently large rotation sphere. Both boundary spheres carry ordinary effective rotations. Here is a direct verification of the assertion for an invariant sphere. Its Euler characteristic forces fixed points. At a fixed point the tangent representation is a planar rotation with integer weight. That weight has absolute value one: otherwise a nonidentity group element would fix the point and have identity differential both on the sphere and on its invariant inward normal, hence would be an isometry with identity first jet and therefore the identity on the connected ambient manifold. At a nonfixed point a finite stabilizer acts trivially on the orbit tangent. It acts trivially also on the transverse tangent line, by orientation, and on the inward normal. The same first-jet argument excludes such a stabilizer. The slice theorem now makes the quotient of the sphere an interval, with an ordinary rotation disk at either endpoint; the free circle bundle over its interior is a product. Consequently the action extends over an ordinary rotation ball.

Cap the two boundary spheres equivariantly, using invariant collars. The resulting closed three-manifold is simply connected. Its orbit space is a compact orientable surface with nonempty boundary, the boundary arising from fixed axes. The only other possible singular orbits are isolated exceptional fibers, represented by interior cone points of finite cyclic order. The orbifold fundamental group of this surface is a quotient of the fundamental group of the three-manifold. Indeed, away from axes and exceptional fibers, loops lift to paths; an endpoint displacement in a fiber can be closed within that fiber without changing the projected loop. Filling an axis tube introduces no nontrivial projected fiber relation. Filling an exceptional solid torus imposes the relation that the corresponding meridian raised to the isotropy order is trivial, as seen in its cyclic disk cover. Van Kampen therefore gives the asserted surjection onto the orbifold group.

An orientable surface with nonempty boundary and cyclic cone points has orbifold group equal to the free product of its ordinary free surface group and the cyclic groups at those points. Triviality of this group implies that the surface is a disk and that there are no cone points. Removing the two cap quotients gives the half-annulus description. Its regular interior is contractible, so its principal circle bundle is trivial.

For the last smoothness assertion, use equivariant normal disks near each axis and a signed transverse polar radius there. Changes of angular reference are smooth functions on the quotient, even in that signed radius. Their local phase lifts can be patched by a partition of unity; the only obstruction is the integral circle cocycle, which vanishes on this orbit rectangle. The same construction relative to a far-end strip preserves the given azimuth there. Thus the product gauge has the stated compatibility with the smooth axes. No isothermal, Weyl, or other metric coordinates have been assumed.□

Proof. The magnetic flux form is closed by \divg_gB=0. Since \Omega retracts onto a sphere, its only two-dimensional de Rham period is the magnetic flux, which is zero after rotation. The form is therefore exact. The decay can be imposed without sacrificing smoothness. On a rescaled annulus, radial homotopy reduces the primitive problem to a two-form on a fixed sphere; its zero integral is exactly the condition for a spherical primitive. These homotopy operators, followed by a fixed spherical elliptic inverse, have uniform bounds through the required derivatives. Scaling back a two-form of Cartesian order r^{-3} gives a primitive of order r^{-2}, with its first two derivatives of the corresponding orders. On consecutive overlapping annuli, the difference of the primitives is a closed one-form and hence is the differential of a function. After subtracting a constant, its rescaled derivatives are bounded. Use cutoff multiples of these functions on the overlaps to glue the annular primitives. The annuli can be chosen with bounded overlap, so these corrections retain the bounds. The same argument glues the end primitive to any interior primitive. All ingredients are smooth on each finite annulus; this avoids any demand for uniform bounds on unspecified higher derivatives at infinity. Averaging over the circle makes \alpha invariant and preserves its equation and decay.

Set \chi=\alpha(\eta). Cartan’s formula gives

The contraction of the momentum constraint with the Killing field, together with \divg E=0, now gives

The divergence theorem proves conservation of Equation (3.2). If \alpha is changed by a closed invariant one-form \beta, then \dd(\beta(\eta))=-\iota_\eta\dd\beta=0. The constant \beta(\eta) is zero on the axis, and thus everywhere. Finally \alpha(\eta)=O(r^{-1}), while E\cdot\nu=O(r^{-2}); the electromagnetic surface term is O(r^{-1}) on the end spheres. This proves its disappearance in the limiting flux. The proof does not identify the bare gravitational flux on S with the flux at infinity.□

Proof. An invariant smooth scalar is even in y. A smooth invariant horizontal vector field has an odd radial component and an even longitudinal component. Since X/y is smooth positive even, X*_hE_H and X*_hB_H have radial component O(y) and longitudinal component O(y^2) with the corresponding parities. Integration from the axis gives the first two assertions.

Smoothness of an invariant symmetric tensor gives an angular–axis component l_x=O(y) and an angular–radial component l_y=O(y^2). After multiplication by X^2 and rotation by *_h, \omega has radial component O(y^3) and longitudinal component O(y^4). Put \delta\psi=\psi-\psi_{\rm ax} and \delta\chi=\chi-\chi_{\rm ax}. Then

The last two terms have the same radial and longitudinal orders as \omega. Integration proves the order-four assertion.

For the end estimates, perform the same argument in rescaled annuli. The leading electric field is the radial Coulomb field, so the leading angular derivative of \psi is Q\sin\vartheta\,\dd\vartheta. The remainder contributes a uniformly bounded function of order \sin^2\vartheta at a pole. The decaying invariant magnetic primitive gives the sharper estimate for \chi: its normalized angular component vanishes linearly in transverse radius, and \chi=\alpha(\eta) vanishes quadratically. Finally X^2l=O(r^{-1}\sin^2\vartheta) as a horizontal covector, with the stronger radial and longitudinal polar orders just proved. Integration along a meridian of a coordinate sphere therefore bounds z and gives its order-two polar difference. For the stronger corrected estimate, fix a dyadic annulus of radius R and use coordinates scaled by R, with signed transverse coordinate y. The rescaled Cartesian tensor R^3K(R\,\cdot) has uniform C^3 bounds and the same invariant tensor parity. The pullback of \omega=X^2*_hl therefore has transverse component O(y^3) and longitudinal component O(y^4), uniformly in R: the factors from X^2 and a covector pullback cancel the overall R^{-3} decay of K. The first two potential estimates and their differentiated versions give these same orders for -(\psi-\psi_{\rm ax})\dd\chi +(\chi-\chi_{\rm ax})\dd\psi. Integrating their sum with \omega from y=0 gives \widetilde z=O(y^4) uniformly. On the unit-scale annulus y is comparable to \sin\vartheta, proving Equation (3.7). Smooth polar coordinate changes with uniformly bounded transverse ratios preserve these estimates. Differentiating the defining equations supplies the radial decay and angular derivative bounds. The assumptions g-\delta=O_4(r^{-1}), K=O_3(r^{-3}) and E,B with three controlled derivatives give all the asserted rescaled bounds.□

Proof. First consider smoothness. In the normalized polar coframe (\dd y,\dd x,y\dd\phi), the angular–radial metric coefficient is even of order y^2, and the angular–longitudinal coefficient is odd of order y. The ratio X/y is smooth positive even. Taking the horizontal Schur complement of the angular coefficient therefore preserves all polar parities and the equality of the radial and angular diagonal coefficients at the axis. The same calculation for the tensor gives the smooth lift of k_0. These statements apply uniformly on rescaled end annuli. Apparent factors y^{-1} in Cartesian differentiation are harmless: the Taylor terms permitted by the polar parities are smooth transverse Cartesian polynomials, and Taylor remainder estimates bound every remaining divided derivative. This also proves the required differentiated end decay. In particular

on the end. On compact axis tubes \mathcal A_y is smooth odd and \mathcal A_x smooth even, so f_a is odd of order y. Smooth invariant vector fields have angular component odd of order y, proving the stated parities of s.

An invariant surface is generated by a meridional curve, and its area is 2\pi times that curve’s length in X^2h. This is the same for g and g_0. First variation of this expression for arbitrary invariant normal speeds shows that its pointwise mean curvature is also unchanged. Its tangent space is generated by one horizontal unit vector and e, so its tangential trace uses only K|_{H\times H} and K(e,e). This proves preservation of future expansion.

For the energy, use the end azimuth fixed in Lemma 3.1. Let \mathcal R be azimuthal reflection. The difference of g from its reflection average is precisely the angular cross tensor, which is odd under \mathcal R. The linear ADM flux on a coordinate sphere is invariant under this Euclidean reflection, and hence is zero on that odd tensor. The difference of the reflection average from g_0 is X^2\mathcal A\otimes\mathcal A in horizontal coordinate components. It has Cartesian order O_2(r^{-2}), since it is quadratic in the normalized cross coefficients O(r^{-1}). Its ADM flux is O(r^{-1}). Thus the energies agree. The polar estimates above make the calculation uniform at both end poles.

It remains to prove the source identities, including their signs. The horizontal Koszul formulas, for invariant horizontal fields a,b, are

Together with the base connection these give

For example the last vanishing also follows from the azimuthal reflection symmetry of (g_0,k_0). The scalar constraint and

now give Equation (3.9) exactly.

For clarity, the normalized first two rows of \mathcal D_X need not individually be smooth three-dimensional covectors at the axis. Their radial coefficients are even and their longitudinal coefficients odd. Their squared norms, and their products with the corresponding odd s_i, are smooth invariant scalars and covectors. The third row is a smooth horizontal covector: its radial coefficient is odd of order y, and its longitudinal coefficient is even of order y^2. Thus all source combinations in Equation (3.9) are smooth. Finally 2|s\cdot\mathcal D_X|\le |s|^2+|\mathcal D_X|^2 implies \mu_0\ge|J_0|_{g_0}.□

Proof. Let (\theta^0,\theta^1,\theta^2) be an oriented Lorentz orthonormal coframe with \theta^0(n)=1, and put \theta^3=X(\dd\phi+\mathbf A). The relevant signs are

Set p=X^{-1}\dd\chi, t=X^{-1}\dd\psi, and v=X^{-2}\omega. Then

Contraction with n and restriction to its spatial slice give exactly E^\flat=\iota_nF and B^\flat=*_g(F|_{T\Omega}). In particular t(n)=-B_e and p(n)=E_e. Writing \dd\mathbf A=\theta^0\wedge a+f_a\theta^1\wedge\theta^2, the mixed connection formula gives l=Xa/2 for the positive future-normal convention. Consequently

This proves the restrictions in Equation (3.11).

For completeness, the integrability of the corrected twist in spacetime follows from the mixed Einstein equation. The mixed Ricci component is

Einstein–Maxwell gives it as -2(*_Ht)_{ij}p^j. These identities are equivalent to \dd\omega=2\dd\psi\wedge\dd\chi, and hence to the closedness of \omega+\chi\dd\psi-\psi\dd\chi. Thus z exists locally, with its constants fixed by the slice.

The horizontal parts of the Maxwell equations give \divg_Ht=2\langle p,v\rangle_H and \divg_Hp=-2\langle t,v\rangle_H, while \dd^2\mathbf A=0 gives \divg_H(X^{-1}v)=0. For a covector \beta, the conformal change in dimension three obeys \divg_\gamma\beta=X^{-3}\divg_H(X\beta). Therefore

The horizontal and fiber curvature contractions of the connection formulas are

One can see the coefficients directly before contraction: the horizontal sectional numerator has correction -3X^2\dd\mathbf A(a,b)^2/4, and the horizontal–fiber block is -X^{-1}\nabla_a\nabla_bX+ X^2(\dd\mathbf A)_{ak}(\dd\mathbf A)_b{}^k/4. The Lorentz Hodge identity

shows that the two Einstein–Maxwell Ricci blocks in Equation () equal, respectively,

Thus the fiber equation reads X^{-1}\Box_HX=-|p|_H^2-|t|_H^2-2|v|_H^2. Since u=\log X and \gamma=e^{2u}H, this becomes

The conformal Ricci formula

then cancels the remaining scalar multiples of H and gives

Equations () and (3.16) are exactly the four Euler–Lagrange equations of the target metric in Equation (3.4). This proves Equation (3.12).

Conversely, start with Equation (3.12), put H=X^{-2}\gamma, and define p,t,v as above. The first equation in () makes 2X^{-1}*_Hv closed, so it has a local connection primitive \mathbf A. Define F by Equation (3.13). The remaining divergence equations and the identities \dd^2\psi=\dd^2\chi=0 give both Maxwell equations. The identity \dd\omega=2\dd\psi\wedge\dd\chi supplies the mixed Einstein equation. Reversing the horizontal and fiber calculations supplies the other Einstein equations.

The slice reconstruction is explicit. Its horizontal tensor is the second fundamental form of its slice in H; the other components are

The spatial potential equations recover E_H,B_H, and the first two velocities recover B_e,E_e. The third recovers the spatial connection curvature. A chosen connection on a simply connected meridian is then fixed up to angular gauge. Finally the inverse constant duality rotation recovers the original ordered pair of electric and magnetic fields.□

Proof. A path of bounded G-length has bounded u. The same length then controls the variations of \psi and \chi. Since these coordinates are bounded, control of \dd z+\psi\dd\chi-\chi\dd\psi also controls the variation of z. A finite-length escaping path is therefore impossible, which proves completeness. The underlying manifold is \R^4.

Let E_0,E_1,E_2,E_3 be the frame dual to (\dd u,e^{-u}\dd\psi,e^{-u}\dd\chi,e^{-2u}\omega). Its nonzero brackets are

In the bivector order (01,02,12,03,13,23), Koszul’s formula gives the curvature operator

Its quadratic form on (a,b,c,d,e,f) is -(a+f)^2-(b-e)^2-(c+d)^2-3(c-d)^2, so it is nonpositive. The geodesic and convexity assertions follow from the Cartan–Hadamard theorem and the second variation formula.□

Proof. Substitute 4M^2=R^2+2Q^2+C/R^2 and a^2=J^2/M^2. Clearing denominators gives the first two identities. Ordinary differentiation gives the third and also

At the lower endpoint b\to0 (also when C=0, where the endpoint is R=0); as R\to\infty, b\sim R/2.□

Proof. The axial radius and quotient are

Both Maxwell contractions in Equation (3.11) have zero time component. The same holds for *_H\dd\mathbf A, since \mathbf A=-a(2Mr-Q^2)\dd t/W has curvature of the form \dd t\wedge a spatial one-form. Thus the map is time independent.

An explicit check of all potential constants is useful. With x=\cos\vartheta, they are

These expressions satisfy the contractions and corrected twist in Equation (3.11), by differentiation. In particular the axis values are (-Q,0,2aM) and (Q,0,-2aM). To check the angular sign directly, the shift is

Stationarity and the positive convention K=\tfrac12\mathcal L_ng give K=-\mathcal L_\beta g/(2N) on the canonical slice. Hence

The leading term Q\dd t/r of \mathfrak a_* gives F_{tr}=Q/r^2+O(r^{-3}) and therefore positive electric charge Q with E=\iota_nF. The magnetic flux is zero.

The horizon is r=M+b, or \sigma=0. There W=(r^2+a^2)^2=R^4 by Equation (3.18); its area density is R^2\sin\vartheta\,\dd\vartheta\dd\phi, proving the area and the boundary value of q_*. In the Euclidean end radius \varrho=(b/2)e^\sigma, one has r=\varrho+M+b^2/(4\varrho). Expanding the spatial metric in the associated Cartesian chart gives

and hence ADM energy M. The shift and potential give K_*=O_3(\varrho^{-3}), the electric Coulomb leading term, and B_*=O_3(\varrho^{-3}), with uniform polar derivatives.

Put \rho_*=bw. Equation (3.21) is static, with spatial metric e^{2q_*}(\dd\sigma^2+\dd\vartheta^2). The wave-map equation in Equation (3.12) is therefore \divg_G(\rho_*\dd\Phi_*)=0, giving the first equation in Equation (3.20). Its spatial Ricci trace is

Since \Delta\rho_*=0, this proves the second equation.

Finally all the displayed spatial coefficients and potentials are smooth even functions of \sigma at zero, with the ordinary polar parity in \vartheta. The apparent lapse division in the canonical K_* and fields is removable. Indeed the angular velocity \omega_*=a(2Mr-Q^2)/W is constant on the horizon: \omega_H=a/R^2. Thus its \vartheta derivative is O(\sigma^2) and its \sigma derivative is O(\sigma), while the lapse is \sigma times a smooth positive even function. The electrostatic potential on \partial_t+\omega_H\partial_\phi is also constant on the horizon; its nonconstant remainder is O(\sigma^2). These orders give smooth canonical tensor and field components after division by the lapse. The metric has no conical defect at the axes, and the explicit potentials give precisely the order-two and corrected order-four axis behavior established earlier. For a two-sided extension across \sigma=0, the stationary lapse is taken with its smooth sign, b\sinh\sigma\sqrt{P/W}; the stationary Killing field changes time orientation across the bridge. This supplies a smooth future normal and tensor, whereas using the absolute value of that lapse would only reflect the chosen stationary time orientation. This proves all the regularity assertions.□

Proof. All losses below can be made arbitrarily small. We first remove the extra puncture tails, while retaining the original weight elsewhere. Take pairwise disjoint puncture neighborhoods and modify the weight only where t>T. A positive bounded lower extension exists there: in the main strip, the puncture comparison implies

Indeed \sin\beta=\operatorname{Im}Z/\abs{Z-a_j} and \abs{\dd\log(Z-a_j)}=\abs{Z}\abs{Z-a_j}^{-1} \abs{\dd\log Z}. Thus a small positive smooth quotient can replace the divergent quotient near a_j, with the transition wholly in t>T.

