Conformal flow and the Riemannian Penrose inequality with minimizing frontiers

Proof. Lemma 2.1 supplies a minimizing enclosure of full perimeter a_g(B). By Lemma 2.3, its frontier is detached from B, locally perimeter minimizing, and outer minimizing; its closed end exterior is connected and complete. The restricted metric retains the scalar-curvature and end hypotheses, and is smooth through the frontier. The regularity and area assertions of the same lemmas therefore permit application of Theorem 1.1 to this exterior, giving the displayed bound.□

Proof. Confinement and existence. First minimize P_a among enclosures of K_0, either globally or with an outer constraint L\subset B_Q. The competitor K_0 bounds the minimizing perimeter by A_0. Choose R_0 containing K_0 and the compact core, and choose \gamma\ge1 such that \gamma^{-2}\delta\le g_0\le\gamma^2\delta in the coordinate end. Put

For almost every R>R_0, truncation by B_R is admissible and cancellation of the common interior perimeter gives

For the constrained problem this is used when R<Q; beyond Q the tail is empty. Set V(R)=|L\cap\{|x|>R\}|_\delta. Euclidean isoperimetry applied to the cut tail, including its spherical boundary, and the coarea formula imply

At a regular cutting radius R_1\in[R_0,R_0+1], a second use of Euclidean isoperimetry gives

Indeed, the original end perimeter is at most a_-^{-1}A_0 and the added cutting boundary is a portion of that sphere. While V>0,

Consequently the tail vanishes by the radius

We use the canonical closed representative L=\supp(\chi_L\,dV_{g_0}); zero tail volume therefore means actual containment in B_{R_*(c)}. No density estimate, connectedness assertion, or potential estimate was needed.

Minimize first in each bounded exhaustion domain B_Q. BV compactness and weighted lower semicontinuity give a constrained minimizer. The radius in (4.5) is independent of Q, so a sequence with Q\to\infty has a limit on one fixed support ball. Every fixed compact competitor is admissible eventually; the limit is thus a global minimizer. All global minimizers have the same support bound. Choose one of maximal background volume by compactness once more. Submodularity and minimality show that the union and intersection of any two minimizers are again minimizers. The maximal-volume one must therefore contain every other minimizer, first almost everywhere and then canonically. This is the unique outermost hull. The same argument constructs the minimizing enclosure of any compact finite-perimeter obstacle J for any continuous weight bounded above and below by positive constants: use P_a(J) as the competitor bound and choose R_0 to contain J. Equality of the weight to one on J is unnecessary for this individual existence assertion.

If the prescribed initial enclosure J was replaced by its outermost minimizing hull K_0, the replacement also preserves local minimality, not only area. For L\mathbin{\triangle}K_0\Subset U in an initial variation ball, apply the local minimality of J to L\cap J and the enclosure minimality of K_0 to L\cup J. Submodularity gives

Thus the same initial comparison radius applies to K_0.

Removal of the initial obstacle. Let K be any minimizing enclosure. For a local competitor L\mathbin{\triangle}K\Subset B_r(x) with r<r_0, initial local minimality and a=1 on K_0 give

Global enclosure minimality applies to L\cup K_0. Combining it with the local submodularity inequality yields

Thus K is a local background quasiminimizer with factor Q=c^{-1}. The comparison includes balls crossing the initial obstacle and uses its local minimality against removals. Cancelling unchanged perimeter also localizes the comparison to smaller variation domains.

Two-phase density before boundary regularity. Choose uniform small background coordinate balls away from a smaller buried collar. Smoothness on the compact core and the differentiated asymptotic-flatness bounds on the end give uniform local absolute and relative isoperimetric constants, sphere-area bounds, and scaled C^{1,\alpha_0} bounds for the divergence-form coefficients of \Delta_{g_0}, for some \alpha_0\in(0,1). No uniform bounds for derivatives of all orders are required. Shrink their radius below r_0. Balls missing \partial K_0 have zero initial local perimeter and cause no exception. We may use geodesic balls in the following slicing argument by localization of the exact weighted comparison.

Let K be one of the hulls and x\in\supp|D\chi_K|. Put

Both numbers are positive for every r>0, since otherwise the perimeter measure would vanish near x. Delete or fill B_r(x), first comparing in a slightly larger ball, and then decrease the comparison radius. BV slicing and quasiminimality give, for almost every r,

If I is the local absolute isoperimetric constant, the perimeter bound and isoperimetric inequality give

Integrating the resulting inequality for m^{1/n} from its zero limit at radius zero, and doing the same for the exterior phase, gives

Also m'(r)+e'(r)=\Area_{g_0}(\partial B_r(x))\le Sr^k, so adding the two comparison inequalities gives P_{g_0}(K;B_r(x))\le QS r^k/2. Perimeter-measure monotonicity extends the upper bound to every small radius; relative isoperimetry and the two phase volumes give the lower bound. Thus, with constants depending only on the stated data,

For the canonical representative K=\supp(\chi_K\,dV_{g_0}), these estimates identify \partial K with \supp|D\chi_K|. They hold at every frontier point, not only almost every reduced-boundary point. Null-set additions or artificial isolated points are not part of the representative.

A bounded complementary component of a minimizing hull can be filled, removing its strictly positive perimeter. This contradicts minimality; equivalently one may use the finite-perimeter component decomposition. The canonical representatives have no measure-zero cracks producing artificial interfaces. Since the manifold has only one end and the conductor is compact, the remaining exterior is its connected end component.

The weak capacitary problem. The harmonic function is constructed before any continuity at the frontier is assumed. For an individual compact conductor K, let \mathcal D^{1,2}(M) be the completion, in the gradient norm, of smooth functions compactly supported relative to M. Such functions may be nonzero on the physical boundary. The asymptotically flat Sobolev inequality, for n\ge3, embeds this space in L^{2n/d}. Minimize

over its closed convex subset consisting of functions whose quasi-continuous representative is at least one quasi-everywhere on K. Here quasi-everywhere means outside a set of Newtonian capacity zero. A smooth function equal to one near K makes the class nonempty. The Hilbert-space direct method gives a unique minimizer W_K; truncation gives 0\le W_K\le1 and W_K=1 quasi-everywhere on K.

Variations of both signs supported in M\setminus K prove harmonicity there. Nonnegative variations supported away from the physical boundary remain admissible, and give

Consequently the escape potential q_K=1-W_K, extended by zero on K, is locally H^1, satisfies 0\le q_K\le1, and obeys

Sobolev integrability of W_K and interior estimates on end annuli imply W_K\to0 uniformly at infinity. A radial supersolution r^{-\beta}, 0<\beta<d, and interior gradient estimates give, after increasing a radius depending on K,

No uniform support radius is assumed in this individual estimate. Thus q_K\to1 at infinity. The homogeneous space is important: W_K need not belong to unweighted L^2 in dimensions three and four. Since K contains a collar of the physical boundary, W_K=1 and q_K=0 there, and the boundary contributes no test or flux term to this exterior problem.

The exterior-indicator boundary contraction. Fix x\in\partial K and work in uniform background coordinates, using Euclidean coordinate radii in this paragraph. By (4.7) and metric comparability, |K\cap B_r(x)|\ge d_* r^n. Choose \kappa\in(0,1/2) so small that |B_{\kappa r}(x)|\le d_* r^n/2. Coarea supplies a radius \rho\in[\kappa r,r] at which the Sobolev traces exist and

At this radius the zero extension of q_K has zero trace almost everywhere on the conductor portion of the sphere. Set

Let H be the g_0-harmonic replacement in the full ball with boundary trace q_K. Weak subharmonicity in (4.8) and the weak maximum principle give q_K\le H. If P_\rho is the Poisson kernel, then

The datum is the exterior indicator \chi, not 1-\chi: only the former dominates the possibly positive trace of q_K. For the fixed smooth background operator, scaled smooth-ball Green-function estimates give

The constant is uniform under the local ellipticity and coefficient bounds already fixed; the kernel integrates to one. Its missing mass on the conductor portion of the sphere is therefore at least a uniform \delta_*>0. Since B_{\kappa r/2}(x)\subset B_{\rho/2}(x), it follows that

This comparison does not assume continuity at the frontier: the harmonic replacement uses an actual Sobolev trace at an eligible radius, whereas the indicator is used only in the Poisson integral and is not asserted to belong to H^{1/2}.

Iteration, starting from q_K\le1, gives some \alpha\in(0,1) for which

For example, decrease to an exponent smaller than one and no larger than \log(1-\delta_*)/\log(\kappa/2). Interior smoothness now supplies a continuous representative equal to zero at every frontier point. The strong maximum principle gives 0<q_K<1 on the connected exterior. Uniqueness among continuous exterior solutions with these boundary and end values follows by applying the maximum principle on expanding truncated exteriors.