These modifications cannot decrease the crossing infimum by a fixed positive amount as T\to\infty. To prove this, suppose that modified crossings of length at most a fixed M did produce such a loss. Enlarge each original side by all the modified tails attached to that side. Between the last encounter with the initial enlarged side before the first encounter with the terminal enlarged side, keep the intervening path. It meets no modified tail in its interior. If an endpoint is in a tail, this path must traverse the unchanged collar T/2<t<T to leave its puncture neighborhood. On that traversal,

At a point attaining, or arbitrarily approaching, this minimum, attach the path to the nearer local ray by a constant-t angular interval. Its original length is at most

Both local rays at a_j lie on the same original global side. Attaching at the two ends of the retained path therefore produces an original opposite-side crossing with only O(T^{-2}) additional length. The finitely many disjoint collars make this estimate uniform. This contradicts a fixed loss. Taking M>L suffices, since longer paths cannot violate any proposed lower bound below L.

At the negative end, uniform convergence permits replacement by a smooth even profile below b_-, with arbitrarily small relative loss. A transition can be made within a region where the original quotient is uniformly close to b_-. At the positive end, fix a small \delta>0 and take V large enough that

Blend downward to (1-\delta)c_+e^{2v} before v=V, and thereafter use (1-\delta)c_+e^{2\min(v,V)}. Radial projection (v,\alpha)\mapsto(\min(v,V),\alpha) converts a path for this lower weight into a path for the preexisting weight with length increased by at most (1+\delta)/(1-\delta). Below V this follows from the same ratio comparison on the transition; above V, projection deletes the radial component, and the angular weight at V is at most that ratio times the saturated weight. Projection preserves the two sides. Consequently saturation loses only this arbitrarily small factor in the crossing lower bound.

The resulting positive continuous quotient is bounded above and below, and is constant in v on both far ends. Approximate it uniformly from below by a smooth quotient, even at the sides and still product on the ends. Such approximation follows by even reflection, convolution on the compact remaining rectangle, and a small downward adjustment; the positive lower bound converts uniform error to relative error. The product profiles can be approximated in the same way before extending them along the tails. The accumulated relative and additive losses can be chosen smaller than L-\ell. This proves the lemma.□

Proof. Choose a lower weight with crossing infimum at least \ell>L', leaving strict room for the losses that follow. Let d_+ denote its distance to the side \alpha=0. The distance inequality gives \abs{\dd d_+}\le W almost everywhere. Since m_0\le b_0\le M_0, angular intervals give

The second inequality follows by appending an angular interval from the point to the opposite side to an almost minimizing path for d_+. For a small h>0, truncate d_+ to [h,\ell-h] and put

Thus \abs{\dd\varphi_0}\le2W/(\ell-2h), and \varphi_0 is identically 1 and -1 on full, uniform collars of the respective sides. Take h small enough to retain strict slack above L'.

Clamp v to a large closed interval with endpoints inside the product tails. This projection is length nonincreasing for W, so composition preserves the gradient bound and makes \varphi_0 independent of v on the far tails. Smooth after even reflection at the sides. The constant side collars remain constant; elsewhere W has a positive lower bound on a compact rectangle, so convolution and a small slack loss preserve the weighted gradient bound. This produces a smooth function. A sufficiently small convex combination with \cos\alpha then gives strict interior values and

Its gradient cost fits within the slack, because \sin\alpha\le W/m_0.

On a product end write W=b_\pm(\alpha)\sin\alpha and set

The vertical crossing on that end gives I_\pm\ge\ell. Both the smoothed function and \varphi_\pm have the same endpoint values and even reflection, so their difference is O(\sin^2\alpha), uniformly on the end. Interpolate to \varphi_\pm sufficiently slowly in v. The angular derivative is bounded by the convex combination of the preceding bounds. The new v derivative is bounded by C\abs{\chi'}\sin^2\alpha, which is arbitrarily small relative to W when the interpolation interval is long. Thus, for some L_0>L',

The angular derivative is strictly negative on the far ends and in small uniform side collars. A compactly supported arbitrarily small Morse perturbation away from those collars leaves these properties intact and leaves only finitely many critical points. We retain the notation \varphi and a slightly smaller L_0>L'.

We next replace its level sets by a foliation without losing the estimate. Delete small open intervals about its finitely many critical values. In a closed remaining regular value band, intersect a level with [-V,V]\times[0,\pi], taking V in the common product tails. There is exactly one level point on each vertical edge and no point on a horizontal edge. A compact regular level is a union of circles and intervals; its two boundary points must therefore be joined by one interval component. This is the unique essential component.

For an entire closed regular band, a vector field satisfying \dd\varphi(V_\varphi)=1, chosen tangent to the vertical edges, flows these essential components smoothly into one embedded rectangle. It has precisely the two extreme essential arcs and the appropriate vertical-edge intervals as its boundary. The Jordan separation theorem therefore identifies its image with the full region between the two arcs. In particular, discarding the other, circular level components creates no holes in this essential band. Essential bands at distinct values are disjoint and ordered, since their order is fixed at both vertical ends and disjoint crossing arcs cannot exchange that order. Near 1 and -1 keep the monotone side collars: outside smaller such collars the values stay uniformly away from the extrema, by compactness and the product-end behavior.

These finitely many essential bands can be completed to a foliation of the whole strip by proper arcs. Here is the extension with its boundary conditions. Slightly shrink each good band, retaining its level coordinate on a neighborhood of the resulting closed band. Between two consecutive bands, their smooth boundary arcs and the two vertical edges bound a disk. Prescribe product parametrizations on collars of all four edges, using the original label where it is already fixed and the existing angular labels on the vertical ends. They agree at the corners because the ends are product. Shrink the collars so that nonadjacent ones are disjoint and round within the compatible corner charts. Smooth Schoenflies charts reduce the extension to a prescribed orientation-preserving collar map of a disk. Here the relative extension can be described explicitly. In polar coordinates its boundary derivative is triangular, with positive diagonal entries. Interpolation to the radial product map is therefore an isotopy on a sufficiently thin collar; extend its generating vector field with a cutoff over the disk. For the resulting boundary map, choose an increasing lift h with h(\theta+2\pi)=h(\theta)+2\pi. The map

where 0\le\chi\le1, \chi=1 near the boundary, and \chi=0 near the center, is a smooth disk diffeomorphism: its angular derivative is 1-\chi+\chi h'>0, and it is the identity near the center. Undoing the extended collar isotopy gives an extension agreeing with the entire prescribed smaller collar. Thus there is no additional corner or orientation condition to impose. Extending by the product ends gives a smooth value function \widehat\varphi, with no interior critical points, agreeing with \varphi on the smaller good bands and on the side collars. It has the same endpoint values and reflection parity. The labels in every intervening strip lie solely in its assigned complementary value interval. Thus discarded circles do not reappear as extra components of a good label.

There is no a priori gradient estimate in an intervening strip, but there are only finitely many of them, they avoid the side collars, and they are product at infinity. Hence \abs{\dd\widehat\varphi}/W has a finite bound there, say M. The total width \delta of their label intervals, including small transition intervals within good bands, can be chosen arbitrarily small. Choose a smooth positive function r of the label, equal to 1 away from those intervals, at most 1 everywhere, and at most \varepsilon on every bad label interval; arrange its transitions inside good bands. Put

The normalization factor is at most 2/(2-\delta). On good bands the old gradient estimate is multiplied by at most this factor. On bad bands it is bounded by this factor times \varepsilon M. First choose \delta small, and then \varepsilon small. The function F\circ\widehat\varphi satisfies Equation (4.2) for some L_1>L'. It has no interior critical points. Since F'>0 at the endpoints, its side jets remain nondegenerate and even. We again call this function \varphi.

Set \Theta=\arccos\varphi. In the interior this is a submersion. At \alpha=0, the even expansion has the form 1-\varphi=\alpha^2 a(v,\alpha^2) with a(v,0)>0; smooth square-root division shows that \Theta is \alpha times a positive smooth even function. The same argument applies to \pi-\Theta at the other side. Thus \Theta has precisely the stated reflection and nonvanishing boundary derivative.

For use in the analytic argument we also record an isotopy. Integrate D\Theta/\abs{D\Theta}^2 from the lower edge, with \Theta as time. The vector field is vertical on the product ends and has bounded coefficients in the remaining compact region. Its flow gives a strip diffeomorphism D(v_0,\theta) with \Theta(D(v_0,\theta))=\theta. Reflection reverses the flow parameter and gives ordinary even/odd parity for the two components of D. In particular D is product on uniform far ends.

Such a diffeomorphism is smoothly isotopic to the identity within this class. First straighten its two end angular diffeomorphisms by convex angular interpolation, whose derivative stays positive, and extend these collar motions by their generating vector fields. Next straighten the parametrizations of the horizontal sides, if necessary, by convex interpolation of their increasing longitudinal parametrizations; these motions are compactly supported in v. After this, the derivative on each side is diagonal with positive entries by reflection. Linear interpolation to the identity in a sufficiently thin collar is consequently a local collar isotopy. Extend its generating field with a symmetric cutoff. The remaining map fixes a neighborhood of the entire boundary of a compact disk. After choosing a smaller fixed boundary collar, the smooth disk isotopy theorem [20, Theorem 4] joins it to the identity through orientation-preserving diffeomorphisms fixing that collar. Reattaching the fixed collars and ends, and taking inverse coordinate maps, gives the required family of angular submersions. All these uses of disk topology concern smooth parametrizations of planar disks; no metric estimate is required during the isotopy.□

Proof. Let \Theta_s, 0\le s\le1, be the isotopy, with \Theta_0(v,\alpha)=\alpha and \Theta_1=\Theta. For fixed (s,\vartheta), the allowed values u=x+iy form the line

It is a smooth proper embedded line with horizontal ends. These lines vary smoothly, periodically in \vartheta, with \Gamma_{s,-\vartheta}=\overline{\Gamma_{s,\vartheta}}. At \vartheta=0,\pi the line is the real axis. Their heights and the locations at which they become horizontal have uniform bounds for s\in[0,1] and \vartheta modulo 2\pi.

Orient each line from its left end to its right end. Its tangent phase has a smooth real lift \beta, obtained by transporting the initial phase zero through the strip diffeomorphisms of the isotopy. It is periodic in \vartheta and odd under reflection. To check the potential integer ambiguity, at the left horizontal end its value is zero. At the right horizontal end its value is an integer multiple of 2\pi; that integer is continuous in the isotopy parameter and is zero for s=0, so remains zero. The same uniqueness of the lift proves periodicity and reflection, with value zero on the straight boundary leaves. On compact sets of positions, the line data, the phase, and all their derivatives are uniformly bounded.

Openness. Work with real-symmetric exterior analytic traces in C^{m,\lambda}, m\ge2, 0<\lambda<1, normalized by U(\infty)=0; the boundary equation takes values in the odd real C^{m,\lambda} functions. At a solution, its linearization in a variation \dot U is

Here c is the magnitude of the label gradient, with sign fixed by the orientation of the strip. At a symmetric trace it is even, while \beta is odd.

The normalized exterior Schwarz problem gives a real-symmetric analytic function a, with a(\infty)=0, whose imaginary boundary value is \beta. The zero-mean compatibility condition is automatic for odd data. The multiplier A=e^a is nonvanishing, real-symmetric, and A(\infty)=1. Substituting \dot U=AH reduces a prescribed linearized value g to

The right side is again odd. The Schwarz problem solves it uniquely: the harmonic conjugate is determined by the periodic Hilbert transform, and the real constant is fixed at infinity. It gives a bounded inverse on the stated Hölder spaces. Multiplication by A preserves the normalization. Smooth composition on the circle now permits the implicit function theorem. Since U=0 solves the equation at s=0, the set of solvable parameters is nonempty and open.

Uniform range bounds. The line family bounds \Im U on the circle, and hence throughout the exterior by the harmonic maximum principle. Write h_{s,+}(\vartheta) for the height of the right horizontal end of \Gamma_{s,\vartheta}. It is a uniformly smooth odd periodic function. Let H_{s,+} be its normalized exterior Schwarz solution and set V=U-H_{s,+}. The functions H_{s,+} are uniformly bounded in every fixed smooth norm. If \Re V had an excessively large positive maximum, it would occur at a point of the circle on an open arc where U is in the right horizontal tail. On that arc \Im V=0. The Cauchy–Riemann equations then give zero normal derivative of \Re V at the maximum, contradicting the boundary Hopf lemma unless V is constant. The constant case has V(\infty)=0 and cannot give such a maximum. One may apply this argument after inversion to the closed disk, where infinity is an interior point. Applying the same argument to the left horizontal ends gives a lower bound for \Re U. Thus every solution, at every s, satisfies

The constant may depend on k; the only effect of \log k is to translate the horizontal thresholds for the line family.

Boundary derivative compactness. Suppose instead that a sequence of solutions has P_j=\max_{\abs\xi=1}\abs{\partial_\vartheta U_j}\to\infty. Choose maximum points e^{i\vartheta_j} and use the half-plane coordinates

The analytic function \xi U_j'(\xi) has its maximum modulus on the circle, so Equation (4.5) and the definition of P_j give

On the imaginary axis its trace lies in \Gamma_{s_j,\vartheta_j+y/P_j}. Differentiate the defining equation. With \beta_j(y) the tangent phase along this trace, the result is

For example, the forcing is (1-\Theta_{s_j,\alpha})/\abs{D\Theta_{s_j}}, evaluated at the trace. The range lies in a fixed compact set, the label gradient is bounded away from zero there, and u_j is uniformly Lipschitz. Hence both \beta_j and g_j have uniform Lipschitz bounds on every fixed boundary interval.

We justify convergence of derivatives all the way to this boundary. Extend \beta_j from a slightly larger interval by a fixed cutoff, and solve the half-plane Schwarz problem with that imaginary trace for an analytic logarithm. Its exponential A_j and its inverse have uniform local C^\lambda bounds for every 0<\lambda<1: the Schwarz integral and Hilbert transform send compactly supported uniformly Lipschitz data to uniformly C^\lambda functions. The analytic derivative D_j=i u_j' divided by A_j is bounded and, by Equation (4.6), has uniformly C^\lambda imaginary trace on the smaller interval. Subtract a Schwarz solution for a localized version of that trace. The remainder has zero imaginary trace and extends across the interval by Schwarz reflection. It is uniformly bounded on a smaller disk, so its derivatives there obey the interior Cauchy estimates. Adding back the Schwarz solution proves uniform local C^\lambda bounds for D_j up to the straight boundary. Consequently a subsequence converges locally in C^1 on the closed half-plane to a bounded holomorphic u, and \abs{\partial_yu(0)}=1.

After a further subsequence s_j\to s_\infty and \vartheta_j\to\vartheta_\infty modulo 2\pi. The limiting boundary trace lies on the single proper embedded line \Gamma_{s_\infty,\vartheta_\infty}. Properness and the bound \abs u\le C place this trace in one compact simple subarc A of that line: take a compact parameter interval containing the inverse image of its intersection with the closed range disk. For every polynomial p, the bounded half-plane maximum principle gives

There is no boundary condition missing at infinity here. Map the half-plane to a disk; the possible exceptional boundary point has harmonic measure zero, and its contribution disappears by boundedness when a surrounding boundary arc shrinks. The displayed inequalities put the image of u in the polynomial hull of A. In one complex variable the polynomial hull of a compact set is the set together with its bounded complementary components. A planar simple arc has connected complement, so its hull is itself. The open mapping theorem therefore forces u to be constant, contradicting its normalized boundary derivative. The assumed derivative blowup is impossible.

Closedness. With first derivatives bounded, the same phase-multiplier argument on fixed boundary intervals gives uniform C^\lambda bounds for them. The phase and forcing, now smooth compositions of C^{1,\lambda} traces, have C^{1,\lambda} bounds. More explicitly, on the fixed circle set D=i\xi U' and let A be the phase multiplier. Then \Im(D/A)=g/\abs A, with g=(1-\Theta_{s,\alpha})/\abs{D\Theta_s} evaluated at the trace. Here D/A is anti-real-symmetric and g is even; this bootstrap uses Schwarz estimates for unrestricted normalized analytic traces, not the odd codomain used for openness. The imaginary boundary mean vanishes automatically because this actual analytic function has (D/A)(\infty)=0, and its remaining real constant is zero by the same normalization. Thus the Schwarz estimates increase the regularity of U by one derivative. Induction gives bounds of every fixed order; interior bounds follow from Cauchy estimates. Thus sequences of solutions have convergent subsequences in the chosen C^{m,\lambda} space, and their limits solve the boundary equation at the limiting parameter. Solvability is closed as well as open. Connectedness of [0,1] proves existence at s=1.

Univalence. For f=k\xi e^U, differentiating the boundary equation gives

Thus its boundary derivative never vanishes. If two boundary values of f coincide, their logarithmic radii agree and their lifted angles differ by an integer multiple of 2\pi. The equivariance of \Theta then says that their parameters differ by the same multiple of 2\pi. The boundary curve is therefore injective. Its winding about zero is one, directly from f=k e^{i\vartheta}e^{U(e^{i\vartheta})}. It is a regular Jordan curve enclosing zero.

For a point w off this curve, apply the argument principle to the exterior domain, including its one simple pole at infinity. If n(w) is the winding of the positively parametrized boundary curve, the number of finite preimages, counted with multiplicity, is 1-n(w). It is one in the unbounded component and zero in the bounded component. No interior point can map to the boundary curve, by the open mapping theorem and the zero count on its bounded side. Hence f is univalent, has no interior critical points, and maps onto the unbounded component. Symmetry implies that a real image point has a real preimage, by uniqueness of preimages. Since the upper half maps to the upper half near infinity, it does so throughout. Finally U(\infty)=0 gives f=k\xi+O(1) with precisely the prescribed leading coefficient.□

Proof of Lemma 4.1. Choose the angular calibration with L_1>L' and solve the analytic boundary problem at the prescribed k. On the upper inner edge,

Consequently Equation (4.2) gives

This already proves Equation (4.1) with strict slack. If an endpoint coincides with an additional puncture, translate f by a sufficiently small real constant. This preserves symmetry, univalence, and the leading coefficient k, and preserves enclosure of zero because the untranslated Jordan curve has positive distance from zero.

The slack survives such a translation, including at the endpoints. Indeed, in Z coordinates the smooth lower length density is

Away from zero it is smooth with odd reflection and has a positive simple zero at the real axis. For the translated smooth symmetric curve, the quotient

extends continuously to \vartheta=0,\pi and depends continuously on the small real translation t. At t=0 it is at least L_1/2. Compactness of [0,\pi] therefore preserves the bound L'/2 for all sufficiently small t. Avoiding the finitely many translations that put either endpoint at an a_j makes the cut miss every extra puncture. Its interior already lies strictly in the upper half-plane. The original density dominates this lower density throughout, so the required estimate follows for B. The statement about remaining marked side points follows from symmetry and exterior univalence.□

Proof. Put C=Q^4+4J^2. The model parameter relations give

Consequently

Here b'(R)>0 by Lemma 3.7. If L/2>\sqrt C, Equation (5.23) shows that the comparison expression increases up to R^2=L/2 and decreases afterward. At that point its logarithmic term vanishes, proving (5.22). If L/2=\sqrt C, use models with R^2\downarrow L/2; then b\downarrow0 and the logarithmic correction tends to zero. The limiting argument is numerical and uses only strictly subextremal comparison models.□

Proof of Proposition 6.1, using Theorem 6.2. We first construct a strictifying direction, including the required elliptic solve. There is a smooth positive invariant function v with

whose Laplacian is integrable and whose end flux has a finite limit. Here \nu points into the exterior and \langle r\rangle is a smooth positive extension of r over the compact part. Choose a smooth invariant majorant d_0\ge|k|_g with d_0=O(r^{-2}), and positive smooth weights \zeta_0,w_0 of orders -2,-3-\delta, respectively, with the corresponding symbol bounds. Solve

on a finite truncation. Continuation from d_0=0 has an invertible linearization: it is a uniformly elliptic operator with bounded drift, homogeneous Neumann data on the inner boundary and homogeneous Dirichlet data on S_R. The maximum principle gives uniqueness and nonnegativity. On each fixed truncation, W^{2,p} estimates and interpolation bound the solution by its supremum. If these suprema were unbounded during continuation, division by them and compactness would give a nonzero homogeneous bounded-drift solution with homogeneous mixed data, contradicting the maximum principle. This proves existence on each truncation.

The bounds persist under exhaustion. On a fixed far region, a sufficiently large multiple of r^{-1}(1-Cr^{-\delta}), multiplied by one plus the compact-core supremum, is an upper barrier: the favorable term in \Delta(r^{-1-\delta}) dominates the metric and drift errors of order r^{-4}. If the core suprema along an exhaustion were unbounded, normalize once more. The barrier makes the limit tend to zero at infinity, while its value is one somewhere on the core. Hence it attains a positive global maximum. It satisfies a homogeneous bounded-drift equation with homogeneous inner Neumann data, so the strong maximum principle and boundary point lemma exclude this. Exhaustion therefore gives a positive solution of Equation (6.7) with the outer condition replaced by decay to zero.

For completeness, rescale on an end annulus by x\mapsto Rx and v\mapsto Rv(Rx). The value bound just proved gives uniform W^{2,p} bounds on smaller annuli for p>3: the rescaled equation has bounded coefficients and at most linear growth in the gradient. The resulting C^{1,\alpha} control makes the right side uniformly Hölder, so Schauder estimates give the stated O_2(r^{-1}) bound. They also give higher symbol estimates through the orders allowed by the coefficients. The equation now gives an integrable Laplacian; the Euclidean and metric Laplacians differ by integrable errors. The divergence theorem yields a finite end flux. These facts prove Equation (6.6).

For a small constant \delta_{\rm p}>0, make the change

and keep the potentials fixed. Identify the old and new unit balls by w_{\rm new}=e^{-2\delta_{\rm p}v}w_{\rm old}. After multiplying the new constraint inequality by e^{4\delta_{\rm p}v}, its ith source row is its old value times e^{-4h_i\delta_{\rm p}v}\le1. Each row is nonnegative on the unit ball, since

This calculation is made off the axis; its left side is a smooth function on the unit-ball bundle and the inequality extends by continuity. Smooth invariant conformal factors preserve the stated polar orders and all these quadratic source combinations. The conformal scalar-curvature and momentum-divergence formulas add to the left side the quantity

Equation (6.6) makes this positive, uniformly on compact sets and at least c_{\delta_{\rm p}}r^{-3-\delta} on the end, after decreasing \delta_{\rm p} if necessary. Inside its positive scaling factor the boundary expansion decreases by 4\delta_{\rm p}. Thus the new boundary is strictly future trapped and Equation (6.1) has a positive residual. Crossing lengths vary by a factor tending to one, scalar-curvature decay is preserved, and the energy tends to E_g as \delta_{\rm p}\downarrow0 by the finite flux of v.

Keep \delta_{\rm p}>0 fixed. Cutting off k sufficiently far out changes the constraints by O(r^{-4}), which is smaller than the strict residual because \delta<1. If the metric does not yet have all end symbol derivatives, smooth only its far end by averaging normalized pullbacks under small constant rotations and dilations. Use smooth kernels commuting with the axial action and the polarization reflection; for example, the rotation kernel may be conjugation invariant. This is smoothing in angle and logarithmic radius and gives symbol bounds of every order. The Euclidean linear part of scalar curvature averages with positive dilation factors; the nonlinear errors are O(r^{-4}) using the original two derivative bounds. Hence its positive r^{-3-\delta} lower bound and its O(r^{-3-\delta}) upper order persist. The source rows are O(r^{-4}), so the full residual persists with a smaller constant. Patch to the original metric on a fixed distant annulus, choosing the averaging aperture small enough that the compact patch errors fit within the residual. Uniform equivalence tends to identity as the aperture decreases. Averaging the linear sphere flux, using rotation invariance and the dilation scaling, also gives convergence of the ADM energies. All preparations preserve polarization.

Choose a smooth positive \mathfrak m no larger than this sourced residual for any |w|_g\le1, with the compact and end lower bounds in Theorem 6.2. Indeed, the residual has the form a(x)+b_x(w) with a a smooth invariant scalar and b a smooth invariant covector. Its minimum on the unit ball is the continuous positive function a-|b|_g, bounded below on compact sets and by c r^{-3-\delta} on the end. A smooth positive minorant is obtained by a partition of unity on the compact part, with a sufficiently small multiple of r^{-3-\delta} on the end, followed by averaging over the circle. No derivative of a row norm is used. The row nonnegativity implies also \mathfrak m\le8\pi(\mu-|J_{\rm con}|_g). The theorem therefore applies to the prepared data. Set

and renormalize the asymptotic coordinates by the constant scale. The identity \bar g_n^{-1}=g^{-1}-w_n\otimes w_n gives, row by row,

Here the calculation again holds off the axis. Since f_n is a smooth invariant function, \bar g_n has the same smooth polar parities as g. The contractions D_1(w_n),D_2(w_n) are smooth odd of order y, and D_3(w_n) is smooth even of order y^2. Thus the completed squares and the new squared row norms extend smoothly across the axis. In particular the resulting scalar-curvature inequality extends there by continuity. Combining the sourced residual with Equation (6.4) leaves the strictly positive quantity \mathfrak m/2. Each row of \cD_{X_r} has in addition the factor e^{-2h_i(t_n+\eps)}\le1. Consequently

Writing the polarized metric as g=h+X^2\dd\phi^2, its meridional area metric satisfies

Thus every new crossing arc has the old lower length bound, and the potential and axis regularity required by Proposition 5.1 are retained. The end energy obeys E_{g_r}\le e^{2\eps}(E_g+o_n(1)). Apply that proposition to each fixed admissible R, first exhaust the truncation at fixed n, then let n\to\infty, then \eps\downarrow0. Finally undo the far-end smoothing and cutoff and let \delta_{\rm p}\downarrow0. The energies and crossing lower bounds converge as just proved, giving Equation (6.2).□

Proof. For the original data, Equation (3.9) supplies Equation (6.1). Its scalar equation, with the axial connection-curvature contribution retained in s, gives R_g=O(r^{-4}) from |\cD_X|+|s|=O(r^{-2}) and k=O_1(r^{-3}). This meets the required scalar decay for every 0<\delta<1. The available two metric derivatives and one tensor derivative suffice even if forming polar components loses a derivative. The crossing-area comparison and the preservation of invariant areas under polarization give L=A/(2\pi), while E_g=m. If A/(4\pi)>\sqrt{Q^4+4J^2}, choose R^2=L/2 in Equation (6.2); its logarithmic term vanishes. At the closed endpoint use subextremal radii R^2\downarrow L/2; then b(R)\to0 and the same conclusion follows by continuity. The formula for F_* in Lemma 3.7 and m>0 give the displayed inequality after squaring. The same choice of R proves the last assertion for sourced data. No classification of equality at the closed endpoint is used here.□

Proof. First show that every exterior support of a small parallel set D_z=\{x:\dist(x,D)\le z\} has expansion at most c+Cz, with C depending on the compact geometry and Q, but not on its curvature. Let such a support touch at x, and take a shortest segment of length z from y\in D to x. Choose a local unit vector field V agreeing at y with its initial tangent, and set F(y')=\exp_{y'}(zV(y')). Prescribe the first derivative of V so that \dd F_y is exactly parallel transport along the segment. For each initial vector this is the Jacobi boundary-value problem with that vector and its parallel translate as the two endpoint values. Rescaling the Jacobi equation shows that the required initial derivative is O(z) times the initial norm. Its component along the segment is zero, since the parallel component of the Jacobi field is constant. Thus it is a permissible first jet of a unit field. More explicitly, if B_i=\nabla_iV, the unit-length constraint on its symmetric second jet is \langle\operatorname{sym}\nabla^2_{ij}V,V\rangle =-\langle B_i,B_j\rangle; its tangential symmetric part can be chosen bounded, and the antisymmetric part is fixed by the bounded curvature commutator. In a smooth orthonormal frame, normalizing a vector-valued polynomial with the prescribed value and first jet realizes precisely this compatibility. Thus V has bounded second derivatives independently of the support. Since F is the identity at z=0, smoothness of the exponential map now gives |\nabla\dd F|=O(z).

If \phi defines the support at x, then \phi\circ F supports D at y: every y'\in D maps a distance at most z from D. The tangent ball at x aligns the support normal with the final segment tangent. Since \dd F_y is exactly parallel transport, comparison of the two tangential Hessian traces after gradient normalization has error only O(z), coming from \nabla\dd F; there is no uncontrolled multiple of \nabla^2\phi. Normal transport and variation of Q contribute another O(z). This proves the claimed parallel-set bound.

Choose 0<a<b so small that c+Cb<c'. Smooth approximation of the distance followed by a regular-value choice gives a smooth enclosure containing D_a and contained in D_b; fill any pockets that meet no designated outer face. Let \eta\ge0 be one on its boundary and supported where \dist(\cdot,D)<b. Replacing Q temporarily by

makes this enclosure a strict inner barrier when A_0 is sufficiently large. The designated outer barriers stay positive. Apply the strict barrier theorem to each complementary component meeting a designated outer face; every such component has both barriers, by connectedness and the strict enclosure. The resulting marginal surfaces together bound a region containing D_a.

If the minimum distance z of that boundary from D were at most b, its boundary at a minimizing point would be an exterior support of D_z. Indeed a path from D of shorter length cannot cross that boundary before reaching it. Its expansion for the original Q is c'+A_0\eta\ge c', contradicting the strict parallel-set bound. Thus the boundary lies beyond D_b, where \eta=0, and has original expansion c'. This proves the enclosure assertion without any regularity hypothesis on D or any support-curvature bound.□

Proof. Consider the auxiliary stationary Lorentz metric -l^2(\dd b-\dd f)^2+\bar g. Its b-slice metric is g, its lapse is U=l\sqrt D, its shift is \beta=Uw, and its slice tensor is

Set \mathfrak B(A,B)=A:B-(\tr A)(\tr B), P_A=A-(\tr A)g, and Q_1=K-k_s. The Gauss and normal-expansion identities, with stationarity, U\tr k_s=-\divg\beta, and, for the future unit normal n_{\rm L},

give the two expressions for the spacetime scalar density

The right side is the static warped-product expression in the coordinate b-f. Moreover,

Substitute the constraint equations and P_K:\nabla\beta=-U\mathfrak B(K,k_s) into Equation (7.17). The result is

The conformal terms in \frac12e^{4t}R_{\hat g} are -4\bar\Delta t-4|\dd t|_{A_d}^2. Differentiating Equation (7.6) gives

Combining these formulas leaves the quadratic tensor expression

This expression is smooth at w=0; for example (1-\chi)q=\frac{p+4}{4+pv}S_1(w,w). Expanding with \tr T=F-\chi q gives Equations (7.11) and (). Completing the mixed and normal squares gives Equation (), hence Equation (7.14). When \dd t=0, the normal quadratic form in (\sqrt\chi q,F) has eigenvalues 1 and (3-\chi)/4, which lie between 1/2 and 1, proving Equation (7.15). Finally, at a constant-f face, (\Hess f)|_{TB}=(\partial_\nu f)\mathrm{II}_B. Substituting this into the trace equation gives the first boundary identity; the graph and conformal mean-curvature formulas give the second.□

Proof. At an inner boundary contact in the first three stages, j(t)=0. Equations (7.25) and (7.16) therefore give 4\partial_\nu t=-H-N_B<0, incompatible with a minimum when \nu points into the domain. The preceding outer estimate excludes the outer face.

Suppose there is an interior contact. Then \dd t=0 and \Hess t\ge0. The graph Gauss equation in the Riemannian warped product g+l^2\dd b^2 gives

where, for a symmetric tensor A,

In fact l^{-1}\Hess l=p\Hess t at contact, so the warping term -v\tr_\perp(l^{-1}\Hess l) and the conformal term -4\bar\Delta t have favorable signs; H^f is the graph second form there, up to the immaterial choice of normal.

Directly from Equation (), at \dd t=0,

Use \chi\le d and |dFq|\le\chi q^2/4+(d^2/\chi)F^2 together with d/\chi\le1+n/4 and Equation (7.15). This gives

Replacing S_1 by H^f=S_1-K introduces only products involving transverse components and dq. Young’s inequality and Equation (7.15) absorb an arbitrarily small fixed multiple of \mathcal T, at polynomial cost in n for |K|^2. Thus, for a fixed c_*>0 independent of n,

When K=0, improve the last error to \Pi_n vF^2. To see this, if (1+p)v is sufficiently small, d and \chi are uniformly close to one and the normal quadratic form in Equation (7.31) is uniformly positive definite; its value at d=\chi=1 is \frac32q^2-Fq+\frac12F^2. Otherwise v\ge c/(1+n), so the previous error is bounded by a new polynomial times vF^2.

At contact, differentiating w multiplies the vector slot of H^f by A_d, and hence |\nabla w|\le\Pi_n(|K|+\sqrt{\mathcal T}). View F as a smooth function of (x,w,h,p). Its explicit x,w derivatives have compact support and polynomial bounds on the range in Equation (7.28); also \nabla p=0 at contact. Since F_h\ge1, for any fixed \eta_0>0,

Here \mathbf1_{\rm comp} denotes the indicator of a fixed sufficiently large compact set; it can equally be replaced by a fixed compactly supported majorant. The difference between the physical flux and V_\lambda costs at most \eta_0\mathcal T+\Pi_n\mathbf1_{\rm comp} after taking its divergence and dividing by u. Indeed differentiate uA_\chi K(w,\cdot) using the preceding estimate. The remaining factor \nabla\log u=H^f(w,\cdot) contracts against A_\chi K(w,\cdot) and involves only \chi H^f_{ee} and H^f_{e\perp}, which are controlled by Equation (7.15).

Compare Equations (7.11) and (7.30), and use Equations (7.32) and (7.33). Outside the fixed compact set, K=0, F=h=O(r^{-\gamma}), and \Ric_g=O(r^{-3}). Choose \eta_0 and then \delta_0 sufficiently small compared with c_*. The resulting lower bound is

Choose C_n so that \mathcal P dominates the last error plus 2\rho everywhere. This requires only polynomial growth, since \rho_0 is positive on the compact set. At the floor m_0=1, and subtracting the prescribed source from this geometric lower bound gives at least

contradicting Equation (7.24).