For completeness the same estimate gives a full spatial Hölder modulus, not just decay at boundary points. If two points a distance \ell apart have one within 2\ell of K, both values are bounded by C_*\ell^\alpha. Otherwise a local ball joining them stays in the exterior. At distance \varrho from K, interior gradient estimates and the boundary decay give |\nabla q_K|\le C_*\varrho^{\alpha-1} when \varrho<r_*/4; at larger distance fixed-radius interior estimates suffice. Integration between the two points again bounds their difference by C_*\ell^\alpha. Boundedness handles larger separations. Hence the zero extensions satisfy

The constants depend on c, the background geometry, and the initial comparison radius. No derivatives of the weight enter this estimate.

The comparison for a Hölder weight. Suppose \operatorname{osc}_{B_r}a\le C_a r^\alpha. If P_{g_0}(L;B_r)\ge P_{g_0}(K;B_r) the conclusion is immediate. Otherwise the density upper bound gives both perimeters at most C_*r^k. The exact weighted comparison yields

This proves the final assertion.□

Proof. The fixed-obstacle minimizing property. Every K_j^\epsilon is also the outermost h_j^\epsilon-minimizing enclosure of the fixed K_0. Indeed, suppose this holds at step j-1, and let L\supset K_0 be any compact finite-perimeter competitor. The two consecutive metrics agree on K_{j-1}^\epsilon, because v_{j-1}^\epsilon vanishes there. The boundary of L\cap K_{j-1}^\epsilon is supported in that same closed set. Hence

Perimeter submodularity therefore gives

The set on the left is an admissible competitor for K_j^\epsilon, which proves global minimality. If L is any global minimizer, the same inequalities show that its union with K_{j-1}^\epsilon is a constrained minimizer. Outermostness of K_j^\epsilon then gives L\subset K_j^\epsilon. The induction starts at the actual outermost initial hull, not at an arbitrary almost-minimizing frontier.

Set A_j^\epsilon=P_{h_j^\epsilon}(K_j^\epsilon). Freezing on the previous conductor and its admissibility at the next step imply

The metric at an intermediate time need not make the frozen K_j^\epsilon globally minimizing. Its area nevertheless equals A_j^\epsilon, since v_j^\epsilon=0 on that frontier; comparison with an arbitrary competitor always uses h_j^\epsilon. On a fixed finite interval h_j^\epsilon and g_t^\epsilon differ uniformly by O_T(\epsilon) for j\epsilon\le t<(j+1)\epsilon.

Closing the induction and its uniform estimates. At step j, the constructed continuous positive factor defines the weighted hull problem. The fixed-obstacle minimizing property just proved lets us apply Lemma 4.2 with a=(u_j^\epsilon)^p and c=4^{-pT}. It gives the density estimates and the continuous potential

In particular v_j^\epsilon vanishes on the whole canonical conductor. Thus the next factor is continuous and agrees with u_j^\epsilon on K_j^\epsilon, including its perimeter traces. This closes the induction underlying the freezing and fixed-obstacle comparisons.

The exact discrete integral identity gives a uniform spatial Hölder bound for u_t^\epsilon on [0,T], while |v_j^\epsilon|\le1 gives the time Lipschitz bound. Hence \operatorname{osc}_{B_r}(u_j^\epsilon)^p\le C_T r^\alpha. The final assertion of the static lemma now gives the additive almost-minimizing estimate with error C_T r^{k+\alpha}.

Uniform finite-time support. The global grid-minimality induction permits application of (4.5) with the fixed obstacle K_0, the fixed area bound A_0, and c=4^{-pT}. Hence, enlarging the resulting radius if necessary,

It follows directly from the static truncation comparison and is independent of the number or shape of the conductor components.□

Proof. Global minimality in the preceding grid metric, with K_j^\epsilon as competitor, gives

The third line follows from Bernoulli’s inequality, with v_{j-1}^\epsilon\le0.

Separate the grid transitions into those for which

and the remaining transitions. Discrete nesting and (4.12) bound the number of the first kind by \epsilon^{-1/2}, so their total area loss is O_T(\sqrt\epsilon). For a transition of the second kind, the two phase-density inequalities imply

once \epsilon is small. Indeed, a boundary point separated from the other frontier by a ball of radius \rho contributes at least c_T\rho^n to the swept volume: use the filled phase at a point of the new frontier and the exterior phase at a point of the old one. The old velocity vanishes on the old frontier; its Hölder modulus therefore bounds its supremum on the new frontier by C_T\epsilon^{\alpha/(2n)}. Summing the \epsilon-weighted losses over at most T/\epsilon+1 transitions gives

uniformly for j\epsilon\le T. The same assertion holds at intermediate times because the current frontier has frozen area. Only discrete nesting has entered this calculation.□

Proof of Lemma 4.1. Compactness and the limiting minimum area. The functions u_t^\epsilon are uniformly Lipschitz in time and uniformly Hölder in space. Choose a mesh sequence tending to zero for which they converge locally uniformly, jointly in space and time, to a positive u_t, and put g_t=u_t^{4/d}g_0. The support bound also controls the end normalization. Put \beta=d/2 and enlarge R_T until \Delta_{g_0}r^{-\beta}<0 for r\ge R_T. The function 1+v_t^\epsilon/a_j^\epsilon is harmonic there, lies between zero and one on the inner sphere, and tends to zero at infinity. The maximum principle gives

Writing a_\epsilon(t)=a_j^\epsilon(1-t+j\epsilon) on the current step and integrating yields

The constants a_\epsilon(t) converge uniformly to e^{-t} on [0,T]. These tail bounds turn local uniform convergence of the factors into uniform convergence on [0,T]\times M and retain the constant -e^{-t} for every fixed-time limit of the velocities. Thus the compactness passage preserves the end normalization.

We claim that, for every t, the minimum g_t-perimeter among enclosures of K_0 equals A_0. Fix a time t and a bounded finite-perimeter enclosure L\supset K_0. Grid minimality at j=\lfloor t/\epsilon\rfloor, the uniform convergence of the lagged metric, and (4.13) imply

All discrete velocities vanish on K_0, so u_t=1 there and P_{g_t}(K_0)=A_0. Hence A_0 is exactly the minimum enclosure perimeter for g_t.

Common support and the background perimeter bound give a subsequential BV limit \widehat K_t of the grid conductors. Lower semicontinuity in the uniformly converging positive weights gives P_{g_t}(\widehat K_t)\le A_0. Since this is itself an enclosure of K_0, (4.14) gives the reverse inequality. Thus every such limit is a g_t-minimizer of perimeter A_0. This also proves perimeter convergence. The density estimates then upgrade convergence of the canonical sets to Hausdorff convergence of their frontiers.

Define K_t to be the outermost g_t-minimizing enclosure of K_0. It has perimeter A_0 and contains every \widehat K_t. Lemma 4.2 bounds its support by a finite-time common radius and gives exact local weighted minimality, since u_t=1 on K_0. It also supplies the uniform density, local area, and background almost-minimizing estimates. A bounded component of its exterior could be filled with a strict decrease in perimeter; thus no such component occurs. The unique asymptotically flat end accounts for the remaining exterior component. These observations also apply to every \widehat K_t.

Nesting of the limiting outermost hulls. For s>t, the monotonicity of the discrete conformal factors passes to the limit, so u_s\le u_t. The already established minimum areas give

It follows that u_s=u_t almost everywhere on the perimeter of K_t. Continuity of the factors and the positive perimeter density at every frontier point improve this to

At this stage no freezing throughout K_t has been asserted.

Take any joint subsequential limit (\widehat K_s,\widehat v_s) of the grid conductors and velocities at time s. The boundary Hölder estimate, interior harmonic compactness, and the fixed-sphere tail comparison identify \widehat v_s as the harmonic function with zero extension to \widehat K_s and end value -e^{-s}. It is strictly negative on its connected end exterior. On the other hand, the discrete integral identity and (4.11) give

Passing to this subsequence yields

Together with (4.15), this implies \partial K_t\subset\widehat K_s. The connected exterior of \widehat K_s agrees with the exterior of K_t near infinity and cannot cross \partial K_t, so it is contained in that exterior. Therefore

This proves nesting using only the discrete integral identity and the previously proved area equality.

Identification of the integral flow. On each finite interval the function V(t)=\Vol_{g_0}(K_t) is bounded and nondecreasing. Its positive left jumps occur at only countably many times. Let \mathcal J be the union of these exceptional sets over integer time intervals, together with \{0\}. If t\notin\mathcal J, every fixed-time grid limit satisfies, by (4.16),

The union may be taken over rational r<t, and its volume is the left limit of V. At a nonjump time this equals V(t). The canonical identity K_t=\supp(\chi_{K_t}\,dV_{g_0}) therefore gives

a point of K_t outside that closure would have a neighborhood of positive K_t-volume disjoint from the union, contradicting equality of volumes. Since \widehat K_t is closed, it equals K_t.