In the fourth stage, f=0, t=Z, and v=0. As long as the common scale factor is positive, the inner floor-contact derivative is that factor times (-H-N_B)/4<0. At an interior floor contact, \mathcal T is compactly bounded by K; enlarge C_n so that \delta_0\mathcal T+\rho-\mathcal P<0 there. The divergence of 4u\nabla t is nonnegative at a minimum, giving a contradiction. At scale zero, the unique mixed-problem solution is Z=0. This also cannot meet the floor.□

Proof. The quantifier sufficiently large truncations allows a threshold R_0(n) depending on n. If the first conclusion fails, then for every \eta'>0 and every n_0 there is an n\ge n_0 with failures at arbitrarily large R. Choose successively \eta'=1/j, n_j\ge\max\{j,n_{j-1}+1\} and R_j\ge\max\{j,R_{\rm floor}(n_j)\} among those failures, with R_{\rm floor}(n) the outer threshold in Lemma 7.3. There are points x_j in the fixed compact set with h_j(x_j)\le b_-+1/j. Pass to x_j\to x_* and take the lower spatial relaxed limit \underline h, defined by infima over tail indices and shrinking balls, followed by the lower semicontinuous envelope. Near a white face first extend each h continuously by the constant b_+ behind the face. This does not assert continuity of the relaxed limit. By construction, \underline h(x_*)\le b_-.

Away from black faces, every smooth lower test \phi for \underline h at a value sufficiently near or below b_-, with \nabla\phi\ne0, has oriented level expansion for K at most \underline h+C_b. Indeed make the touching strict by a fourth-order perturbation and transfer it to local minima of h-\phi along a subsequence. The contacts are interior, because white boundary values are higher. There \nabla f=\tau_0^{-1}\nabla\phi, so

The lower Hessian comparison in the trace equation therefore gives the level expansion bound. The second-stage correction has D_0\le0 at these low values, so it has the favorable sign.

We pass from this lower-test statement to exterior supports of closed sublevels. Fix b_-<k<k' sufficiently close to b_- and put D_k=\{\underline h\le k\}. Apply the increasing reparametrizations

Their lower relaxed limit on \underline h is the function u_k equal to zero on D_k and one off D_k. At a point outside the closed set, lower semicontinuity provides a local positive gap from k; on the set itself the unchanged point gives limiting value zero. Let \phi define an exterior support at x_0, with \phi(x_0)=0, \nabla\phi(x_0)\ne0, and D_k locally on its nonpositive side. Choose a small closed ball with |\phi|<1/4, and set \phi_\eta=\phi-\eta|x-x_0|^4, where \eta>0. Then u_k-\phi_\eta has its unique minimum zero at x_0 and a positive gap on the ball’s boundary. Minimize the lower semicontinuous function r_j(\underline h)-\phi_\eta on this ball. Because r_j(\underline h(x_0))\le r_j(k)\to0, the lower relaxed limit property gives interior minimizers y_j\to x_0, with minimum values c_j\to0. Their transformed contact values a_j=\phi_\eta(y_j)+c_j=r_j(\underline h(y_j)) tend to zero and, for each finite j, lie strictly between zero and one.

On a sufficiently small neighborhood of y_j the smooth function \psi_j=r_j^{-1}(\phi_\eta+c_j) is therefore a lower test for \underline h. The inverse is single-valued and strictly increasing; moreover \nabla\phi_\eta(y_j)\to\nabla\phi(x_0)\ne0, so \nabla\psi_j(y_j)\ne0 for large j. Since r_j(k')\to1, one also has \underline h(y_j)<k' for large j. Increasing reparametrization preserves the gradient direction and the tangential Hessian divided by gradient length, regardless of the size of its derivatives. Apply the already proved lower-test inequality to each fixed \psi_j, and then let j\to\infty. The quartic perturbation has zero first and second jets at x_0. Thus every exterior support of D_k, away from black faces, has expansion at most k'+C_b. The original solution limit is taken first for each fixed test; the reparametrization limit is a separate subsequent limit.

This property excludes contact with a white face: the reversed white face itself would be an exterior support, with black expansion -c_w>0. The radial height bound makes D_k compact on the end, since k<0. Apply Lemma 7.1 in the original connected truncation \Omega_R, with the original boundary as required inner face and S_R as its only designated outer face. Its compact set is exactly D_k\cup\mathcal B_{c_b}: neither the white regions nor omitted bounded components are adjoined. Every frontier point of this union outside \mathcal B_{c_b} lies in the reduced exterior away from white faces, and hence has the just-proved support bound. At any component of the smooth frontier of \mathcal B_{c_b}, including components facing omitted regions, a support of the union also supports that black region, and comparison gives the bound c_b<k'+C_b. Thus the union contains the required inner collar and meets all hypotheses of the lemma, whether or not it is connected. The lemma encloses it in a smooth trapped region at any threshold slightly larger than k'+C_b and still negative. It is therefore contained in the full black region at that larger threshold.

Right continuity at c_b=b_-+C_b lets us choose a threshold just above c_b whose full region still misses C_*. Then choose k,k' close enough to b_- that the preceding enclosure uses a smaller threshold. A purported sequence in C_* with limiting height at most b_- places a limit point in D_k\cap C_*, a contradiction. This proves uniform strict separation by the usual subsequence criterion.

For the other color apply the argument to -h,-K,-C. Extend the original heights constantly by their low boundary values locally behind black faces before taking the reverse relaxed limit. The reversed black face is a forbidden support for the high sublevels. The ambient domain is now the fixed black complement in \Omega_R, connected after the black pocket filling, with all black frontiers and S_R designated as outer faces. Adjoin only the full white region to the high sublevel. Supports on any full white frontier, including hidden components, are controlled by its smooth threshold equation. Visible black faces are excluded as just explained. A hidden black component has positive distance from the closure of the reduced domain: a fixed collar along its connected smooth face lies in a single adjacent complement component. Thus the union is disjoint from every designated outer face. Omitted components are not adjoined and introduce no other support frontier. Any pocket filling in the enclosure lemma is relative to these designated faces. Hausdorff right continuity at c_w gives the same contradiction. If a selected color is absent, right continuity means nearby larger-threshold regions are empty, and the enclosure contradiction is unchanged.□

Proof. In a black collar, test with h_0=b_-+\psi(s), where \psi(0)=0 and \psi'>0. At a comparison point the gradient is common to the solution and test, so t,l,d,\chi are computed using that gradient and the given value of Z. The trace of the test is

where H_s and P_s use \nu_s. For slopes bounded above and below by positive constants,

Use Lemma 7.4 on the far leaf of a fixed short collar, and choose a small positive slope so that a lower test stays below the solution there. In this direction D_0 vanishes. The stable-leaf inequality and C=C_b-k_Bs give a positive trace-minus-right-side residual at least k_Bs/2-O(d), apart from the last logarithmic-slope term in Equation (7.36). Choose

nonnegative and smoothly cut off to zero by s=2/n, with a fixed sufficiently large A. By Equation (7.37) its contribution dominates O(d) on the initial interval. From s\ge1/n onward, the s margin dominates the exponentially small d error. The slope changes by at most the fixed factor e^{2A}; choose its initial size after A, keeping the whole test small. This constructs a lower barrier with strictly positive boundary derivative.

All constants in this choice are independent of n and R. The far leaf is fixed with the domain; height separation gives one fixed \eta' there for every sufficiently large n and R\ge R_0(n). Choose A from the fixed geometric residual constants first, and only then choose the initial slope using \eta', k_B, the collar width and the fixed ratio e^{2A}. Thus these comparisons introduce no arbitrary fixed-n constants into the polynomial bounds \Pi_n.

For an upper barrier choose a large positive slope and the opposite logarithmic derivative. The smooth leaf and offset variations are O(s), whereas the height increase dominates them; the negative logarithmic-slope term controls the initial d errors. The global height bound supplies comparison on the far leaf. Strict height monotonicity gives the comparisons throughout the collar. Equality at its initial face then gives the two black boundary derivative bounds in Equation (7.35). Reverse signs for white faces. These barriers near the appropriate faces, together with height separation off their own-color collars, give b_-\le h\le b_+ on the required compact core.

In the third stage, start at the current assigned height (1-\zeta)b and use lower and upper tests with large negative and positive slopes, respectively. Taper the magnitude of each slope using the negative logarithmic derivative just described. The limiting directional inequalities (7.26) give the respective residual signs at the face. At fixed boundary height the errors are O(s+d+\chi); in particular, the term \tr_{A_\chi}K is compared using the common leaf normal. The height shift controls the s error, and the logarithmic-slope term, using Equation (7.37), controls d+\chi. This gives both boundary derivative bounds in the third stage. All uses of l\ge\ell occur at a common test gradient, so a ceiling for Z is unnecessary.□

Proof. Set W=w/\sqrt D. By the two-sided normal-slope bounds (7.35), w\cdot\nu has a definite sign on each face and a\ge1/2 there. The corresponding signed fluxes of uW=Lw and \sqrt D W=w are La and a. Apply the divergence theorem to these fields, multiplied by the collar cutoff and \varphi, reversing the sign on the white collars. The estimates needed for the resulting volume terms are

For the second inequality, direct differentiation gives

In \divg(Lw)/u every derivative is multiplied by \sqrt d; in particular the longitudinal Hessian contribution is controlled by \sqrt d\,|q|. Derivatives of L contribute (p+2)\sqrt d\,w(t). These and the transverse terms are bounded by (7.14); replacing H^f by S_1-K costs a bounded compactly supported term. The same calculation applies to the second quotient in (7.42). Derivatives of the fixed collar cutoffs have bounded cost. This proves both trace estimates.

At assigned first-stage heights, the two colors both satisfy w_\nu(F-P_B)=aH. Consequently the physical boundary expression is

The omitted normal drift in the first homotopy stage has size O(\chi)\le O(d), because \dd f is normal on a face and A_\chi has normal eigenvalue \chi. The prefactor in the homotopy boundary condition lies in [0,1]. Thus its flux has size O(ud). Finally, (7.35) and l\ge\ell give D^{-1/2}\le C\tau_0/\ell, proving (7.40).□

Proof. Only an upper bound for Z remains to be proved. Indeed Z=t+\tfrac18\log D\ge-\eps, and an upper bound for Z, together with t\ge-\eps and l\ge\ell, bounds D, t and \sigma. The height estimate already bounds f=h/\tau_0 at fixed n.

An integral estimate above a fixed height. There is M_0=M_0(n) such that, in the first three stages,

In fact the penalty is supported, on the allowed side of the floor, where -\eps\le t\le-\eps+1/n. There l and L are comparable to \ell, so large Z forces arbitrarily large \sigma and hence arbitrarily large a\sigma. The bounded coefficient \mathcal P=C_n(\rho_0+v\rho_1) is therefore absorbed into half of the indicated height term.

In the first stage choose a smooth increasing k_0 which vanishes below M_0 and equals one above M_0+1, and test the divergence equation with k_0(Z). Integration by parts gives the nonnegative principal contribution 4\int u k_0'|\dd Z|_{A_\chi}^2 on the source side. The drift cross term is bounded by half of this contribution plus C\int u k_0'|K|^2. The latter is bounded: it has compact support, and in the transition strip Z\in[M_0,M_0+1] the floor bounds D, u and all zeroth-order coefficients. The inner flux costs at most

Use (7.38). On the fixed collars, positivity of \rho implies 1+\sqrt{\mathcal T}\le C(\rho+\delta_0\mathcal T), and \Pi_n\tau_0/\ell\to0. Thus these terms absorb into the positive source in (7.43). The remaining trace term contains k_0'|\dd Z|_{A_\chi}; Young’s inequality absorbs its quadratic part into the principal contribution, with another bounded transition-strip remainder. We obtain

Let \Gamma denote the ordinary graph of f in g+\dd b^2, with measure \dd\Gamma=\sqrt{1+\sigma^2}\,\dd V_g. On every fixed compact set, (7.44) bounds e^Z in L^2(\dd\Gamma). To see the density comparison, use

The part where k_0=1 is controlled by (7.44), since \rho has a positive lower bound on that compact set. On the complementary part Z\le M_0+1 the floor bounds the entire density. Moreover \ell\le l\le\exp(1) and d/\chi\le1+n/4, so the graph gradient norm is comparable, at fixed n, to |\dd\cdot|_{A_\chi}.

Sobolev inequality on the graph, including its boundary. On compact base patches, including patches meeting an inner face,

for smooth \omega supported away from the other patch edges. Here the constant is independent of the particular graph. Embed a fixed smooth extension of the base patch isometrically in a Euclidean space, using Nash’s theorem [14], and use the product embedding for the graph. The Euclidean mean-curvature vector of this graph obeys

Indeed its graph-normal component is the trace of \nabla^2f/\sqrt{1+\sigma^2} with longitudinal coefficient (1+\sigma^2)^{-1}; all its transverse entries and this longitudinal trace are controlled by (7.14). The second fundamental form of the fixed base embedding adds a bounded term, since it is contracted against the inverse graph metric.

The Euclidean submanifold Sobolev inequality of Michael–Simon [13], in the boundary form of [5, Theorem 1], therefore gives

for nonnegative \varphi of the stated support. The graph has constant height on each inner face, so its boundary measure is exactly \dd A_g. The last term is controlled by (7.39) and the graph comparisons above. Taking \varphi=|\omega|^4, then applying Cauchy–Schwarz to |\omega|^3|\nabla_\Gamma\omega| and (1+\sqrt{\mathcal T})|\omega|^4, proves (7.45).

Iteration to a supremum bound. For a fixed base cutoff \eta test the first-stage equation with \eta^2e^{2kZ}/L, where k\ge k_*(n) will be large. After integration by parts the positive differentiated-test contribution is

The source has lower bound \delta_0\mathcal T-\Pi_n. Since \dd\log L=(p+2)\dd t, coercivity and Young’s inequality bound its cross terms with 4\dd Z+\lambda K(w,\cdot) by

Take k_*(n) large enough to absorb the gradient-square part; control drift and cutoff terms by the same inequality. After cancelling L, the boundary term is bounded by C\int_B\eta^2e^{2kZ}. Apply (7.39) to \varphi=\eta^2e^{2kZ}. Its \sqrt{\mathcal T} contribution is absorbed into a small part of the source. Its differentiated exponential contributes C(n)k|\nabla_\Gamma Z|, which costs a small part of k|\nabla_\Gamma Z|^2 and C(n)k by Young’s inequality. Thus

Applying (7.45) to \omega=\eta e^{kZ} and using (7.46) gives the iteration from exponent 2k to 6k. For nested patches with separation s, the multiplier in this step is at most

The product of these multipliers converges when k is multiplied by three and the patch gaps decrease geometrically. Hence on two nested patches U'\Subset U (relative to the inner boundary),

with C(n,s) bounded by a fixed inverse power of s. Interpolation with the already bounded L^2 norm gives

Young’s inequality, with any desired small coefficient for \sup_Ue^Z, and a second iteration on geometrically nested patches absorb the larger supremum. The terminal term vanishes because each individual smooth solution has a finite supremum; the resulting bound is uniform over solutions. This proves Z\le C(n) on the compact sets containing the drift support and all inner faces.

Outside the drift support, a high interior maximum contradicts (7.43) by the divergence equation. Comparison, with zero outer data and the compact bounds just proved, completes the first-stage ceiling. In the second and third stages the drift is zero everywhere, so this maximum argument applies directly. At a boundary maximum with Z sufficiently large, the upper normal-slope bound of Proposition 7.5 bounds D on the face, hence forces t>1. Then j(t)=1, so the prescribed conormal flux is zero. The Hopf boundary lemma excludes such a high maximum. The same argument applies in the fourth stage, where f=0 and the source and flux are scaled together. At scale zero the mixed homogeneous problem has only the zero solution.

The required choices of n and R_{\min}(n) are compatible with Proposition 7.5. More explicitly, failure of a uniform positive height gap for all n\ge N and R\ge R_{\min}(n), for every choice of N and finite R_{\min}, would give, for each j, an n_j\ge j admitting arbitrarily large failing radii at gap 1/j. Choose such an R_j\ge j. This is precisely the contrary sequence excluded in Lemma 7.4. A finite number of exceptional fixed n can be discarded by increasing N; infinitely many with unbounded failure tails would give the same contrary sequence. After fixing the uniform gap, eventual validity at each remaining n defines a finite R_{\min}(n), which we may enlarge to be at least n. No uniform estimate in n is asserted for the subsequent ceiling constant.□

Proof. We organize the proof so that neither a derivative nor a continuity modulus of \chi enters.

Reflection and a Hessian estimate near exponent two. At a constant Dirichlet face subtract the boundary value and use boundary normal coordinates. Reflect z oddly and the metric isometrically. Tangential first derivatives vanish on the face and the normal first derivatives of the odd extension agree. Thus the extended z belongs locally to W^{2,p} for every finite p for each individual smooth solution; its second derivative has no surface atom. The reflected metric is Lipschitz and smooth on each closed side, with uniform bounds there. Equation (7.47) holds almost everywhere on both sides, with the reflected axis and the odd reflected source. A linear coordinate change at the center makes the metric Euclidean there. On sufficiently small rescaled balls the coordinate principal matrix, denoted by A, obeys

Indeed its defect at a point in an orthonormal frame is precisely 1-\chi, and the coordinate metric error can be made smaller than \chi_0/2. This statement also holds in a reflected patch.