Thus at every nonexceptional time all subsequential grid limits agree. Compactness proves convergence of the entire chosen sequence in L^1 and in Hausdorff distance of canonical sets and frontiers. The latter assertion follows directly from the two phase-density estimates: a boundary point remaining away from the limiting boundary would have a ball wholly in one limiting phase, contrary to the corresponding positive discrete phase volume and L^1 convergence. The reverse direction uses the two positive limiting phase volumes. Common support prevents boundary points from escaping in this argument. The boundary Hölder estimates, interior harmonic compactness, uniqueness, and uniform tail comparison then give

where v_t is the harmonic function on E_t, zero on K_t and tending to -e^{-t}. The static density and capacitary arguments construct this selected potential at exceptional times as well. Nesting and the maximum principle make v_t nondecreasing in time, so it is measurable. Dominated convergence in the exact discrete integral identity gives

In fact the convergence of these integrals is uniform in space and on [0,T], since

The scalar norm is measurable by taking the supremum over a countable dense spatial set; it is bounded by two, so the displayed almost-everywhere convergence justifies the last dominated-convergence step. The same exceptional set therefore works for every spatial point. The possible failure of grid velocities to converge to the selected v_t at a jump time has no effect on this integral. Now, and only now, nesting and the integral identity give u_s=u_t throughout K_t for s\ge t, since each later velocity vanishes there.

The end normalization gives v_s\ge-e^{-s}, so the same identity improves the finite-time lower bound to

It also shows that u_t is g_0-harmonic in E_t: each earlier velocity is harmonic there, and interior estimates justify integration on compact subsets of that exterior. All conductors contain the fixed collar of the physical boundary, so v_t=0 and u_t=1 on that collar. No boundary condition or mean-curvature assumption at the buried boundary is added. The conformal transformation of the scalar Laplacian gives

Consequently w_t=1+v_t/u_t is the g_t-harmonic capacitary potential: its boundary value is one and its end value is zero. The inequalities -e^{-t}\le v_t\le0 and u_t\ge e^{-t} give 0\le w_t\le1 directly. Its constant extension over K_t equals one on the same buried collar.

Finally choose the convergent mesh subsequence diagonally over T=1,2,\ldots. The true hull at each time is determined by its limiting metric, so the constructions on overlapping finite intervals agree. This gives the flow for all t\ge0. Equation (4.2) already holds at every time by the minimizing-value argument, including jump times. The local weighted comparison and its Hölder-weight conversion give (4.3) uniformly on each finite interval, as asserted.□

Proof. The proof has two parts. First, ordering prevents a second frontier from developing excess beside a flat regular patch. Then the harmonic equation and the minimal graph equation improve the resulting common graphical bounds together.

A common graphical neighborhood. We use the following compactness consequence of the power almost-minimizing estimate. A sequence with uniform local perimeter and two-phase density bounds has a locally L^1-convergent subsequence with locally convergent perimeter measures. If the limit has a regular point, the boundaries converge as single C^{1,\theta} graphs on a smaller neighborhood of that point, with uniform bounds. Strict perimeter convergence excludes additional sheets; L^1 convergence alone would not suffice. These are the almost-minimizer compactness and flat-boundary regularity conclusions used in [5, Theorem 2.27].

Fix the reference regular point p\in\Sigma_t and choose coordinates in which its tangent plane is horizontal. Consider any sequence of frontiers \Sigma_{\sigma_j} with points p_j\in\Sigma_{\sigma_j} tending to p. After a subsequence, the filled sets all lie on the same ordered side of K_t. The compactness just stated gives a limiting filled set K_*, and two-phase density puts p on its perimeter support. Ordering passes to the limit. Every tangent filled set of K_* at p is therefore ordered with the tangent half-space of K_t; its multiplicity-one minimizing boundary cone is supported in a closed half-space.

Such a cone is a plane. Indeed, the normal coordinate is nonnegative on the cone and satisfies \Delta_{\rm link}x_\perp=-(k-1)x_\perp on its spherical link. Integration forces that coordinate to vanish. Local area growth and the singular-set codimension at least seven supply cutoffs with vanishing tangential L^1 gradient, so the integration includes the singular set. Multiplicity one and the two-phase density bounds give the nonempty plane with density one. Density-one regularity thus makes \partial K_* a single C^{1,\theta} graph near p, tangent to the reference sheet. Regular-patch convergence gives the same fixed graphical neighborhood and a uniform bound for \Sigma_{\sigma_j} when j is large.

Here is the uniformity conclusion. Fix a small slope threshold and choose a corresponding bound for the scaled C^{1,\theta} graph norm in the flat-boundary regularity theorem. If no neighborhoods U\Subset V worked for the whole time family, choose radii r_j\downarrow0 and frontiers meeting B_{r_j^2}(p) at points p_j that fail this graph bound on the radius-r_j cylinder centered at p_j. The preceding subsequence instead converges on a fixed graphical neighborhood of p. Its slopes at p_j tend to the horizontal slope, and their oscillation on a radius-r_j cylinder tends to zero under the uniform C^{1,\theta} bound. A fixed shrink of the cylinders allows their centers to move from p to p_j. Restricting the fixed graphs then contradicts the chosen failures. This proves the common graphical neighborhood.

For a rescaled family, repeat this contradiction in a fixed regular cone chart. The reference graphs converge to its smooth sheet, the rescaled smooth background metrics have uniform bounds, and the power errors tend to zero. The same ordered-limit and density-one argument gives fixed inner and outer graphical neighborhoods, uniformly in the rescaling index. A finite cover gives the assertion on a compact regular cone patch. None of these arguments uses separation for frontiers in different conformal metrics.

One-sided derivatives and the first graph equation. Choose three nested neighborhoods V_0\Subset V_1\Subset V_2 within this common graphical region. For any time \sigma, if \Sigma_\sigma meets V_1, its uniform C^{1,\theta} graph and the zero Dirichlet trace of v_\sigma give one-sided boundary estimates

Here and below norms on graph domains use their uniformly controlled charts. If \Sigma_\sigma misses V_1, its distance from V_0 is bounded below. On each relevant component v_\sigma is then harmonic on a whole neighborhood or identically zero, and interior estimates give the same bound. It would not suffice merely to know that the frontier misses V_0. Nesting gives E_t\subset E_\sigma for \sigma\le t, so all these exterior estimates can be integrated on the common later exterior.

For the first stationarity argument, flatten the graphical boundary and reflect the negative exterior function v_\sigma oddly. Its zero trace implies that tangential first derivatives vanish at the graph, and odd reflection matches the normal first derivative. This gives a uniformly controlled C^{1,\alpha} extension, nonnegative on the interior side, which dominates the original zero extension. Where a frontier is absent, keep the harmonic or zero function. On a smaller common neighborhood, fixed finite charts and a partition of unity preserve agreement on the exterior and domination on the interior. Integrating these extensions gives

Shrinking the neighborhood preserves a uniform positive lower bound. Thus \widetilde g_t=\widetilde u_t^{4/d}g_0, d=n-2, is a uniformly controlled C^{1,\alpha} extension of the one-sided metric.

These integrals require only measurable, not continuous, time dependence. The original harmonic velocities are pointwise monotone in time, interior derivatives are limits of measurable difference quotients, and nested graph heights are measurable. Reflection in the graphs and the finite chart constructions preserve this measurability. Equivalently, one can first integrate the derivatives directly on the common later exterior by dominated convergence and use extensions only for the graph equation.

By Lemma 4.1, each K_t is locally perimeter minimizing in g_t, including for variations across K_0. Since \widetilde g_t\ge g_t and the two area functionals agree on \Sigma_t, it is also locally minimizing in \widetilde g_t. The graph is stationary in this extension metric. The minimal graph equation and uniform elliptic regularity improve the graphs to C^{2,\alpha} on smaller disks. The constants depend on the graph and metric bounds. The common local minimizing comparison makes the equation valid on the whole regular graph disk, including where that disk meets the initial conductor. At time zero the metric is the smooth reference metric and the same local minimizing comparison holds. Thus the estimate is uniform for the whole time family, including zero.

Closing the bootstrap. We now have a common C^{2,\alpha} graph bound, including at time zero. Boundary estimates on these improved domains give uniform C^{2,\alpha} estimates for all harmonic velocities. Integrate them on each common later exterior to improve all conformal factors, and apply the minimal graph equation again. Repetition on nested neighborhoods gives every prescribed finite order. At these later stages use a bounded extension operator for uniformly regular graph domains. Higher-order extensions need not dominate the metric: stationarity is already established, and agreement of the boundary value and first derivatives gives the same minimal graph equation. In particular, no uniform positive Hopf lower bound is needed to control the sign of higher-order extensions.