The strict-defect Hessian absorption below is a Cordes-type estimate; for the linear framework see [21, Section 2, Lemma 1, and Section 2.1, Theorems 2–3]. The reflected near-two estimate and the subsequent nonlinear gradient and Morrey bounds are developed here.

For a compactly supported function on \R^3, the operator D_x^2\Delta^{-1} from a scalar source to the full Hessian array has L^2 norm one: its Fourier multiplier has squared Frobenius norm \sum_{i,j}\xi_i^2\xi_j^2/|\xi|^4=1. Its L^p norm is bounded for 1<p<\infty, and interpolation makes that norm arbitrarily close to one when p is sufficiently close to two. Absorbing (I-A):D_x^2v using (7.49) therefore gives, for some 2<p_0<3,

Here and below a fixed smaller pair of concentric balls may replace the displayed pair. To localize, apply the compact-support estimate to a cutoff times v and absorb first derivative terms by \|D_xv\|_p\leq\eta\|D_x^2v\|_p+C_\eta\|v\|_p. The rescaled Christoffel terms are handled by the same inequality. Using nested cutoffs gives an arbitrarily small multiple of the larger Hessian norm, with a polynomial cost in the inverse gap; iteration on nested balls absorbs that term. The proof also gives the exponent-two version of (7.50). It applies to the piecewise smooth reflected functions just described.

A flux identity and a drop or affine alternative. Normalize the gradient bound to one on a rescaled unit ball and make the source and geometric errors at most \delta. Geometric errors here include the metric deviation from its constant value, its piecewise first derivatives, the smooth-side curvature, and the scaled second fundamental form of a reflection face. Put P=\nabla z, H_z=\Hess z, and s=1-|P|^2\geq0. The equation gives \Delta z=G_0+(1-\chi)H_z(e,e) and therefore the exact identity

This identity remains valid when P=0. Since 1-\chi\leq1-\chi_0<1, Young’s inequality, with a fixed parameter depending on \chi_0, gives

For example, choose \eta>0 with (1+\eta)(1-\chi_0)^2<1 and use (G_0+(1-\chi)H_z(e,e))^2 \leq(1+\eta)(1-\chi_0)^2|H_z|^2+C_\eta G_0^2. The divergence of the right side of (7.51) is

Consequently, on the smooth sides,

The differentiation is of the right side of the exact identity; it does not require differentiating \chi or G_0 separately.

There is also a controlled distributional version across a reflected face. There P=(\partial_\nu z)\nu, and the normal component of the right side of (7.51) is

The tangential Hessian of a function constant on the face is, up to the normal-sign convention, its normal derivative times the second fundamental form. Thus the flux jump has magnitude at most C\delta, irrespective of the second normal derivative of z. Incorporate the volume density in coordinate divergence. A bounded plane density q(x') is the divergence of the bounded field q(x')\mathbf1_{\{x_3>0\}}\partial_{x_3} locally. Hence the plane error in (7.53) is another divergence-field error of size C\delta. All subsequent weak inequalities have bounded uniformly elliptic principal coefficients and scalar and vector errors tending to zero with \delta.

Consider a sequence with \delta\to0 and \inf_{B_{1/2}}s\to0. Dropping the favorable Hessian term, the inhomogeneous weak Harnack inequality for nonnegative divergence supersolutions gives s\to0 in measure on interior smaller balls [7]. Fixed radii in this argument can be chosen so as to include any prescribed ball compactly contained in B_{1/2}. We also have D_xs\to0 in L^1 locally there. To see the latter claim, test the supersolution inequality by \eta^2(b-s)_+, where b>0 and \eta is a compactly supported cutoff. Ellipticity and Young’s inequality give

The gradient bound and (7.50) already give a uniform local L^2 bound for D_xs. On \{s\geq b\}, Cauchy’s inequality and convergence in measure therefore give convergence of its L^1 integral to zero. On \{s<b\} use (7.54), then let b\downarrow0. Testing the full inequality (7.53) by a fixed nonnegative cutoff now yields H_z\to0 locally in L^2. Since the scaled Christoffel symbols tend to zero, D_x^2z\to0 there as well. After subtracting constants, Poincaré’s inequality makes the gradients converge in L^2 to a constant vector. Its length is one because s\to0 in measure. The gradient bounds give local uniform convergence of a subsequence to the corresponding affine function.

It follows that for every sufficiently small prescribed b_*>0 there are fixed numbers 0<r_*<k_*<1 and an error threshold such that one of the following holds:

Indeed, fix r_*<1/4. Failure for every k_*<1 and every sufficiently small error threshold would give a sequence with gradients approaching length one somewhere in B_{r_*} and no such affine approximation. The preceding compactness conclusion contradicts this. Increasing k_* if necessary ensures r_*<k_*.

Improving a nonzero affine approximation. Rescale the affine branch to a unit ball. Suppose

the gradient has a fixed bound, and the forcing and scaled geometric errors are at most b\delta. Set Y=(z-L_1)/b. The Christoffel contribution of the affine L_1 is included in these errors, so |A_\chi:\Hess Y|\leq C\delta. Equation (7.50) gives a uniform interior W^{2,p_0} bound for Y. Let e_0=D_xL_1/|D_xL_1| be the fixed Euclidean axis. On an interior ball the difference between the actual principal matrix and I-(1-\chi)e_0\otimes e_0 tends to zero in each finite L^q norm as b\to0, uniformly in the displayed slope range. Indeed D_xz=D_xL_1+bD_xY, the last term tends to zero in measure, and the coefficient difference is bounded. The small metric error has the same property. The set where D_xz=0 causes no problem: it is contained in the set where b|D_xY|\geq1/4. Choose q by 1/q=1/2-1/p_0. Hölder’s inequality then gives

The coefficient \chi remains measurable and spatially variable. The smallness here is uniform for all b below one fixed threshold. Indeed Sobolev embedding bounds D_xY in L^{p^*}, where p^*=3p_0/(3-p_0). Splitting according to b|D_xY|\leq1 gives

where A_0=I-(1-\chi)e_0\otimes e_0 and B_\chi is the actual coordinate principal matrix. This error need only be below the fixed tolerance for the normalized correction Y, not below b\delta. To track all coordinate terms, write A_0=I-(1-\chi)e_0\otimes e_0 and let B_\chi be the actual coordinate principal matrix. With q_L=D_xL_1, the equation used here is

The notation B_\chi:\Gamma q means B_\chi^{ij}\Gamma_{ij}^{k}q_k. The last term is bounded in L^2 by Cb\delta, and the other source and connection terms by C\delta.

For completeness, the fixed-axis inhomogeneous equation on B_{3/4} with zero Dirichlet data has a unique W^{2,2}\cap W^{1,2}_0 solution. Write it as \Delta v=f+(1-\chi)\partial_{e_0e_0}v and iterate the Dirichlet Laplacian inverse in the norm \|\Delta v\|_2. The convex-ball Hessian inequality

makes the contraction factor at most 1-\chi_0. Subtract this solution for the source in (7.56). It has small W^{2,2} norm, hence small uniform norm in dimension three. The remainder Y_0 satisfies

and has a uniform W^{2,2} norm.

This fixed-axis equation has an interior C^{1,\alpha_1} estimate even for measurable \chi. To prove it, put v_1=\partial_{e_0}Y_0. Distributional differentiation in the distinguished direction gives

Here v_1\in W^{1,2} and \chi\partial_{e_0}v_1\in L^2; the formula makes no assertion that \partial_{e_0}\chi exists as a function. De Giorgi’s estimate makes v_1 Hölder continuous, with some 0<\alpha_1<1. Caccioppoli applied after subtracting its value at the center gives, on smaller balls,

Since \Delta Y_0=(1-\chi)\partial_{e_0}v_1, its Poisson source q_0 satisfies

Localize this source in a fixed smaller ball and take its Newton potential. The gradient kernel and its derivative have sizes C|x-y|^{-2} and C|x-y|^{-3}. For two points a distance h apart, the near and far annuli therefore bound the gradient difference by

where the sums are over dyadic radii within the fixed ball. The difference between Y_0 and this potential is harmonic locally and has uniform interior bounds. This proves the asserted estimate for every component of D_xY_0.

Choose 0<\alpha_2<\alpha_1, then a sufficiently small fixed \lambda_0\in(0,1), and finally \delta and b sufficiently small. The affine approximation of Y_0 at the center, together with the small uniform correction, produces an affine L_2 with

More explicitly choose \lambda_0 so that the C^{1,\alpha_1} remainder is at most \tfrac12\lambda_0^{1+\alpha_2}, then make the correction smaller than the same quantity. This also explains the order of the choices.

Iteration and the Hessian mass bound. Choose b_* small enough for (7.59) and so that

Then fix the constants in (7.55). If L_0+Q_0=0 the conclusion is immediate. Otherwise divide the dependent variable by L_0+Q_0 and start on a sufficiently small uniform spatial scale. At each occurrence of the gradient-drop branch, divide the spatial scale by r_*^{-1} and the gradient normalization by k_*^{-1}; equivalently replace z(y) by z(r_*y)/(r_*k_*), up to an additive constant. The normalized source is multiplied by r_*/k_*<1. Geometric errors also decrease. If this branch persists, the constant approximants give the required approximation with every \alpha\leq\log k_*/\log r_*.

If the affine branch occurs, rescale its ball and iterate (7.59) with errors b_j=b_*\lambda_0^{j\alpha_2}. At each step divide the dependent variable by the spatial ratio, preserving the gradient bound, and retain the full affine slope. The total slope drift is less than 1/4, so all slopes remain in the range required for the comparison. The forcing and geometric errors shrink as \lambda_0^j, whereas the required tolerance is b_*\lambda_0^{j\alpha_2}\delta. Because \alpha_2<1, an initial scale that enforces this relative smallness at entry enforces it at every later step. The same initial choice works after any number of drops, since r_*/k_*<1.

Thus at each center and all small radii there is an affine approximant with error at most C(L_0+Q_0)r^{1+\alpha}, where

The ratios of successive scales are fixed and the slopes are bounded. Comparing approximants at successive scales shows convergence of their slopes; comparing approximants on overlapping balls bounds the difference of limiting slopes by C(L_0+Q_0)|x-y|^\alpha. For a smooth z each limiting slope is D_xz at the center. This gives the first estimate in (7.48). All previous arguments apply to the reflected Lipschitz metric, since its only additional flux error was already included.

Finally subtract the affine approximant on B_{2r} and apply the exponent-two estimate (7.50) at scale r. Its value error contributes C(L_0+Q_0)^2r^{1+2\alpha} to the Hessian-square integral. The bounded forcing and the Christoffel term acting on the bounded affine slope contribute at most C(L_0+Q_0)^2r^3. Since \alpha<1, this is bounded by the same claimed power for small r. Adding back the Christoffel term proves both versions of the Hessian estimate.□

Proof. Use uniform patches on the compact core and on the end, including boundary normal patches. Boundary components carrying different conditions are separated; the patches are chosen to meet at most one face. At every stage the trace equation is

On the bounded variable range, l/\sqrt D is bounded below and \chi has a positive lower bound at fixed parameters. The right side is bounded, and the face values of f are componentwise constant. Lemma 7.8 consequently gives a uniform Hölder bound for D_xf and

The second equation as an equation with a controlled measure. Write its coordinate divergence form as

Here a_0 includes the volume density and the principal tensor 4uA_\chi; it is bounded, symmetric and uniformly elliptic. The bounded flux summand b_0 includes the prescribed homotopy drift. The expression for t is smooth in (x,Z,D_xf) on the bounded range. Its first derivatives are bounded by C(1+|D_xZ|+|D_x^2f|), and the tensor expression for \cT is smooth and quadratic in these derivatives. These facts prove the displayed growth bound, including where D_xf=0.

For a flux face, flatten the boundary and reflect Z evenly. For the separate outer Dirichlet face, reflect Z oddly after subtracting its constant value. If S_0 is Euclidean reflection in the flattened plane, reflect a_0 as S_0a_0S_0 in both cases. Reflect b_0 as S_0b_0 in the even case and as -S_0b_0 in the odd case. For the even extension the bounded prescribed conormal flux gives a bounded plane density in the divergence. For the odd extension the normal component of the total flux agrees on the two sides, so there is no plane source. The smooth solution on each closed side justifies these distributional formulas directly. By (7.61), the extended equation has a signed measure right side obeying

where \mu_0 is a positive measure. The upper restriction \beta\leq1 includes the O(r^2) mass of a bounded plane density.

An exponential transformation and oscillation decay. Let \mathcal L_0=-\operatorname{div}_x(a_0D_x) and V_\pm=e^{\pm kZ}. Since Z is bounded, these functions and their reciprocals have fixed bounds for any fixed k. The product rule in (7.62) gives

Indeed the negative term containing k^2a_0 absorbs the Ck|D_xZ|^2 term from (7.63) when k is fixed sufficiently large; Young’s inequality absorbs the bounded-drift cross term. Products with the plane measure are legitimate because Z is continuous across the reflecting plane.

On a ball B_h in the extended patch solve the zero Dirichlet problem

Its solution satisfies

For the bounded divergence field, the scaled vector-source estimate with any exponent greater than three gives the term Ch. For the positive measure, the Dirichlet Green function of a bounded measurable uniformly elliptic divergence operator in dimension three has upper bound C|x-y|^{-1} [12]. If necessary extend coefficients to a doubled ball before using that bound. The growth in (7.63) yields

The volume source has the smaller bound Ch^2. Existence in W^{1,2}_0 may be seen by smoothing the restricted positive measure with its uniform ball-growth bound, using these uniform sup estimates, and passing to a limit. Testing the smooth problems by their solutions bounds the energies, since the measure term is at most the sup norm times its total mass and the vector term is controlled by Cauchy’s inequality. This also justifies the correction for a measure source.

Equation (7.64) now shows that

are nonnegative weak supersolutions of \mathcal L_0. Write M_h=\sup_{B_h}Z and m_h=\inf_{B_h}Z. On the bounded range of Z, the positive exponential deficit is uniformly comparable to M_h-Z, and the negative exponential deficit to Z-m_h. At each point their ordinary counterparts add to M_h-m_h. Therefore one of the two exponential deficits is at least a fixed multiple of M_h-m_h on at least half of B_{h/2}. Weak Harnack applied to that S_\pm, and then (7.65), bounds the corresponding deficit below throughout B_{h/4} by c(M_h-m_h)-C\varepsilon_h. In the positive case this lowers the upper value of Z; in the negative case it raises the lower value. Consequently

Iteration gives a uniform C^{0,\theta} bound for some \theta>0. The same proof in the reflected charts gives the estimate to both kinds of boundary face.

Absorption of the quadratic gradient term and bootstrap. Now Z and D_xf are Hölder continuous, and all coefficients of (7.60) are smooth functions of those variables and x. Interior and Dirichlet Schauder estimates give a uniform C^{2,\alpha'} bound for f, for some \alpha'>0. Expand (7.62) on the original, unreflected patches. The principal coefficients are Hölder continuous, and the right side is bounded in absolute value by C(1+|D_xZ|^2), since D_x^2f is now bounded. On an inner face df is normal; hence A_\chi preserves the normal direction, and its positive normal eigenvalue converts the prescribed conormal flux into

where \mathcal G is smooth on the bounded range. The outer condition is Dirichlet. There is no junction of these conditions within a patch.

Fix an exponent p>3. The local linear W^{2,p} estimate for these Hölder principal coefficients follows by freezing coefficients and absorption, using the usual Dirichlet or normal-derivative estimate at a smooth face [2, 7]. On nested uniform patches U\Subset U' it gives

For the boundary term, extend the smooth composition \mathcal G into a collar. Its W^{1,p} norm is bounded by C(1+\|Z\|_{W^{1,p}}) because f is already C^{2,\alpha'}. The trace theorem then bounds its required W^{1-1/p,p} norm and explains the last term in (7.67).

The existing positive Hölder modulus makes the quadratic term absorbable. On a ball or boundary half-ball of radius h_0, subtract a constant from Z. First derivative interpolation, including the scaled lower-order term [16], gives

No homogeneous normal boundary condition is needed for this interpolation. In a flattened half-patch, choose c\in[\inf Z,\sup Z] and extend u=Z-c by Eu(x',-t)=3u(x',t)-2u(x',2t) on a smaller collar and cut off on a larger patch. Its value and first normal derivative match across the face, its sup norm is at most 5\operatorname{osc}Z, and its W^{2,p} norm has the usual bounded extension estimate after scaling. Whole-space interpolation yields Equation (7.68), including its lower-order scaled term. The fixed smooth charts change only the constants. The oscillation on each such patch is at most Ch_0^\theta. Cover U' by these smaller patches with bounded overlap, using a slightly enlarged patch for each interpolation estimate. Sum their estimates in \ell^p. This bounds the quadratic term by Ch_0^\theta\|D_x^2Z\|_p+C(h_0) on a slight enlargement. Ordinary interpolation treats the W^{1,p} term in the same fashion. Take the supremum of the resulting W^{2,p} norms over patches of one fixed size throughout the truncation. Each enlargement is covered by a bounded number of other patches, independently of the truncation radius. This supremum is finite for each smooth solution on its finite truncation. Choose h_0 once so that the coefficient of the supremum on the right is less than one half and absorb it. We obtain a uniform W^{2,p} bound for Z and hence a uniform C^{1,\theta'} bound for some \theta'>0.