For each fixed order only finitely many nested neighborhoods are used. On a compact regular cone patch the rescaled smooth background metrics have uniform bounds of every such order, while the lower bound for u_t and the bound for v_t are unchanged. All the estimates are therefore uniform in the blow-up index. Together with the original C^1 graphical convergence, interpolation gives convergence in every lower fixed differentiability order. In particular the one-sided working metrics converge to their constant contact-point values with the derivatives needed in the Jacobi expansion.□

Proof. Normalize z_i by a_i. Interior Harnack chains and the boundary Carleson estimate bound this normalized function on a smaller cylinder. Boundary Schauder estimates then give the stated C^{2,\alpha} bound. An interior Harnack chain supplies a fixed positive comparison value on a sphere in a uniform interior tangent ball. The tangent-ball barrier gives the positive lower bound for the normal derivative on the central boundary patch; the upper bound follows from the derivative estimate. These are the standard local boundary estimates for uniformly regular elliptic Dirichlet problems; see [16].

Join corresponding points of the two graphs by the common graph coordinate segment, which lies on the exterior side of the first graph. The Hessian bound and convergence of the normals show that the error in transferring the derivative is at most

For large i this is less than half the first positive lower bound. The same factor a_i multiplies the bound and its error, even if a_i\to0. This proves the relative assertion.□

Proof. Suppose the supports meet at the origin, and put k=n-1\ge7.

A common cone with connected regular part. The power almost-minimizing comparison gives local strict perimeter compactness and almost-monotonicity of density. Hence every simultaneous blow-up has a subsequence converging to multiplicity-one minimizing boundary cones for g_{\rm ref}(p). They remain nested. Since g(p) is a constant scalar multiple of g_{\rm ref}(p), both cones are stationary also in the normalized Euclidean metric. Their supports are connected, because radial segments join all points to the vertex; their singular sets have dimension at most k-7. The same-metric strong maximum principle of [36, Theorem 1.1 and Remark (3)] identifies the two supports, since both contain the vertex. Multiplicity one identifies their perimeter measures. This works for every simultaneous blow-up, not only the sequence selected below.

The regular part of the common cone C is connected. To see this, restrict its varifold to a regular component. Its stationarity extends across the singular set by the local area bound and cutoffs with vanishing tangential L^1 gradient, since \mathcal H^{k-1}(\operatorname{Sing}C)=0. Each regular component is dilation invariant: the radial dilation orbit is a path in that component. Its closure therefore contains the vertex. Distinct component closures intersect only in the singular set, because a regular point has a connected smooth-sheet neighborhood. This would give intersecting stationary integral varifolds whose intersection has zero \mathcal H^{k-1} measure, contrary to [36, Theorem 3.2 and the proof of Theorem 1.1, pp. 8–9]. These are same-metric statements and do not assume the different-metric conclusion being proved.

A median chosen before the regular patches. Choose a fixed smooth coordinate chart normalized so that the earlier working metric at the contact point is Euclidean. Let D(y)=\dist_{\R^n}(y,\Sigma'), let \mu_r denote the perimeter measure of \Sigma\cap B_r, and define the fixed median

Atoms are allowed. Both \{D\le q(r)\} and \{D\ge q(r)\} have at least half the measure. The contact set has zero perimeter measure, since there is no regular contact and the singular sets have zero perimeter measure. Thus q(r)>0 for small positive r. Also q(r)\le r, since the origin belongs to \Sigma'.

In fact q(r)/r\to0. If not, choose a contradicting sequence and then a simultaneous minimizing-cone subsequence. Compactness and the density bounds give local Hausdorff convergence of both rescaled supports to the same cone. On B_1, distance to the second support therefore tends uniformly to zero, contradicting the assumed lower bound for the rescaled medians.

Put f(r)=q(r)/r. Given any R_i\downarrow0, choose 0<r_i\le R_i such that

The supremum is finite and positive; neither continuity of the median nor attainment of the supremum is required. For every 0<\rho\le1, this choice gives

Pass to a simultaneous cone subsequence at these radii, and put

Every graph separation on every patch will be divided by this same a_i.

From ambient distance to normal height. The record scales and the normalizing numbers a_i are now fixed. We next relate their ambient-distance definition to the graph variable in (6.1). On a compact regular core of C, choose a slightly larger regular neighborhood. For large i, \Sigma'_i is the g_i-normal exponential graph of \zeta_i>0 over \Sigma_i, where g_i is the rescaled earlier working metric. Their relative slopes tend to zero. For a point in the smaller core, its nearest point on \Sigma'_i lies in the larger neighborhood: the local normal candidate has distance tending to zero, while the complement of that neighborhood stays a fixed positive distance away. The Lipschitz-graph distance estimate in coordinates flattening \Sigma_i gives

uniformly on the smaller core. The normal exponential segment has Euclidean length (1+o(1))\zeta_i and direction approaching the Euclidean normal, because the contact metric was normalized and the rescaled metrics converge there in C^1. Thus the same relative Lipschitz-graph estimate applies to its endpoint. This remains a relative estimate even when \zeta_i is much smaller than the blow-up error; an additive comparison with the cone would not suffice. Figure 1 distinguishes these two measurements and the use of one median on all regular patches.

Choose a compact regular core inside C\cap B_1 whose omitted area is less than one eighth of its total area, and enlarge it slightly while keeping away from the singular set and sphere. Such cores exist because those two sets have zero k-measure on the cone. Graphical perimeter convergence makes the omitted area less than one quarter for large i. Each median set therefore retains positive measure on this fixed core.

Local bounds with the same normalization on every patch. Set W_i=\zeta_i/a_i. The lower median set and (6.4) provide a point with W_i\le2 in one of finitely many regular core patches. Choose that patch along a subsequence. Weak Harnack for the nonnegative supersolution L_iW_i\le0 gives a uniform local L^p bound for some p>0, before any bound for F_i/a_i has been proved. To handle the zero-order term, write

and replace c_i by \min\{c_i,0\}, obtaining \widehat L_i. Since W_i\ge0, we still have \widehat L_iW_i\le0. The usual weak Harnack inequality with nonpositive zero-order coefficient applies with uniform constants on every fixed patch.

An L^p bound on an overlap of fixed positive measure supplies a bounded-value point for the next patch. Weak Harnack there propagates the bound. Connectedness of \operatorname{Reg}C gives finite chains from the anchor patch to every fixed compact regular set. Consequently

The constants may grow with K. No uniform Harnack chain through the singular set has been asserted.

Controlling the forcing before passing to a limit. The local L^p bounds have been obtained without dividing a right-side estimate by the possibly small number a_i. They now let us control that quotient. In a ball where F_i\le-c\lambda_i, solve

Uniform ellipticity, coefficient bounds, and the maximum principle give \phi_i\ge c_0>0 on a smaller ball. Since \widehat L_i\zeta_i\le F_i and \zeta_i\ge0 on the boundary, comparison gives

Combining this with (6.5) yields \lambda_i/a_i\le C_K. The assumed upper bound for |F_i| and a finite cover now give

The order is important: first weak Harnack, then the positive Dirichlet comparison, and only then bounded-right-side elliptic estimates.

The weak Harnack exponent need not exceed one. The local boundedness estimate for the now bounded-right-side equation bounds the supremum on an inner ball by the L^2 norm on a larger ball and the right-side bound. Interpolate the L^2 norm between the bounded L^p norm and the larger-ball supremum, apply Young’s inequality, and iterate on nested radii to absorb that supremum. For each i the preliminary finite suprema exist by local regularity. This gives uniform local L^\infty bounds; interior elliptic estimates then give uniform W^{2,q} bounds for every finite q. One countable exhaustion and one diagonal subsequence give a single function W, with local C^1 convergence on all of \operatorname{Reg}C, satisfying

The upper median set also retains positive measure on the fixed regular core. Equation (6.4) gives points there with W_i\ge1/2. Local uniform convergence makes W nonzero. Since \Delta_CW\le-|A_C|^2W\le0, the strong maximum principle and connectedness make W>0 throughout the regular part. No separately normalized patch limits are used.

The infimum near the vertex. We have obtained one positive limit on the entire regular part. The record-scale choice now gives information on its behavior near the vertex without changing its normalization or taking new patchwise limits.

Fix 0<\rho\le1. The lower median set in \Sigma_i\cap B_\rho has half its area and, by (6.3), normalized ambient distance at most 2\rho. Choose a compact regular core in C\cap B_\rho omitting less than one quarter of its area. Perimeter convergence and (6.4) give a point in this core with W_i\le3\rho for large i. The already chosen local uniform convergence applies to this core. Thus the same limit satisfies

It is enough to use a countable sequence of \rho\downarrow0. The constant 3 does not depend on the core or on \rho: for each fixed radius the relative error in (6.4) tends to zero before this limit is taken. Therefore the infimum of W in every ball about the vertex is zero. Its essential infimum there is also zero, since every small regular value has a neighborhood of positive perimeter measure with comparably small values.

A bounded supersolution across the singular set. To apply a mean-value inequality on the whole minimizing cone, we need a Sobolev supersolution across its singular set. First discard the nonnegative Jacobi potential, obtaining \Delta_CW\le0, and then set h=\min\{W,1\}. Approximation by increasing concave truncations shows that h is a bounded positive weak Laplace supersolution on the regular part. It need not be a Jacobi supersolution. For a regular-part cutoff \eta, use the nonnegative test \eta^2(1-h) to obtain

and hence

The singular set has zero tangential two-capacity on compact regions. Indeed, cover the relevant compact singular set by finitely many balls B_{\rho_j}(x_j), with centers on that set, maximum radius tending to zero, and \sum_j\rho_j^{k-2}\to0. This is possible because its dimension is at most k-7. Let \theta_j vanish on the ball, equal one off its double, and satisfy |D\theta_j|\le C/\rho_j. Put \chi=\min_j\theta_j. Almost everywhere its gradient is one of the active gradients, so local area growth gives

There is no overlap-count factor. These cutoffs vanish near the whole relevant singular set and converge to one on the regular part.

Use \eta=\psi\chi in (6.9), where \psi is a fixed compact spatial cutoff. Fatou’s lemma gives locally finite Dirichlet energy for h across the singular set. Moreover, \psi h\chi\to\psi h in the local surface W^{1,2} norm: the term h\nabla_C\chi tends to zero by boundedness and (6.10), while the omitted \nabla_C h term tends to zero by absolute continuity of its now finite energy. Thus h genuinely extends as a local Sobolev function on the whole minimizing boundary; its values on the measure-zero singular set are immaterial.

For a nonnegative ambient smooth compactly supported test function \varphi, insert \varphi\chi into the regular-part supersolution inequality. The cutoff error is bounded by

The other term converges by dominated convergence. Hence

for tests meeting the singular set as well.

The mean-value contradiction. The bounded function h is now a positive weak Laplace supersolution on the whole minimizing boundary, with locally finite energy and zero essential infimum in every ball about the vertex. These are the conditions needed for the minimizing-boundary weak Harnack theorem of Bombieri–Giusti [7]; the precise form needed here is stated in [12, Theorem 3.1, p. 11]. For an area-minimizing hypersurface in B_R and a positive weak Laplace supersolution, a normalized L^p mean on B_r is bounded by its infimum on B_r when r\le\beta R and 0<p<k/(k-2). The constants depend only on dimension and p. This is an extrinsic estimate on the minimizing boundary, not an assertion that regular-patch Harnack chains have uniform constants through cone singularities.

Apply it to h+\epsilon on the cone, with fixed concentric balls and, for example, p=1. Choose the smaller ball to contain a regular open subset on which h>0; the cone and the same h are defined on any required larger ball. Its conclusion is

The essential infimum of h is zero by (6.8). Sending \epsilon\downarrow0 would make the positive integral on the left zero, a contradiction. This proves separation. No growth selection at infinity or limiting truncation level is needed.□

Proof. The flow assertion was proved in Section 6.2. For the perturbed enclosure, the preceding regular comparison excludes regular contact. The power almost-minimizing estimate, the regular-patch metric and graph bounds, and the displayed forcing equation verify Proposition 6.1, which excludes the remaining singular contact. Compactness of each pair of supports then gives positive distance. The constants and the resulting distance may depend on the fixed time pair or perturbation parameter.□

Proof. If \Sigma is empty, take \Omega_\varepsilon=\Omega. Otherwise we first choose a regular distance level, then take a short offset of its superlevel, and finally smooth the resulting C^{1,1} hypersurface. The offset controls curvature; the final smoothing loses an arbitrarily small amount of that bound.

Write \rho(x)=\dist(x,\Sigma) on \Omega. All distances and geodesics below are taken in the ambient metric. Choose a relatively compact ambient neighborhood of \Sigma and a smaller compact safety neighborhood. Fix r_*>0 much smaller than the distance separating their boundaries and the injectivity radii on the larger compact neighborhood. Short paths used in the proof then stay in the region of controlled smooth geometry, and the squared-distance functions used for pairs less than 4r_* apart are smooth. No completeness of an auxiliary ambient extension is needed.

Regular distance levels. If 0<\rho(x)<r_*, every nearest point of \Sigma to x belongs to \Lambda. Indeed, a shortest segment from x to the full boundary stays in \Omega until its endpoint, and a shorter terminal subsegment supplies an interior tangent ball there. The nearest-point set is compact. It therefore lies in the interior of a compact smooth hypersurface patch Q\subset\Lambda with smooth boundary. Choose Q sufficiently close to that set that \dist(y,q)^2 is smooth for q\in Q and y near x.

Because \Lambda is embedded, there is an ambient open set \mathcal V containing the nearest-point set with \mathcal V\cap\Lambda\subset\operatorname{int}_{\Lambda}Q. After shrinking the neighborhood of x, every nearest point to each such y lies in \mathcal V. Otherwise, compactness of \Sigma\setminus\mathcal V would give a nearest point of x outside \mathcal V. These nearby nearest points belong to \Lambda by the preceding tangent-ball argument, and hence lie in the interior of Q. Thus

on an open neighborhood of x. Apply [32, Theorem 3 and the compact-parameter-with-boundary extension in the proof of Theorem 1]. A countable family of these neighborhoods covers the small distance collar, so the critical values of \rho^2 form a null set. On each compact interval of positive distances, the square-root map is Lipschitz and \partial_C(\rho^2)=2\rho\,\partial_C\rho, where \partial_C denotes the Clarke differential. Thus the positive critical values of \rho also form a null set. Choose s_i\downarrow0 such that

The compact parameter patch is the reason this theorem applies even though the accessible locus \Lambda need not be closed.

Positive reach of the superlevels. The distance function is locally semiconcave away from \Sigma. To obtain smooth upper supports, take a minimizing segment from \Sigma to a point of the collar and retain a short terminal piece. Distance from its earlier endpoint, plus the length of the preceding piece, is a smooth upper support near the terminal point. The local Hessian comparison estimate gives a uniform upper Hessian bound on a sufficiently small fixed ball. Consequently, in a geodesic coordinate ball,

Here the superdifferential of the semiconcave function agrees with its Clarke differential.

We include the regular-superlevel implication underlying the positive-reach criterion associated with [2]. At a point of a regular level, let \mathcal K=\partial_C\rho(x). This is a compact convex set not containing zero. Separation gives a direction v_0 such that \langle p,v_0\rangle>0 for every p\in\mathcal K. The directional derivative of a semiconcave function is \min_{p\in\mathcal K}\langle p,v\rangle. Every direction for which this minimum is positive therefore enters the superlevel. Separation of convex cones implies that the polar of its contingent tangent cone, and in particular each proximal outward normal, is contained in -\operatorname{cone}(\mathcal K). Since \mathcal K is convex, each unit proximal outward normal can be written \mathbf n=-p/|p| with p\in\mathcal K.

Compactness of the level and upper semicontinuity of the superdifferential give a uniform lower bound |p|\ge\eta>0 there. For a nearby point y of the superlevel, Equation (7.4) yields

This gives uniqueness of sufficiently nearby nearest-point projection. Indeed, suppose that z had two distinct nearest points x,y, at the same distance a. In a sufficiently small bounded-geometry ball, the Taylor expansion of squared distance from z gives

For a<\eta/(4C) this contradicts the preceding bound. Enlarging the constants slightly covers the curvature terms. Compactness excludes distant distinct nearest points when a is small and makes this projection radius uniform. Thus

has positive reach. It is closed in the ambient manifold because \Sigma is the full boundary and s_i>0, and its boundary is compact. The same separating direction v_0 shows that its outward normal cone is pointed: it cannot contain opposite nonzero normals. Only the regular-superlevel implication has been used.

Offsets and their curvature. Fix a regular value s=s_i and put F=F_i. For 0<\tau<\operatorname{reach}(F), sufficiently small compared with s and the ambient injectivity scale, define

The closure of O_\tau lies in \Omega, since \rho\ge s-\tau>0 there. Moreover, \Omega\setminus O_\tau lies in the s-collar of \Sigma.

Each point of \Gamma_\tau has a unique footpoint q\in F and can be written p=\exp_q(\tau v), where v is a unit outward normal to F. The same ray realizes distance to F for every positive parameter below \min\{\operatorname{reach}(F),r_*\}. These facts follow from uniqueness of projection, first variation and continuation of a minimizing normal segment; they are also the nearest-point and normal-cone facts used in [4, Section 3.1]. No differentiability of the projection is required.