Schauder estimates for the trace equation and then the expanded second equation, with their respective boundary conditions, now increase regularity successively. At each step the source of the second equation uses at most second derivatives of f, which have already been controlled by the preceding trace estimate. Iteration reaches every required finite order, with constants depending on that order and the fixed parameters. The construction uses smooth three-dimensional patches also at the rotational axis, so no singular orbit-coordinate estimate is required.□

Proof. We first work with smooth input Z and obtain estimates depending only on its C^{1,b_{\rm reg}} norm. Approximation will then give the asserted solve for general inputs in that space. Write the normalized trace equation as

Here F includes the modification appropriate to the current stage. For an auxiliary parameter s\in[0,1], interpolate between this equation and \Delta f=\tau_0f:

Keep the current boundary values throughout this interpolation. At zero gradient A_\chi=I and \tr_{A_\chi}K=\tr K. The strict monotonicity of F in h therefore gives a uniform height bound by the maximum principle, for every s and every bounded input set.

We next establish a global gradient bound for this scalar interpolation. If Z is bounded and \sigma\to\infty, then t\to-\infty. In that range p=n and l=e^{nt}, and the defining relation between t and Z is exactly

It follows, uniformly for bounded Z, that

Because n>4, the exponent 8/(n+4) is less than one. On the entire bounded-height range we consequently have

The same inequalities hold for the longitudinal eigenvalue \chi^{(s)}=1-s+s\chi and the right side G^{(s)}, with uniform constants. The two transverse eigenvalues remain exactly one.

Let r_0 be distance from a boundary component in a fixed smooth collar, and set

Use f_{\mathrm{face}}+\psi as an upper barrier and f_{\mathrm{face}}-\psi as a lower barrier. At a possible contact their gradient has length q'=\psi', and the longitudinal second derivative has magnitude k(q')^2. By (7.72), its signed contribution dominates all forcing and collar-curvature terms:

Choose k large relative to the structural constants, then a collar width \delta' small enough that 1/(2k\delta') exceeds the required slope threshold. Finally choose B_0\delta'\ge1 so large that \psi(\delta') exceeds the height oscillation. Throughout the collar \psi'\ge1/(2k\delta'), so the differential inequalities have strict sign; height monotonicity rules out a positive comparison maximum. This proves

The assigned values are constant on each face, and \delta' can be chosen below the separation of distinct boundary components.

For the interior estimate maximize

over pairs at distance at most \delta'. The distance may be computed in a fixed smooth extension of the metric; choose \delta' below its uniform injectivity radius. Pairs with one point on the boundary are controlled by (7.74), since the distance to that boundary component is no greater than the pair distance. Pairs at distance \delta' are controlled by \psi(\delta')\ge\operatorname{osc}f. A positive maximum would therefore occur at an interior pair with r_0=\dist(x,y)>0. The two gradients there are parallel transports of the common vector of length q'=\psi'(r_0). Simultaneous variations in parallel transverse directions, using parallel trial fields in the second variation of distance, give

Separate longitudinal variations give

Subtracting (7.70) at the two points yields

Our choices of k and the lower threshold for q' make this impossible. Notice that the two longitudinal eigenvalues need not coincide or have a known modulus of continuity. Thus |f(x)-f(y)|\le\psi(\dist(x,y)) for nearby pairs, proving a global Lipschitz bound on each bounded input set.

The interpolation is now uniformly elliptic. Lemma 7.8 applies to its axial matrix and bounded forcing, first giving a positive Hölder modulus for \nabla f. Schauder estimates give C^{2,\theta} and then, by differentiating after this first gain, C^{2,b_{\rm reg}} and C^{3,b_{\rm reg}} bounds. More explicitly, the first differentiated estimate gives C^{3,\min(\theta,b_{\rm reg})}; hence \nabla f is Lipschitz, allowing the original coefficients to be estimated in C^{b_{\rm reg}}, followed by the stated full exponent. All these bounds are uniform in s.

For fixed input Z, the principal matrix is independent of the height f, while

The linearized Dirichlet operator thus has strictly negative zeroth-order coefficient when written with everything on the left. The maximum principle and linear elliptic Dirichlet theory make it invertible [7]. Starting at \Delta f=\tau_0f, openness follows from the implicit function theorem, and the bounds above give closedness. This proves existence for s=1. At a positive maximum of the difference of two solutions, their gradients and principal matrices agree, while strict height monotonicity contradicts the Hessian inequality; this proves uniqueness.

For a general C^{1,b_{\rm reg}} input, choose smooth invariant inputs Z_j with the same zero outer data, uniformly bounded in C^{1,b_{\rm reg}}, and converging in C^{1,b'} for every b'<b_{\rm reg}. Such an approximation follows by extension and smoothing in boundary charts, correction of the outer trace, and averaging under the circle action. Convergence in the full C^{1,b_{\rm reg}} norm is neither asserted nor needed. The smooth scalar solutions have the uniform C^{3,b_{\rm reg}} bounds just proved. Compact embedding gives a subsequence converging in C^{2,b_{\rm reg}} to a solution for Z; the third-derivative Hölder seminorm bound passes to the limit by lower-exponent compactness. Uniqueness identifies this limit and removes dependence on the approximation. Finally, for any convergent sequence of actual inputs in C^{1,b_{\rm reg}}, the same compactness and uniqueness argument gives continuous C^{2,b_{\rm reg}} dependence, also at the matching endpoints of the four homotopy stages. Symmetry follows from uniqueness.□

Proof. Use the Banach space X_R of invariant C^{1,b_{\rm reg}}(\overline\Omega_R) functions with outer value zero. For each input Z, solve the trace equation by Lemma 7.10. Compute t,u,A_\chi,\Xi and all boundary expressions from this input and its solved f. Freeze them, replacing only the differentiated Z in the principal flux by the output unknown \widetilde Z. In the first three stages, for example, the linear problem is

with the prescribed frozen inner flux and zero outer Dirichlet value. The fourth stage uses its stated simultaneous source and boundary scaling.

On bounded input sets the coefficients are uniformly elliptic. The nonempty outer Dirichlet face makes this mixed problem coercive; there is no flux compatibility condition. Its coefficients, drift, and inner flux data are at least C^{1,b_{\rm reg}}, and its source is at least C^{0,b_{\rm reg}}. Linear boundary regularity therefore bounds the output in C^{2,b_{\rm reg}} [2, 7]. Together with continuous dependence of the scalar solve, this defines a continuous compact map T_s:X_R\to X_R, continuous in the full homotopy parameter s. The derivative count is worth specifying: f\in C^{3,b_{\rm reg}} and Z\in C^{1,b_{\rm reg}} imply that t, uA_\chi and the drift are C^{1,b_{\rm reg}}, while the quadratic expression \mathcal T is C^{0,b_{\rm reg}}. Uniqueness of both solves preserves invariance. At a fixed point, Z\in C^{2,b_{\rm reg}} and f\in C^{3,b_{\rm reg}} initially. Then the frozen coefficients improve to C^{2,b_{\rm reg}} and the source to C^{1,b_{\rm reg}}, giving Z\in C^{3,b_{\rm reg}}; the trace solve next gives f\in C^{4,b_{\rm reg}}. Repeating this triangular bootstrap makes each fixed point smooth. Proposition 7.9 supplies the uniform classical bounds for these smooth fixed points.

Let t_s[Z] denote the value of t after the scalar solve and define the relatively open subset of [0,1]\times X_R

Choose M larger than the fixed-point bound supplied by Proposition 7.7 and Proposition 7.9. Lemma 7.3 excludes fixed points on the floor boundary; the classical estimates exclude them on the norm boundary. Fixed points in the closure of \mathcal O form a compact set, because the homotopy maps bounded sets to relatively compact sets. Leray–Schauder degree is therefore invariant on the moving sections \mathcal O_s [11]. One way to see the minor moving-domain point is to cover this compact fixed-point set by finitely many product neighborhoods lying in \mathcal O, subdivide the parameter interval accordingly, and apply homotopy invariance and excision on those fixed neighborhoods. No control of T_s on unbounded input sets is needed.

At the terminal endpoint the trace comparison gives f=0, and the linear second solve has identically zero output. The zero input is interior to \mathcal O_1, so its degree is one. The physical endpoint consequently has a fixed point with t>-\eps. For this fixed n, let R\to\infty through the admissible truncations. The estimates of Propositions 7.7 and 7.9 are uniform on every compact subset, including inner-boundary charts. A diagonal subsequence converges smoothly there, giving the asserted physical solution and the limiting bound t\ge-\eps.□

Proof. Beyond the support of K,C,D_0,D_1, the trace equation is

On the bounded variable range at fixed n its principal matrix is uniformly elliptic and its height coefficient has a positive lower bound. Slowly decaying exponentials Ce^{-c(r-r_*)} are positive supersolutions for the operator obtained by moving the height term to the left, if c>0 is small. Choose C to dominate the data on r=r_*. The zero outer data allow the same barriers on every truncation; applying them to both signs and then exhausting gives exponential decay of f. For the derivative assertion, freeze the coefficients in this homogeneous linear equation along the solution. Proposition 7.9 bounds their C^m norms on uniform unit patches, independently of R, for every fixed finite m. Local homogeneous estimates therefore give \|f\|_{C^{j,\alpha}(B_1)}\le C(n,j)\|f\|_{C^0(B_2)}; on outer half-patches the same estimate uses the zero Dirichlet condition. Each finite j requires only finitely many of the already bounded coefficient derivatives. Thus every required derivative of f decays exponentially; no decay of the derivatives of Z is needed at this step.

For clarity, put E=t-Z=-\tfrac18\log D. Then E, D-1, v, A_\chi-I, and their required derivatives are O_n(e^{-2cr}), while w,H^f,F=O_n(e^{-cr}) with derivatives. The tensor expression for \mathcal T consequently gives

Every remaining term contains two factors from H^f,F,w, or a graph error multiplying bounded derivatives of Z. In particular, u-L(Z) and p(t)-p(Z) have the same exponential bound, and

Here the divergence uses only the radius-uniform unit-patch bound on \nabla^2Z. The height term \tau_0a\sigma and the penalty term v\rho_1 are exponentially small as well. There are no persistent K terms, since K vanishes on this end; derivatives of the profiles have bounded fixed-n coefficients and multiply the stated graph errors. Consequently the end divergence equation takes the form

This also holds on the truncations, with uniform constants. More precisely, the right side is

which is O(r^{-4}) without any prior decay estimate for Z.

Put b(z)=p(z)+2-\delta_0(p(z)+1) and choose \Phi(0)=0, \Phi'(z)=\exp(\int_0^z b(q)\,\dd q). On the bounded solution range, \Phi' and its reciprocal are bounded. Then Y=\Phi(Z) satisfies \Delta Y=O(r^{-4}). For 0<\eta<1, the positive function r^{-1}-r^{-1-\eta} has Laplacian -\eta(1+\eta)r^{-3-\eta}+O(r^{-4}) in the AF metric. Multiples of this function dominate both the source and the bounded inner-end data, and are positive at the zero outer Dirichlet face. Comparison on the truncations proves Y=O(r^{-1}), hence Z=O(r^{-1}) after exhaustion.

The penalty therefore vanishes sufficiently far out. In the transformed equation the remaining nonexponential terms have smooth weight coefficients of order -4 and bounded smooth factors in Z. On rescaled annuli, a W^{2,p} estimate with p>3 first bounds the rescaled gradient and its Hölder modulus. Schauder estimates then bound the rescaled second derivatives. The exponentially small graph errors are controlled in these estimates by the unit-patch bounds of every required finite order. This yields Z,t=O_2(r^{-1}).

It follows that \divg V=u\Xi is integrable and V=4\nabla t+O(r^{-3}). The divergence theorem on \Omega_R gives

because the outward normal of \Omega_R on B is -\nu. This proves existence of the flux limit. The graph term has exponentially decaying ADM contribution. For the conformal factor e^{4t} the ADM change is -(2\pi)^{-1}\lim\int_{S_R}\partial_\nu t, which is precisely -\mathfrak F_n/(8\pi). This proves (7.75).□

Proof. At the physical endpoint, (7.40) bounds the negative inner flux by C(\tau_0/\ell)\int_B L. Apply (7.38) with a cutoff equal to one near B. Since \rho is bounded below on the collars,

there. As \Pi_n\tau_0/\ell\to0, this cost absorbs into, for example, half of the positive bulk integral of u(\rho+\delta_0\mathcal T). These are polynomial estimates; none of the arbitrary fixed-n regularity constants enters this absorption.

On the penalty support, -\eps\le t\le-\eps+1/n, so l and L are comparable to \ell. Directly from the definitions,

Therefore

Young’s inequality against the remaining positive \delta_0\tau_0ua\sigma term yields

The displayed remainder has vanishing total integral. Indeed \rho_0\asymp r^{-4} is integrable, \rho_0^2 is integrable, and \rho_1^2\asymp r^{-4\gamma} is integrable in dimension three because \gamma>3/4. Finally

so both small factors dominate every polynomial loss in n. Insert these bounds in (7.75) to obtain the proposition.□

Completion of the proof of Theorem 6.2. At the physical endpoint F=\tau_0f+C, whence w(F)=\tau_0a\sigma+w(C). Substituting (7.23) into (7.11) shows that the remainder after 8\pi(\mu+J_0(w)) is

The first three terms are nonnegative, and the height and offset bounds give

This is the scalar-curvature conclusion of the theorem. Combining the physical boundary condition with (7.16) gives H+4\partial_\nu t=-N_B, so

The metric inequality \hat g\ge e^{-4\eps}g guarantees completeness with the inner boundary included. Lemma 7.12 supplies the required end regularity and finite energy, and Proposition 7.13 supplies its upper bound. All choices respect axial invariance. The limits are taken in the order established above: fix the prepared data, selected domains, cutoff shapes and \eps; choose large n and form its profiles; solve and exhaust R at that fixed n; then let n\to\infty in the scalar energy estimate. No convergence of the deformation metrics as n\to\infty is needed.□

Proof. Extend the geometry smoothly and equivariantly across S into an auxiliary cap, and require each bounded inner set to contain that cap up to S. Count its full frontier, including coincidence with the obstacle. First impose an outer sphere obstacle. Perimeter compactness gives a minimizer. It has a C^{1,1} frontier, smooth and minimal off the inner obstacle, by smooth-obstacle perimeter regularity in dimension three [8, Regularity Theorem 1.3(iii)].

The outer sphere can be removed uniformly for metrics in the paths under consideration. Large coordinate spheres are strictly mean convex, and the competitor S gives a uniform area upper bound. If a free minimizing frontier had a point at radius r_j\to\infty, its portion in an asymptotic ball of radius \varepsilon r_j, for a fixed small \varepsilon>0, would have area at least c r_j^2 by the interior density estimate for perimeter minimizers. This contradicts the area upper bound. Thus every minimizer stays in one compact set. Filling bounded pockets decreases perimeter and preserves admissibility.

Among minimizers in this compact set choose one with largest filled volume. The perimeter inequality for union and intersection shows that both the union and the intersection of two minimizing inner sets are again minimizers. Rotating a maximum-volume minimizer and taking its union with the original one therefore leaves it unchanged modulo null sets. It is invariant. Smooth outward approximation of its regular graphs, or equivariant smoothing of an outward defining function, gives smooth invariant enclosing cuts with convergent area. Invariant areas are preserved by polarization. The original minimum-area hypothesis consequently gives a(g)\geq A, while S gives the reverse inequality.

For a sequence d_j\to0, choose invariant minimizers T_j for \widetilde g_{d_j}. Compact containment and perimeter compactness give a subsequence converging to a filled inner set T. Uniform equivalence of the metrics, lower semicontinuity, and comparison with a background minimizer show that T\in\cT_0 and that the perimeters converge. Strict convergence of the vector measures representing the perimeter normals then implies convergence of their area and tangent-plane measures. In particular, the integral defining A'_T is continuous under this convergence.

On the common compact set the expansion of area is uniform:

Comparing the perturbed minimum first with a minimizing member of \cT_0, and then with its own minimizer T_d, proves the upper and lower one-sided bounds in Equation (8.5). The same compactness argument at d=0 proves compactness of \cT_0.□

Proof. Let \mathcal B^+ be the one-point compactification of \mathcal B. For a variation v consider the continuous test triple

in

Continuity at the added point follows from Equation (8.9) and \delta<1; continuity of the last block follows from Lemma 8.2.

The linear space L(V) misses the open cone of triples strictly positive everywhere. Indeed a variation in this cone has a positive uniform margin in all three blocks. It has a smooth path realizing its derivative with

uniformly on the indicated domains. For completeness, realize a finite sum of compact directions, the prototype, and the strict direction by first making the linear compact and prototype changes, and then applying the displayed exponential scalings with parameter a_{\rm dir}d. On a compact set the Taylor remainders are uniform, also through the axis by the polar orders. On the end the new metric tails are O_2(r^{-1}); the second-order scalar-curvature remainder is O(d^2r^{-4}), as are or dominate all other constraint remainders. Since r^{-4}=O(w_0), this proves the stated estimate. The end curvature remains O(r^{-3-\delta}), the constraints remain integrable, and the ADM energy is differentiable. All other assumptions of the enlarged class in Corollary 6.3 persist. For small d>0 the path is therefore feasible. Equation (8.5) and the strictly positive objective block make the derivative of

strictly negative at its zero background value, contradicting Equation (8.6).