The normal ray supplies tangent balls on both sides of \Gamma_\tau. The interior supporting radius is bounded below when \tau stays away from zero. For the exterior supporting ball, fix 0<R<\min\{\operatorname{reach}(F),r_*\} first and take \tau<R/2. Continuing the normal ray to R gives an exterior supporting radius R-\tau\ge R/2. Local graph comparison with these spheres gives a compact C^{1,1} hypersurface. For this fixed F, the principal curvatures have a uniform lower bound as \tau\downarrow0; their upper bound may grow like \tau^{-1}.

The offset is now a C^{1,1} hypersurface lying strictly inside \Omega. To finish its construction we must transfer the positive curvature of accessible boundary points to this offset. We do so by comparison with smooth interior test domains. Let a smooth region D\subset O_\tau touch \Gamma_\tau from inside at p=\exp_q(\tau v). Its outward normal at contact is the parallel translate of v along the normal ray.

Suppose first that the unit normal cone of F at q is a singleton. Take a shortest segment from q to \Sigma. Its initial direction is an outward normal to F: its negative is the gradient of a smooth distance upper support, and the superlevel inequality places the initial direction in the polar tangent cone. It must therefore equal v. Its endpoint z\in\Sigma is accessible and smooth. Figure 2 shows the order of the two offsets along this segment. Enlarging D by distance s-\tau touches \Sigma at z. The enlargement stays on the \Omega side, since every point of D has distance at least s-\tau from \Sigma.

Let C_R\ge0 bound the negative part of the ambient Ricci curvature on the fixed neighborhood. Along the touching geodesic, the normal Riccati equation gives

First-contact comparison with the accessible part of \Sigma gives H(\text{enlarged }D\text{ at }z)\ge\gamma.

Here is the endpoint argument if the enlargement is focal at z. Localize D near p and shrink it strictly away from p, preserving the touching segment, so that p is the unique terminal footpoint. In tangent coordinates with outward normal in the positive vertical direction, replace its boundary graph u by u-a|x|^2, where a>0 can be arbitrarily small. This shrinks the domain and increases its outward second fundamental form at contact by 2a times the tangent metric. The boundary index form of the original minimizing normal segment is nonnegative; the perturbation adds 2a|J(0)|^2 for each variation field J. Variations with J(0)=0 have positive index because the segment is shorter than the ambient conjugacy scale. Thus the perturbed segment is nonfocal through its endpoint. Apply the Riccati comparison to the perturbed domain and let a\downarrow0. Its mean curvature at contact tends to that of D, proving also in this case that

Suppose next that the normal cone contains a direction not parallel to v. Convexity supplies a nontrivial one-sided angular arc through v. Exponentiating this arc at radius \tau produces a curve in \Gamma_\tau. The test boundary has the same first derivative at contact. Second-order comparison along the arc, even when it is one-sided, therefore gives a principal curvature at least 1/(2\tau) for small \tau. The exterior supporting ball gives a lower bound -C_F for all the other principal curvatures, independent of \tau for this fixed F. Hence

This exceeds any fixed lower bound when \tau is sufficiently small. The pointedness proved above excludes the remaining apparent possibility of opposite normals without an angular arc.

Choose the regular value s<\varepsilon sufficiently small that C_Rs<\varepsilon/4. Then choose \tau sufficiently small compared with s, the reach of this fixed F, and its curvature constants. The resulting C^{1,1} hypersurface has outward mean curvature at least \gamma-\varepsilon/4 in the barrier sense. At almost every twice-differentiability point of a C^{1,1} graph, quadratic test functions give the same inequality for its classical almost-everywhere mean curvature. The order of these choices is important: neither the reach nor C_F is asserted to remain uniform as s\downarrow0.

Smooth graph approximation. We use the following direct fact. A compact cooriented embedded C^{1,1} hypersurface with almost-everywhere outward mean curvature at least c has smooth embedded approximations converging in C^1 whose mean curvature is at least c-o(1).

To prove it, choose a smooth ambient vector field V uniformly transverse to the hypersurface, by smoothly approximating its continuous unit normal near the compact hypersurface. The C^1 inverse function theorem and compactness give a fixed flow tube for V, in which each sufficiently short flow line meets the hypersurface exactly once. Local graph convolution and a partition of unity give a smooth C^1 approximation \Gamma_0 sufficiently close to lie in this tube and remain uniformly transverse to V. Projection along these fixed flow lines is a global diffeomorphism. The original hypersurface is therefore the flow graph of a C^{1,1} function u over \Gamma_0. This fixed transverse flow avoids any assumption about the normal injectivity radius of an arbitrary C^1 approximation.

In local coordinates its mean-curvature operator has the form

where the coefficients are smooth on a compact set of allowed graph jets. The sign of A depends on the graph convention and is immaterial for the following averaging argument. Choose a finite smooth atlas with subordinate partition of unity \chi_\alpha and smooth u using normalized nonnegative convolution kernels \rho_{\alpha,\zeta}. The functions u_\zeta converge to u in C^1 and have uniformly bounded second derivatives. Their second derivatives equal the partition-weighted averages of D^2u plus a uniformly vanishing term. Indeed, differentiating the partition introduces differences u_\zeta-u and Du_\zeta-Du, which tend uniformly to zero, while the sums of the derivatives of the partition vanish.

Because D^2u is uniformly bounded, the coefficient replacement inside those averages,

gives a uniformly vanishing commutator. The lower-order coefficients converge uniformly as well. Coordinate changes and connection terms are included in B; their additional smoothing errors are controlled by the same C^1 convergence. Consequently

uniformly in x, proving the asserted approximation.

Apply this construction to the fixed hypersurface \Gamma_\tau. Its positive distance from \Sigma ensures that a sufficiently small C^1 perturbation remains inside \Omega. Choose the region on the same side of the smoothed boundary that contains the deep part of \Omega. It differs from O_\tau only in a tubular neighborhood of \Gamma_\tau. Choose the width of this neighborhood smaller than both its distance from \Sigma and \varepsilon-s. The removed collar is then contained in the \varepsilon-collar of \Sigma. Finally choose \zeta so small that the mean-curvature loss is less than \varepsilon/2. The resulting region is \Omega_\varepsilon, and its mean curvature is at least \gamma-\varepsilon. This proves the proposition.□

Proof. Work on the connected component containing the chosen end. Choose a positive smooth function \rho, equal to r^{-n-\beta} on each distant end, with 0<\beta<1. The global Sobolev inequality on a manifold with finitely many Euclidean ends, followed by exhaustion, gives the positive solution of

More explicitly, \rho\in L^{2n/(n+2)}, so the energy completion of compactly supported functions gives a weak solution by the coercive Dirichlet form. Positivity follows by testing the negative part. The local elliptic estimates and end barriers give a bounded smooth solution; the homogeneous decaying maximum principle gives uniqueness. On each harmonically flat end, Uv satisfies a Euclidean Poisson equation with right side -U^{(n+2)/(n-2)}\rho. The Newton potential, split into inner, comparable, and outer annuli, gives

Equivalently, subtract the Newton kernel’s monopole and use the finite \beta' moment of the right-hand side. The contribution of the finite inner boundary is a decaying harmonic function with the same type of expansion. Scaled interior estimates give all fixed orders of derivatives. Thus (1+t v)^{4/(n-2)}g satisfies Equation (8.1), is complete, and has strictly positive scalar curvature for every t>0, since

Apply the cited positive mass theorem and let t\downarrow0. The ADM mass changes by 2tA at the chosen end, which proves the claim.□

Proof. If \Gamma is empty, its capacity is zero and the assertion is Lemma 8.1. Assume henceforth that \Gamma is nonempty. Let \phi=1-w, where w is the potential in Equation (7.1), and write c=\mathfrak c(\Gamma,h). On the doubled manifold define \psi=\phi on the first copy and \psi=-\phi on the second. In signed normal distance across the seam, \psi and its first derivative agree: its value is zero and both normal derivatives equal \partial_\nu\phi. For 0<a<1, put

where \bar h is the doubled corner metric. The factor is bounded below by a, tends to one at the first end and to a at the second, and is weakly harmonic. Consequently h_a is complete, has two harmonically flat ends after constant rescaling at the second, and has nonnegative scalar curvature away from the seam.

Choose the common transverse normal from the second copy to the first. The two mean curvatures of \bar h with this normal are -H_\nu and H_\nu. Their jump in the corner condition is therefore -2H_\nu\ge0. The normal derivative of V_a is continuous, so its contribution cancels in the conformal jump formula; the jump for h_a has the same sign. We now explain the analytic smoothing step to avoid extending the dimensional or topological statement of a corner theorem without justification.

Apply the local Gaussian-collar construction of [26, Section 3, Proposition 3.1 and Equation (36)]. Its tensor convolution uses the coorientation of this collar and is independent of the topology or number of ends outside it. It gives uniformly comparable C^2 metrics, equal to h_a off a shrinking collar, whose negative scalar curvature has a uniform bound. For each fixed collar parameter, approximate this metric in C^2 by a smooth metric, keeping it unchanged outside a slightly larger collar. Scalar curvature is continuous in the metric’s C^2 jets, so choosing the approximation sufficiently close preserves the negative-scalar bound up to a fixed enlargement. Denote these smooth metrics by h_{a,\delta}. Their negative scalar curvature is supported in a collar whose volume tends to zero. Choose a smooth q_\delta\ge(R_{h_{a,\delta}})_-, uniformly bounded and supported in a slightly larger shrinking collar. Solve

Here is the required existence and mass estimate on this two-ended manifold. Put a_n=4(n-1)/(n-2), 2^*=2n/(n-2) and p=2n/(n+2). A fixed complete metric with these ends satisfies \|f\|_{2^*}^2\le S_a\int|df|^2: combine the Euclidean end inequalities with a compact-core Poincaré estimate having trace on one fixed end annulus. Uniform metric comparison gives the same inequality for h_{a,\delta}, uniformly in small \delta. Since \|q_\delta\|_{n/2}\to0, the form

is coercive on the homogeneous energy completion of compactly supported smooth functions. Solving B_\delta(u_\delta,f)=\int q_\delta f gives

Testing the negative part gives u_\delta\ge0. Local elliptic regularity and the harmonic exterior equation give a smooth solution z_\delta=1+u_\delta\ge1, tending to one on both ends; the end limits also follow from u_\delta\in L^{2^*}. The corrected smooth metric z_\delta^{4/(n-2)}h_{a,\delta} has nonnegative scalar curvature. It is still harmonically flat at both ends. In separately normalized end coordinates write u_\delta=A_{i,\delta}r_i^{2-n}+O_j(r_i^{1-n}); the coefficients are nonnegative because u_\delta\ge0. Flux integration over both ends gives

Thus each coefficient tends to zero, and each end mass converges to that of h_a. Lemma 8.1 proves nonnegativity of this limiting mass. This argument is local at the seam and accommodates both ends; it does not invoke the one-ended global statement of [26, Theorem 1] outside its stated class.

On the retained end,

so the conformal ADM formula gives

Let a\downarrow0. Both ends remained complete for every application of the smooth positive mass theorem; no theorem about a compactified point was used.□

Proof. Fix a slice with normalized metric h, and write m,c,w for its mass, capacity, and capacitary potential in normalized end coordinates. For \lambda>0 small set p=1+\lambda w and h_\lambda=p^{4/d}h. Conformal harmonicity and the end expansions give

This is the earlier separating perturbation with \lambda=\varepsilon/(1-\varepsilon). Its detached smooth enclosure \Gamma has H_\nu<0. Capacity monotonicity and Lemma 8.2, applied to its smooth exterior, give

Sending \lambda\downarrow0 proves m\ge c on the original time slice.

For the evolution formula, write the initial end as g_0=H^{4/d}\delta, where H>0, \Delta_\delta H=0, and H\to1. The products Hu_t and Hv_t are Euclidean harmonic on the end. Expanding them there and dividing by H gives, for any fixed 0<\beta<1,

Flux identification and normalization of the end coordinates give

Since B is locally absolutely continuous, so is m, and differentiation gives m'=-2\mu almost everywhere.□

Proof. The integrated mass identity. Since B'=b and B(0)=0,

Consequently

This supplies integrability of the nonnegative \mu. Set a=m-2\mu=2c_t-m. Nesting and the maximum principle make e^tv_t nondecreasing: the later potential is zero on the larger filled set and both have limit -1 at infinity. Its end coefficient e^tb(t) is therefore nondecreasing. Since

the same holds for e^{2t}a(t). This implies, distributionally, Da+2a\,dt\geq0. Substitution of the integral identity gives monotonicity of 3m-2\mu+2\int_0^t m (up to a constant). Thus for h>0,

This one-sided slope bound and integrability imply \mu(t)\to0: a value at least \varepsilon>0 forces an interval immediately before it on which the integral is bounded below by a positive constant depending only on \varepsilon,m(0). If m(0)=0, nonnegativity and mass monotonicity give m=0; the same slope bound excludes positive point values of \mu.

The escaping-point ball. Let a=e^{2/d} and R(s)=R_{\max}e^{2s/d}. Suppose that containment holds for 0\le s\le T but there is a point p\in\Sigma_{T+1} with |p|\geq aR(T). Use the ball

Here the balls are Euclidean balls in the original end coordinates. Choose R_{\max} so the indicated exterior avoids the initial obstacle and \tfrac12\delta\leq g_0\leq2\delta there.

To estimate the metric on this ball, use its total harmonic conformal factor. Write g_0=H^{4/d}\delta on the initial end, where \Delta_\delta H=0 and H\to1. Then Hv_s is Euclidean harmonic outside \Sigma_s. If \Sigma_s\subset B_{R(s)}, its boundary value on S_{R(s)} is nonpositive and its limit at infinity is -e^{-s}, so

Using v_s\leq0 on the remaining time interval gives, for r=|y|\geq R(T),

Choose H-1\leq C_0r^{-d} and R_{\max}^d\geq C_0. The last expression is at most 2e^{-T}. Taking also H\geq1/2 gives the concrete bounds

Every finite-perimeter variation supported in B_\rho(p) remains an admissible enclosure. Comparison of area elements therefore makes \Sigma_{T+1} Euclidean perimeter-quasiminimizing there, with ratio

The isoperimetric proof of the density estimate, using the competitors that fill or delete a ball, applies at every measure-theoretic boundary point, including a singular point. It yields

It follows that the fixed area satisfies

All time factors cancel. The constants are independent of T and of further enlargement of R_{\max}, giving the required contradiction. To cover all times, enlarge R_{\max} to obtain containment on [0,1] from the finite-time support bound. If containment holds on [0,N], apply the argument at each t\in[N,N+1] with T=t-1. This proves Equation (10.1) at every real time.□

Proof. Rescale the metric by R^{-2} and the coordinates by R^{-1}. The harmonic coordinate radius becomes one, and the mass becomes m/R^d. We apply the argument of [23, Theorem 1.3] at this scale. Its formal hypothesis concerns every complete asymptotically flat metric on the underlying manifold; we verify that the positive-mass statements used in its proof follow here from Lemma 8.1. The one-end assumption also identifies the mass flux in Lee’s Lemma 2.1 with the flux in the divergence integral over the entire manifold.

The scalar-flat reduction of [23, Lemma 2.5] gives a metric k=f^{4/d}h with f>0 and f\to1. The factor is bounded above and below by positive constants, so k remains smooth and complete. Its end factor Zf is Euclidean harmonic. For this scalar-flat reference metric, Lee uses the variations

where the cutoff \zeta is compactly supported and \widehat k_s is scalar-flat. Outside the support of \zeta, the product of the old end factor and u_s is Euclidean harmonic. The conformal scalar-curvature identity gives

Both scalar-curvature coefficients in the operator definition and inhomogeneous equation preceding Lee’s Lemma 2.4 must have this value, as in the earlier conformal equation in that paper. Lemma 2.4 estimates \Delta_k, and the ensuing perturbation argument applies with the corrected fixed dimensional coefficient. Its exterior estimates give u_s-1=O(|s|) uniformly. On the compact region away from the perturbation, u_s-1 is harmonic, so the maximum principle extends this bound throughout the manifold. For sufficiently small |s|, 1/2\le u_s\le3/2. The perturbed and conformally corrected metrics are therefore smooth and complete.

The scalar-flat argument uses nonnegativity of the varied mass at a small negative parameter [23, proof following Lemma 2.2]. This follows from Lemma 8.1, since every varied metric has a harmonic end outside a compact set. The same lemma gives m_{\rm ADM}(k)\ge0 after the initial scalar-flat reduction. Lee’s Lemma 2.5 gives m_{\rm ADM}(k)\le m_{\rm ADM}(h) and controls the end-factor difference by the mass difference. Thus all the required positive-mass inputs hold, without a spin or orientability assumption. The constants in Lee’s estimates depend only on dimension and the target error at the normalized radius, not on the compact interior. Restoring the scale R gives the asserted implication.□

Proof of Proposition 11.1. We use the following elementary harmonic estimate. If q\ge0 is Euclidean harmonic on \{r>R\}, tends to zero, and has expansion q=b r^{-d}+O(r^{-d-1}), then

Indeed, its spherical average is exactly b r^{-d}, and the Harnack inequality on scaled annuli bounds its maximum on a sphere by a dimensional constant times this average. In particular, the constant does not depend on the geometry inside B_R.