Separation of an open convex cone from a linear subspace gives a nonzero continuous positive functional on \mathcal X annihilating L(V). This separation requires no closed-range assertion for L. Riesz representation gives three nonnegative finite measures. The measure of the objective block has positive total mass: otherwise evaluation on the strict direction, whose first two blocks are strictly positive, forces both remaining measures to vanish. Divide by that positive mass to obtain \omega. If \overline\Lambda denotes the resulting first measure, set \dd\Lambda=w_0^{-1}\dd\overline\Lambda on \mathcal B. It is locally finite. Its atom at the added point gives z_0a_{\rm dir}, since the first test block has value 8a_{\rm dir} there. This proves Equation (8.10). Averaging the identity over the compact symmetry group gives the last assertion.□

Proof. Let \sigma_0 be an arbitrary smooth invariant normal speed supported strictly inside \Omega. Take a short local Einstein–Maxwell Cauchy development of the original data about its compact support. Local hyperbolic existence and uniqueness propagate the symmetry. A compact change of its spacelike slice, followed by polarization, gives exact constraint variations

To justify the second identity, differentiate the orthogonal horizontal projection defining h. Terms from differentiating the horizontal lifts pair to zero by orthogonality. Differentiating the fiber length gives the remaining fiber component, so the polarized variation is precisely 2\sigma_0k.

The potentials in these tests can be chosen with compact variation and unchanged axis constants. Maxwell’s equations give the closed spacetime orbit forms defining \psi,\chi, which are integrated normally in the short slab. With these potentials fixed, the orbit form defining z is closed as well: its exterior derivative vanishes on every spacelike meridional plane by the axial momentum equation on each tilted slice. Such planes test every two-form locally in the three-dimensional orbit space. Normal integration therefore extends z with its initial constants. These constructions preserve the polar orders. Only the interior neighborhood of the chosen support is used.

All end and boundary variations, and a_{\rm dir}, vanish in these tests. Equations (8.10) and (8.11) give

The common-plane property above makes the tangential trace a single measurable function on the union of free frontiers. The aggregate area measure is positive. Equation (8.12), first for invariant tests and then for all tests by averaging, says that this function vanishes almost everywhere for the aggregate measure. Integrating its absolute value and using Fubini shows that, for \omega-almost every T, the trace vanishes on almost every point of its free frontier. Continuity makes it vanish everywhere on each smooth free portion. Such portions are minimal and hence satisfy the MOTS equation.

The same equation holds almost everywhere across contact with S. In tangent graphs the difference between the frontier and the obstacle, and its first derivatives, vanish on the contact set. Since the graphs are C^{1,1}, differentiability and the density theorem imply that their second derivatives also agree at almost every contact point. Their outward orientations agree by obstacle inclusion. The frontier therefore solves the smooth uniformly elliptic MOTS graph equation almost everywhere across contact. In a graph chart, write this as

The C^{1,1} graph is a strong W^{2,\infty} solution; bounded slope makes the principal matrix uniformly elliptic. Evaluated on (x,f,\dd f), both the coefficients and the right side are Lipschitz. Interior Schauder regularity for this linear equation therefore gives f\in C^{2,\alpha} for every \alpha<1 on a smaller chart. Differentiation and elliptic bootstrap then make the graph smooth. Strong comparison implies local coincidence whenever it touches S. Connectedness then forces a touching component to be all of S; its total area is A, so it has no further component.

If the frontier is disjoint from S, it is a smooth interior enclosing MOTS. For an invariant surface the original and polarized expansions agree. The filled-frontier convention removes any spurious pocket component, so this is an enclosing surface forbidden by the exact outermostness hypothesis of Theorem 2.2. Thus almost every minimizing cut in the multiplier is S.□

Proof. Take the divergence of Equation (8.13) together with Equation (8.14). The scalar principal symbol is |\xi|^2; the vector symbol is

All couplings are lower order. This is an elliptic system with smooth coefficients on the rotational three-space. Distributional elliptic regularity therefore proves interior smoothness, including at the axis.

We record the additional metric equation to justify continuation and the end estimates. For a pure metric variation the second-order term is \Hess N-(\Delta N)g. The gravitational lower-order terms are smooth linear expressions bounded in norm by

This follows directly from

the algebraic variation of (\tr k)^2-|k|^2, and the \nabla k*\dot g+k*\nabla\dot g terms in the momentum variation. Terms due to varying the volume disappear because the background constraint residual is zero.

The matter terms, while initially written off the axis, have smooth tensor coefficients. Up to constant factors and smooth trace terms, they contract against

Free compact variations of s_i give

Thus the second tensors become -Ns_i^2e\otimes e, which are smooth by the polar orders. For i=1,2, the radial and azimuthal coefficients of the first tensor have the same leading value on the axis and smooth even parity; their difference is O(y^2). The radial-longitudinal coefficient is smooth odd. In Cartesian polar disks these are exactly the conditions for a smooth invariant tensor. For i=3, the higher vanishing orders directly give the same conclusion. This proves smoothness of the full metric equation through the axis; the matter terms are bounded by C|N|\sum_i(|D_i|^2+s_i^2).

Taking the trace of the metric equation and substituting back determines all components of \Hess N from the first jets of N,Y. Differentiating Equation (8.13) and commuting covariant derivatives likewise expresses \nabla^2Y in those first jets, with coefficients formed from k,\nabla k,\operatorname{Rm}. Hence

satisfies the asserted homogeneous first-order system. In a collar of S, integrate its normal ordinary differential equation from an interior collar slice. Smooth dependence on the tangential parameters gives a smooth one-sided extension to the boundary.

There might initially remain a separate boundary-supported part of the moment measures. Take \dot k=0 and a compact metric variation whose value and first jet vanish on S, with arbitrary second normal derivative of its tangential trace there. Neither the area nor the expansion derivative sees this second jet. After integration of the smooth interior adjoint, its boundary terms also vanish. In the residual at S, only the scalar-curvature variation tests the prescribed second jet. Equation (8.10) therefore forces the boundary part of N to vanish. The measure inequality |Y|\leq N then removes the boundary part of Y.□

Proof. Set F=(N,Y). The first-jet equations and the original end bounds give

Here the matter coefficients have order at least r^{-4}, so any q'\in(1/2,1) is allowed. Radial integration with Gronwall’s inequality, applied to |F|/r+|\partial F|, first gives F=O(r) and \partial F=O(1). The Hessian bound then gives a limiting gradient along each ray. Integrating the Hessian along paths of length O(r) on coordinate spheres shows that these limits agree. A nonzero linear term in N would contradict N\geq0 in opposite asymptotic directions. Once that term vanishes, |Y|\leq N eliminates every linear term of Y as well. Consequently

Reinsertion into Equation (8.18) gives \partial^2F=O(r^{-1-2q'}). The gradient limits are zero, so \partial F=O(r^{-2q'}). Since 2q'>1, radial integration gives finite value limits, and sphere paths show that there is a single such limit. One final insertion gives

Use the energy prototype in Equation (8.10). It vanishes near S, and a_{\rm dir}=0. Integrate the smooth adjoint over a large truncation. Scalar linearization gives the limiting constraint flux 16\pi N_\infty E'_g. Terms containing \nabla N, the nonconstant part of N, and momentum metric-variation boundary terms tend to zero by the preceding decay. The residual tail is integrable. Since E'_g=1/2\neq0, the identity gives N_\infty=1.

To eliminate the constant spatial translation, write g=\delta+H_0 and put a=Y_\infty for this paragraph only. Equation (3.9) and scalar linearization give

For an affine function V define the Euclidean flux covector

It satisfies \partial_iU(V;h')_i=VLh'. The shift equation gives \cL_Yg=O_1(r^{-3}). Subtracting \partial_aH_0 and the pure Euclidean gauge \cL_Y\delta leaves errors O_1(r^{-2-q'}). The affine flux of these errors tends to zero. The affine flux of \cL_Y\delta is identically zero after a smooth extension to the interior, because its scalar linearization vanishes. Thus the flux of U(V;\partial_aH_0) tends to zero.

On the other hand, differentiation of the definition of U gives

Euclidean integration by parts, with arbitrary smooth interior extensions, consequently yields the exact sphere identity

The first term tends to zero; the second tends to -16\pi m\,\partial_aV. Since m>0, testing all affine V gives a=0. This calculation uses just the hypotheses stated in the last part of the lemma.

Finally the prolonged shift equation, now with zero limits, gives

Starting with Y=O_2(r^{-q'}), integration from infinity first improves this to Y=O_2(r^{-1-q'}), and then to Y=O_2(r^{-2}).□

Proof. Choose a compact smooth invariant meridional vector field W with W|_S=\xi_0\nu. Extend the data smoothly across S and take their Lie derivatives along W. These are allowed compact first variations in Equation (8.10); they need not form a feasible path. Covariance gives C'=\cL_WC=0 on the exterior. This also holds at S, since the exact residual vanishes on the interior and all its one-sided jets vanish at the boundary. No constraint condition on the extension is required. Naturality identifies the fixed-surface expansion of the pulled-back data with the expansion of the surface moved by W in the original data. Consequently the first area variation is \int_S H\xi_0\,\dd A_g, and the expansion variation is L_S\xi_0, where L_S is the MOTS normal stability operator with principal part -\Delta_S. Equation (8.10) and \dd\lambda_S=2N\,\dd A_g therefore give

Averaging allows arbitrary scalar tests. Outer area minimization implies H\geq0 by outward first variation; the polarized and original mean curvatures of S agree. Thus L_S^*N\geq0 with N\geq0. To check the sign in the strong minimum principle, write locally L_S^*=-a^{ij}\partial_i\partial_j+b^i\partial_i+c_0(x). Choose a constant C_0\geq\max(0,\sup_Sc_0). Replacing c_0 by C_0 preserves the inequality because N\geq0. The function v=-N\leq0 then satisfies

Its zeroth-order coefficient is nonpositive, and a zero of N is a nonnegative interior maximum of v. The strong maximum principle forces local constancy. Connectedness of S proves the dichotomy.□

Proof. Let \xi=\partial_{t_0}. Stationarity of \Phi and \Ric_\gamma=2\Phi^*G imply \Ric_\gamma(\xi,\cdot)=0. The three-dimensional Killing twist

is a scalar. Differentiating with the parallel volume tensor, using \nabla_i\xi_j=-\nabla_j\xi_i and the Killing second-derivative identity, gives \dd\mathcal W=\pm2*_\gamma (\xi^\flat_\gamma\wedge\Ric_\gamma(\xi,\cdot))=0. Under \gamma=X^2H_3, the covector wedge acquires the factor X^4 and the Hodge star on three-forms the factor X^{-3}. Consequently \mathcal W is X times the corresponding twist for H_3. The latter is bounded near any ordinary interior axis point: H_3 is smooth and nondegenerate there because N>0. Since X vanishes there, the constant \mathcal W is zero. Substituting \xi^\flat_{H_3}=-\mu_*\dd t_0+\alpha_Y into Frobenius gives

This identity extends smoothly to the doubled meridian.

Where \mu_*>0, Equation (9.3) makes -\alpha_Y/\mu_* closed and gives the local static form (9.2). In these coordinates \gamma=-\rho^2\dd t^2+X^2h_+. The tt Ricci equation is \rho\Delta_{X^2h_+}\rho=0; conformal invariance of the two-dimensional scalar Laplacian then gives \Delta_{h_+}\rho=0.

We next exclude an interior null set. First there is a positive collar of the inner boundary. The boundary alternatives from Proposition 8.1 are exhaustive. If N>0 on S, then Y=N\nu, \operatorname{sym}\nabla Y=-Nk, and (8.3) give

If N=0 on S, then Y=0 there. Its tangential derivatives vanish, and the symmetric-gradient equation makes its remaining first derivatives vanish as well. At inward boundary distance s,

These collars can be chosen equivariantly and are uniform through the poles by compactness. The end also has \mu_*>0, since N\to1 and Y\to0.

Suppose that the compact interior set \mathcal Z=\{\mu_*=0\} on the double is nonempty. Proposition 8.1 gives \mu_*\ge0 everywhere and N>0 in the interior, so Y\ne0 on \mathcal Z. Let Z be the h-orthogonal quarter-turn of Y, using the orientation of the double. Evaluation of (9.3) on (Y,Z) yields, on a neighborhood of \mathcal Z,

The coefficient is smooth. Uniqueness for this scalar ordinary differential equation shows that the flow of Z preserves \mathcal Z. A trajectory starting there exists for all time, because it stays in a compact set on which Z is smooth. Its omega-limit set has no equilibrium. Extend Z smoothly in the plane around this compact set and apply the planar Poincaré–Bendixson theorem: \mathcal Z contains a periodic orbit, a simple closed curve C.

Every component of \{\mu_*>0\} on the bounded side of C, intersected with either open regular half-meridian, would carry a positive harmonic function \rho=X\sqrt{\mu_*}. This function extends continuously to the compact closure and vanishes on its entire frontier: the frontier lies in an axis, the inner boundary, or \mathcal Z. A positive maximum would therefore occur at an interior point, where h_+ is smooth and elliptic. The local strong maximum principle gives a contradiction. No ellipticity at the frontier is needed in this argument. Intersections of C with a reflected axis cause no additional frontier: first restrict to either open half and then take a component; its relative frontier is contained in the listed sets, with C itself already contained in \mathcal Z.

If C surrounds the inner obstacle, the positive collar supplies such a component. Otherwise C bounds a disk disjoint from the obstacle. The same maximum argument, together with \mu_*\ge0, forces \mu_*=0 on that entire disk; a positive point on an axis would have positive off-axis neighbors. Hence Y, and therefore Z, is nonvanishing on the closed disk. Its restriction to C is a tangent field to a simple periodic orbit and has winding number one, whereas a nonvanishing field extending over a disk has winding number zero. This last contradiction proves \mu_*>0 throughout the interior. Simple connectivity now gives the global primitive f in (9.2).□

Proof. In the zero-lapse case, the preceding Taylor expansions give \mu_*=s^2(\kappa^2+O(s)). Taylor division on s\ge0 shows that \lambda=s(\kappa+O(s)), \alpha_Y/\mu_*, and \alpha_Y^2/\mu_* are smooth, the last vanishing to order two. Thus h_+ extends smoothly with the stated boundary value and normal derivative, and \dd f extends smoothly. Its primitive also extends, after fixing an additive constant. This argument only requires one-sided smoothness.

In the positive-lapse case, \mu_* itself is a boundary defining coordinate. At S, \dd\mu_*=2N\kappa\,\nu^\flat and \alpha_Y=N\nu^\flat, so in a smooth collar (\mu_*,z) we can write

where a_1 and the tangential one-form \eta_0 are smooth. Pull back by \mu_*=\lambda^2. Then

The pulled-back h has even tangential and normal coefficients and odd mixed coefficients under \lambda\mapsto-\lambda. The one-form in parentheses changes sign under this reflection. The metric is consequently invariant, with vanishing mixed coefficients and normal coefficient 4a_0^2=\kappa^{-2} at the boundary. Its tangential block is the positive boundary block of h, proving nonsingularity and the asserted derivative of \lambda. Moreover

The smooth remainder is closed, because \dd f is closed. Integration in the collar gives (9.5).

Choose all collars equivariantly under meridional axis reflection. The preceding divisions then preserve even and odd polar parity also at the poles. On an ordinary axis away from its endpoints the added term in h_+ has zero transverse radial component. Thus the usual no-cone condition is retained: |\dd X|_{h_+}=1 on such an axis.

We make the joint corner regularity explicit. Near a pole in the positive-lapse case choose equivariant coordinates (\mu_*,y), with y a signed transverse axis coordinate. The signed extension of X is

The coefficients h_{\mu_*y} and \eta_y, where \eta_0=\eta_y\dd y, are odd in y, while h_{yy}(\mu_*,0)=A_X(\mu_*,0)^2 by the original no-cone condition. In (\lambda,y), Equation (9.6) has components

They extend smoothly under both commuting reflections \lambda\mapsto-\lambda, y\mapsto-y. At the corner the metric is diagonal and positive, with normal entry \kappa^{-2}, and along y=0 its yy entry is A_X(\lambda^2,0)^2. Thus no-cone regularity persists jointly up to the pole. The zero-lapse case uses the original smooth boundary coordinate s and the same y-reflection, without requiring an s-reflection.□

Proof. Reflect the static meridian across the axes. Its conformal filling at infinity is a disk with a smooth boundary. Uniformize this disk, sending infinity to its center and the axis reflection to complex conjugation; inversion gives an exterior circular coordinate, and its logarithm gives the asserted strip. Smooth isothermal and boundary conformal regularity give nonsingular charts through the ordinary axes and up to the inner boundary, including their intersections. This uses the smooth boundary established in Lemma 9.2. In the positive-lapse case its two commuting corner reflections are carried to the coordinate reflections. Near \vartheta=0, their smooth even and odd parities give

with smooth coefficients; near the other pole use \pi-\vartheta instead. This follows by dividing each odd function by its coordinate and applying smooth even-function factorization in the two variables. It does not require analyticity. Thus \sigma/\lambda, \sin\vartheta/y, and X/\sin\vartheta are smooth positive ratios jointly at the corner. In the zero-lapse case the same polar conclusion follows from the smooth chart in (s,y), with the y-reflection.

For clarity, the end normalization in this construction retains an initially unknown positive constant. We use the reflected isothermal expansion (5.10), constructed in Section 5.3. The estimate Y=O_2(r^{-2}) gives h_+-h=O_2(r^{-4}) on the end. After inversion of an asymptotically Euclidean meridional chart, the reflected conformal structure extends across the origin with Hölder coefficients of every exponent q'<1. Its isothermal rectification has a nonzero first derivative there; rescaled annulus estimates give the differentiated remainder. Reflection supplies the same expansion in ratio at the axes. It follows that, for some c_\infty>0,

uniformly, the first quotient interpreted by its continuous polar extension. Since \lambda\to1, the same first limit holds for \rho. The harmonicity proved in Lemma 9.1 becomes flat harmonicity in the strip, and \rho vanishes continuously on each finite side.

Put b'=2c_\infty. For any \eps>0, at a sufficiently distant line \sigma=L the functions (b'\pm\eps)\sinh\sigma\sin\vartheta bound \rho. They have the same zero values on the finite sides. Maximum comparison on the finite rectangles, followed by L\to\infty and then \eps\downarrow0, gives

At the inner boundary, differentiation toward the domain gives \partial_{\nu_+}\rho=\kappa X, because \rho=X\lambda and |\dd\lambda|_{h_+}=\kappa. As \nu_+=P_+^{-1/2}\partial_\sigma, this says

The original and static metrics agree tangentially there. The area of the original invariant boundary is therefore

Since \kappa=b/R_0^2 and A=4\pi R_0^2, we obtain b'=b.□

Proof. Let \Phi_* denote the model map in (3.20), with the same ordered axis constants as \Phi, and set

Both maps satisfy the harmonic-map equation with weight w. The target is complete, simply connected, and nonpositively curved, as established in the axial reduction. Joint convexity of squared distance along its geodesics, applied to the two weighted harmonic-map equations, gives

in the open strip; derivatives and divergence in this display are Euclidean.

We verify the boundary controls needed to use this inequality. By Lemma 3.3, on every bounded radial range the two axial lengths are comparable to the same ordinary transverse radius. Their logarithmic difference is consequently bounded. If y is that radius, the electromagnetic potential differences from their common axis values are O(y^2), and the adjusted central potential differences are O(y^4). In particular, writing \Delta\psi=\psi-\psi_*, \Delta\chi=\chi-\chi_*, the central displacement

is O(y^4). To bound distance, first change the height u=\log X, then join the remaining three coordinates at fixed height. The target length of the latter path is controlled by

These quantities are bounded, indeed vanish at a regular axis. Lemma 9.2 preserves the ordinary transverse radius by a smooth positive ratio at the horizon–axis corners, so the same argument applies there. Away from the axes smoothness of both maps gives boundedness directly. Thus D is bounded on every compact radial range, including all finite boundary faces and corners.

At infinity the matched scale in (9.7) gives u-u_*\to0 uniformly. The end estimates (3.6) for the normalized potentials, including their polar ratios, give respectively

Here the common axis constants remove the otherwise singular polar terms. The preceding path estimate shows that D\to0 uniformly as \sigma\to\infty.

Fix \delta>0 and set v=(D-\delta)_+. It vanishes on a sufficiently distant tail and is bounded. Test (9.10) with \chi_{\mathrm{cut}}^2v, where \chi_{\mathrm{cut}} is a compactly supported cutoff in the open strip. The weak chain rule and Cauchy’s inequality give

There is no outer error if the outer transition of \chi_{\mathrm{cut}} is placed in the zero tail of v. At each finite side, with Euclidean distance x to that side, use a cutoff that rises from zero to one on \eps^2<x<\eps with derivative (x|\log\eps|)^{-1}. On a fixed radial range w\le Cx, and

Products of these cutoffs treat the corners; their gradient energies are bounded by the sum of the individual energies. Letting \eps\downarrow0 in (9.11) proves \nabla v=0 throughout the open strip. Its zero tail makes v=0. Finally \delta\downarrow0 gives \Phi=\Phi_*.

It remains to determine a metric coefficient; map equality alone does not yet do this. Put

where P=r^2+a^2\cos^2\vartheta is the model coefficient. The static quotient has spatial metric e^{2q}(\dd\sigma^2+\dd\vartheta^2). Its scalar constraint, or directly (3.12), gives -\Delta q=|\partial\Phi|_G^2; hence q-q_* is harmonic. It vanishes on \sigma=0 by (9.8). At an ordinary axis the no-cone condition says \sqrt{P_+}=|\partial_\vartheta X|. Since X=X_*, the two nonsingular conformal scales have the same axis values. Inside the strip their logarithmic difference is precisely

It extends boundedly through the corners, because the static base metrics are smooth and positive, and tends uniformly to zero at infinity: both square-root scales are asymptotic to (b/2)e^\sigma. Maximum comparison now gives q=q_*. Thus P_+=P, X=X_*, and \lambda=\rho/X=\lambda_*, proving the proposition. The oriented crossing order of the axes, future static time, and increasing original azimuth have been preserved throughout; the Hodge-star conventions in (3.11) therefore agree.□

Proof. Proposition 9.4 identifies the horizontal Lorentz metric, the map, and the map’s normal velocities on the graph t=f. In particular, the curvature identity in (3.11), including

identifies the pullback of \dd(-\omega_*\,\dd t) with \dd\mathcal A. Consequently \dd(\mathcal A+\omega_*\,\dd f)=0. The meridian is simply connected, so the real primitive \beta in (9.12) exists globally. On the graph the model’s fiber form becomes

Its horizontal metric is h_+-\lambda^2\dd f^2=h by (9.2). The induced metric is therefore exactly the original g.

We also identify the full second fundamental form. The identities (3.17) make this explicit. The horizontal quotient tensor, together with n(\log X), determines the horizontal block K|_h, by the tensor relation used in Proposition 8.1. The fiber-fiber unit entry is K(e,e)=n(\log X). Finally the spatial identity

determines the mixed block. All these quantities agree under the map, with the same future normal, so no component of K is discarded in undoing polarization.

Likewise, \dd\psi=X*_hE'_H and \dd\chi=X*_hB'_H recover both horizontal electromagnetic components. Their normal map velocities recover the remaining components through E'_e=s_2, B'_e=-s_1. Thus the rotated Maxwell fields agree. If Q>0, apply

if Q=0, use the identity rotation. The model then has the ordered original charges (Q_e,Q_b). The constant rotation preserves the stress tensor and E_f\times B_f, so it also preserves the total angular-momentum normalization. The model choice a=-J/M consequently has exactly the prescribed J with the positive convention for K.

Every interior meridional change above respects axis reflection. The differential defining the angular rotation has ordinary invariant polar parity, so its primitive gives a smooth rotation of each normal polar disk. This extends the map through the axes. Finally \dd f=-\alpha_Y/\mu_*=O(r^{-2}), and the global static meridional coordinates tend to the model’s spatial infinity.□

Proof. In the zero-lapse case the static meridional coordinates and f are already smooth up to S. In the positive-lapse case, the isometric reflection in Lemma 9.2, followed by conformal reflection across the inner circle, shows that \sigma is odd and \vartheta even as functions of (\lambda,z). Hence

extend as smooth even functions, and therefore as smooth one-sided functions of the original coordinate \mu_*=\lambda^2. At the poles, the joint smooth positive ratios in (9.9) give the original polar extension.

The apparent singularity of the angular variable is removed in co-rotating coordinates. Put

On the graph its angular adjustment has differential

The model expansion is \omega_*-\omega_H=\sigma^2\omega_2(\sigma^2,\vartheta), with smooth polar coefficients. This differential is therefore smooth also in the positive-lapse case, where \dd f=-\dd\mu_*/(2\kappa\mu_*)+\dd f_0. It is closed and has a smooth primitive up to the boundary, with the same axis parity as before.

Use local horizon coordinates

They satisfy the exact identities

To check that these are smooth nonsingular extension coordinates, the model has

where L and all the remaining model coefficients are smooth in \sigma^2 and the ordinary angular variables. The leading horizontal block is

The error \sigma^4L\dd t^2 is a smooth quadratic expression in V\dd U-U\dd V. The fiber form in co-rotating coordinates is \dd\phi_\sharp-(\omega_*-\omega_H)\dd t, which is smooth by (9.15). Thus the complete Lorentz metric extends nonsingularly, using ordinary angular patches at the poles. For the electric model potential, its coefficient on the horizon generator is constant. Subtract that constant times \dd t; the remaining time coefficient is O(\sigma^2), smooth in \sigma^2, so its field also extends smoothly. The same holds after the constant duality rotation.

On the graph, if N|_S=0, both U and V vanish smoothly at S. This is the bifurcation sphere. If N|_S>0, use (9.5) to write instead

Both are smooth in the original variables, with U=0, V>0 on S. This is the future horizon branch: in the exterior U<0<V, and the future stationary generator has components \kappa(-U\partial_U+V\partial_V). Figure 1 summarizes these two attachments.

The extended map preserves the smooth positive original metric. It is thus an immersion and is spacelike up to S; the future unit normal then extends smoothly as well. It is injective in the interior, since its static meridional coordinates form a global coordinate change and each fiber map is a rotation. On the boundary the map to the horizon cross-section coordinates is a diffeomorphism, so distinct boundary points remain distinct. No interior point maps to the horizon. Finally compactness on bounded radial sets and properness at the end make the injective map proper onto its image, hence an embedding. The induced K,E_f,B_f, already identified in the interior, agree at S by smooth extension.□

Proof. Write r_x=|x| in the prescribed slice chart, let T be the stationary time Killing field future at infinity, and decompose

Since the end approaches ambient spatial infinity, N_T>|Y_T|_g eventually and N_T^2-|Y_T|_g^2=-\bar g(T,T)\to1. We first obtain

No prior bound on the tilt of the embedding is needed. The Gauss equation bounds the tangential ambient curvature components in the slice frame by O(r_x^{-3}); Codazzi bounds the components with one normal index by the same order. The remaining normal-normal block follows by contraction with the ambient Ricci tensor, whose components in this frame are O(r_x^{-4}) by the given electromagnetic fields and the Einstein–Maxwell equation. Thus \overline{\operatorname{Rm}}=O(r_x^{-3}) in the entire slice frame.

The Killing first-jet equation \bar\nabla^2T=\overline{\operatorname{Rm}}*T, split into tangential and normal components, therefore has the end bounds used in Lemma 8.6. The splitting adds only K*\bar\nabla T and (\nabla K+K*K)*T; the skew symmetry of \bar\nabla T controls all its components by the tangential derivatives. In slice coordinates the Christoffel symbols and their derivatives are O(r_x^{-2}), O(r_x^{-3}). The radial first-jet integration and causality argument of that lemma consequently give constant limits with O_2(r_x^{-q'}) errors, for any fixed 1/2<q'<1. Here the normalization is determined by -\bar g(T,T)\to1, rather than by a variational prototype.

For the original unpolarized data, the constraints and prescribed falloff give

The pair (N_T,Y_T) satisfies \operatorname{sym}\nabla Y_T=-N_TK, with K=O_1(r_x^{-3}), bounded N_T, and the constant asymptotics just proved. Since m>0, the general assertion of Lemma 8.6 gives Y_{T,\infty}=0. The norm limit then gives N_{T,\infty}=1. Prolonging \operatorname{sym}\nabla Y_T=-N_TK gives

Twice integrating from the zero end limits, and reinserting the improved bounds, yields (9.18).

Project along T to stationary spatial coordinates \bar x of the ambient exterior, and restrict to \bar r=|\bar x| sufficiently large. This projection is a local diffeomorphism on the slice, because a timelike vector cannot be tangent to a spacelike hypersurface. It is proper over that spatial tail: the end goes to ambient spatial infinity, and a compact core has bounded projected radius. Explicitly, for every ambient radius R there is a slice radius R_x(R) such that

Thus the preimage of the ambient tail contains one whole connected intrinsic coordinate tail, and the preimage of any compact subset of the ambient tail is a closed subset of an intrinsic compact set. Its image is therefore open and closed in the connected tail, so it is surjective and a covering. The spatial tail is simply connected. Each component of its preimage covers the whole tail and hence reaches arbitrarily far out in the slice end: otherwise its closure would lie in an intrinsic compact set, whose projected radius is bounded. All such components must therefore intersect the one connected full intrinsic tail just specified. There is just one component and one sheet. This proves the single-graph assertion.

The stationary orbit metric on this three-dimensional end pulls back to

By (9.18), it differs from g by O_2(r_x^{-4}) and has the same mass. In the ambient stationary chart, the same metric differs from the canonical spatial metric by \bar g_{ti}\bar g_{tj}/(-\bar g_{tt})=O_2(\bar r^{-4}). The canonical mass is M.

Here the equality of masses in the two charts can be checked directly. Since both represent the same uniformly Euclidean metric, D\bar x and its inverse are bounded. Distances in the two ends show that r_x and \bar r are comparable. The transformation law for the Christoffel symbols then gives D^2\bar x=O(r_x^{-2}). Integrating along rays and sphere paths yields a constant orthogonal matrix O_* with

The ADM flux comparison separates a pure Euclidean symmetric gradient, whose linear flux vanishes, from the original perturbation. Quadratic errors tend to zero. The Euclidean scalar linearization is integrable (O(r_x^{-4})), so asymptotically round deformed spheres give the same limit as round spheres. Thus m=M.

In the \bar x-chart the graph satisfies

Its vanishing tilt leaves the leading Coulomb fluxes unchanged, giving Q_e^2+Q_b^2=\widetilde Q_e^2+\widetilde Q_b^2; possible orientation changes do not affect this identity. The graph tangents and the normalized future normal covector proportional to -\dd t+\dd F give, with the positive convention for K,

For every Euclidean rotation vector R,

Its round-sphere integral is zero because R preserves the sphere area form. The trace-reversed Euclidean term also vanishes, since R is tangential; all metric, normal, and area errors tend to zero at these decay orders. Hence the angular flux vector in this chart has the canonical norm |M\widetilde a|.

Under (9.20), rotation vectors agree, up to the constant orthogonal change, with errors O(\log r_x). Their flux errors are O(\log r_x/r_x), since K=O_1(r_x^{-3}). The corresponding spheres differ in radius by O(\log r_x), and their normals and relative area elements by O(\log r_x/r_x); these also give vanishing angular-flux errors. In the prescribed slice chart, axisymmetry forces the two transverse components of the angular flux vector to vanish. Its axial component is the prescribed total J, because the electromagnetic correction tends to zero at infinity. Thus its norm is |J|, proving J^2=M^2\widetilde a^2.□

Proof. This is a direct boundary argument, independent of the adjoint boundary dichotomy. In the smooth co-rotating coordinates (9.14), write the section as U=0,\ V=v(x)\ge0, where x ranges over the sphere of generators. Such a global graph exists because generator projection is a diffeomorphism. Smoothness of the model coefficients in -UV, with the weights imposed by the horizon Killing field \kappa(-U\partial_U+V\partial_V), gives on U=0

and

Here q is a positive sphere metric and all coefficients are smooth functions of x. The last identity follows from the stronger form \bar g_{VV}=U^2a_2(UV,x); merely knowing \bar g_{VV}=0 on the horizon would not suffice.

Set \ell=\partial_V, and choose the transverse future null normal k with \bar g(k,\ell)=-1. The section tangents are e_A^{\,S}=\partial_A+v_A\partial_V; orthogonality gives

The remaining component is fixed by nullness and is immaterial for the expansion because the horizon’s outgoing second fundamental form is zero. Direct contraction of -\bar g(k,\bar\nabla_{e_A^{\,S}}e_B^{\,S}) with q^{AB} gives the linear expression

For example the relevant lower Christoffel coefficients are

combining these with k_V=-1, k_A=v_A yields (9.23). The VV coefficients vanish on the horizon, so there is no quadratic gradient term.

Outer area minimization, applied to arbitrary nonnegative outward variations, implies H\ge0. Since \theta_+=H+\tr_SK=0, the other future expansion is \theta_-=\tr_SK-H=-2H\le0. The outward future normal n+\nu is a positive multiple of \ell, so n-\nu is a positive multiple of k. Consequently \theta_k[v]\le0. Equation (9.23) is a uniformly elliptic linear inequality for the nonnegative smooth function v. The strong minimum principle applies even if its zeroth-order coefficient has either sign: in the operator -\theta_k, replace that coefficient by its positive part, which preserves the supersolution inequality because v\ge0. Hence a zero of v forces v\equiv0 on the connected sphere. Otherwise v>0 everywhere, proving the two alternatives.□

Completion of Theorem 2.2. Corollary 6.3 proves the numerical inequality on the closed physical branch. On the strict branch, Proposition 8.1 supplies the stationary quotient and its boundary conditions. Lemmas 9.1–9.3 and Proposition 9.4 identify that quotient with the subextremal model determined by Lemma 3.7. Propositions 9.5 and 9.6 give the required smooth embedding of the original metric, tensor, and electromagnetic fields, with its future normal and its stated boundary image. The horizon area formula and Lemma 9.7 prove the converse, and the canonical examples verify sharpness within the stated class. No classification on the extremal equality branch is asserted.□