A smooth reduction from two ends to one. Let (N,k) be a smooth complete connected boundaryless manifold with exactly two harmonically flat ends and compact end complement, and suppose R_k\ge0. Designate one end for retention. There is a harmonic function \chi tending to one on that end and zero on the other, with 0<\chi<1. To construct it, choose a smooth function constant with these values outside a compact set and minimize its Dirichlet energy after addition of the homogeneous energy space. The global Sobolev inequality gives the solution; truncation and the strong maximum principle give its bounds. Exterior elliptic estimates give the stated end limits and expansions.

Write k=Z_i^{4/d}\delta on either end. Then Z_i\chi is Euclidean harmonic. On the end to be filled,

Positivity of b_2 follows by comparison with a positive multiple of r^{-d} from any fixed coordinate sphere. Under inversion x=y/|y|^2, the factor

extends harmonically across y=0 with value b_2. Thus \chi^{4/d}k extends to a smooth metric after adding one point. The filled manifold is complete, has one end, and has nonnegative scalar curvature, since R_{\chi^{4/d}k}=\chi^{-4/d}R_k away from the added point and is zero near that point.

At the retained end write \chi=1-A r^{-d}+O_j(r^{-d-1}). If M=m_{\rm ADM}(k), the filled metric has mass M-2A. The function Z_1(1-\chi) is nonnegative Euclidean harmonic with monopole A. Consequently

where the first inequality uses Lemma 8.1 on the smooth filled manifold. The estimate depends only on the retained-end coordinate radius and mass; the second end may vary with the compact interior.

A common coordinate tail. Use the normalized metric G_t and total harmonic factor F_t from Section 9. In this subsection, \Sigma_t,w_t denote its frontier and capacitary potential in these coordinates; the latter is the pullback previously written w_t\circ\Phi_t. A coordinate ball includes the compact core when the whole exterior is under discussion. The mass, capacity, and deficit remain m(t), c_t, and \mu(t)=m(t)-c_t. The preceding radius bound and harmonic estimates give \Sigma_t\subset B_{R_c}, the factors F_t are uniformly bounded on a fixed sphere outside B_{R_c}, and

Indeed, take T=t-1 in the radius estimate: U_t\le4e on the normalized tail r\ge R_{\max} for t\ge1. The initial bounded time interval is covered by the finite-time flow bounds. Multiplication by the uniformly bounded initial harmonic factor gives the asserted bound for F_t. Increasing R_c once includes that sphere. Let w_t be the capacitary potential equal to one at \Sigma_t and zero at infinity.

For 0<\lambda<1 put p=1+\lambda w_t and h_{t,\lambda}=p^{4/d}G_t. The detachment and smoothing construction above provides a smooth enclosing boundary \Gamma with H_\nu^\Gamma<0. It can be chosen in a ball B_{R_1} whose radius is independent of t and \lambda. Here is the confinement argument. On the common coordinate end the positive harmonic factor F_tp has a uniform bound on S_{R_c} and tends to one. The maximum principle and scaled gradient estimates give

Thus all coordinate spheres beyond a fixed R_* have positive outward mean curvature in h_{t,\lambda}. Write K_t for the original conductor and L for its detached minimizing enlargement. The functional minimized by L is

Let X be the outward unit normal to the coordinate spheres. For almost every s>R_*, the finite-perimeter divergence formula on L\cap\{r>s\} gives

Both the perimeter and the nonnegative volume penalty decrease under clipping, strictly if the removed tail has positive volume. Minimality therefore confines L to B_{R_*}. The subsequent smooth boundaries lie in shrinking collars, so they lie in B_{R_1} for one fixed R_1>R_*. Fix such a smoothing index before doubling. Thus the coordinate tail is fixed even when the separation distance and smoothing parameters vary.

The capacity comparison and the completed end. Suppress t temporarily and put h=G_t, F=F_t, w=w_t, c=c_t, m=m(t), \mu=\mu(t). The potential and capacity of the original conductor for h_\lambda=p^{4/d}h are exactly

These identities follow from the harmonic conformal transformation formula and the end expansions. If w_\Gamma is the potential of \Gamma for h_\lambda and c_\Gamma its capacity, enclosure monotonicity and Lemma 8.2 give

On the common tail the function Fp(w_\Gamma-w_{\Sigma,\lambda}) is nonnegative Euclidean harmonic with monopole D_\Gamma. Its sign follows from the maximum principle on the exterior of \Gamma.

For 0<a<1 the positive two-ended doubling construction in Lemma 8.2 has retained end factor

and retained end mass

The Miao smoothing and its conformal correction produce smooth complete two-ended metrics with R\ge0 and retained factor Bz_\delta on \{r>R_1\}. In the notation of that proof, z_\delta\ge1, and

is Euclidean harmonic on the common tail with monopole A_\delta\to0. Their retained masses are M_\delta=m_B+2A_\delta\ge0.

The auxiliary Euclidean end factor already defined above is J=F(1-w/2), with t suppressed. Direct subtraction gives

Each of the three harmonic functions appearing on the right is nonnegative before its displayed sign, with monopoles respectively c, D_\Gamma, and c_\Gamma. Equation (11.2) therefore yields

The capacity difference controls the end-factor error independently of the distance between \Gamma and \Sigma_t.

For each t, choose \lambda_t=a_t=(t+2)^{-1}; choose the detachment and boundary smoothing parameters next; finally choose \delta_t so that A_{\delta_t}\le(t+2)^{-1}. The Sobolev constants used in the seam correction may depend on these fixed choices. This order suffices because Lee’s constants do not depend on the compact interior. The preceding bounds imply

Apply the one-end reduction (11.3) to each smooth completed metric. Its underlying double is connected because the original exterior is connected and \Gamma is nonempty; it is boundaryless with exactly two ends and compact end complement. Denote the resulting masses and harmonic end factors by \mathcal M_t and \mathcal Z_t. These smooth connected one-ended metrics have their harmonic coordinate end on the same \{r>R_1\}, and the first two assertions of the proposition follow from Equation (11.3). Lemma 11.2, followed by the end-factor comparison, now gives

for any one fixed R_2>\max(2,\ell_n)R_1. The exterior maximum principle also gives weighted convergence on a slightly larger fixed tail.

The normalization and the inner tail. The initial harmonic factor in the normalized coordinates is h_t=H\circ\Phi_t. Thus

The initial end expansion gives h_t-1=O_j(e^{-2t}r^{-d}) on any fixed exterior tail. Consequently the preceding conclusion implies W_t\to1 there.

It remains to extend the weighted convergence to each fixed radius s>R_c. The functions J_t are positive Euclidean harmonic on \{r>R_c\}, with spherical averages 1+\tfrac12\mu(t) r^{-d}. Their bounded masses therefore give uniform Harnack and elliptic bounds on each compact subannulus. Every subsequential harmonic limit equals one on the distant tail, and hence everywhere on \{r>R_c\} by unique continuation. Thus J_t\to1 uniformly on S_s. Applying the maximum principle to J_t-1 outside S_s yields

The same conclusion holds for W_t, since h_t\to1 with the stated decay. This proves Equation (11.1).□

Proof. The conformal flow preserves A and has decreasing mass m(t)\to M\ge0. Proposition 11.1 controls its auxiliary factor on a common tail. Put q=2(n-1)/d.

Harmonic normalization and characteristics. Recall the total harmonic factors F_t=h_tU_t, Z_t=h_tV_t, and J_t=h_tW_t from Section 9. The flow metric has factor F_t, and the auxiliary metric has factor J_t; their monopole coefficients are m(t)/2 and \mu(t)/2. The background factor h_t=H\circ\Phi_t contributes the initial-mass term: the r^{-d} coefficient of U_t is e^{-t}B(t), while that of F_t is m(t)/2=e^{-t}B(t)+e^{-2t}m(0)/2.

Set

The evolution and mass identities give

For y_s=e^{2(t-s)/d}x, the chain rule therefore gives

Here is the required quantitative bound for j_t. Equation (11.1) gives \sup_{r\ge s_0}r^d|J_t-1|\to0 for every fixed s_0>R_c. Choose s_0=2\max\{R_c,R_{\max},1\}. The Kelvin transform

extends harmonically across zero, has value K_t(0)=\mu(t)/2, and is uniformly small on |y|\le s_0^{-1}. The interior gradient estimate on the half ball therefore gives |j_t(x)|\le C\varepsilon r^{-d-1} for all sufficiently large t and r\ge2s_0. For any fixed 0<\beta\le1, this proves, for s\ge t_0 and r\ge2s_0,

Integrating the characteristic equation gives the explicit estimate

The first term tends to zero with the initial time t_0 fixed. First taking t\to\infty and then \varepsilon\downarrow0 proves the desired exterior convergence of the total factors. More explicitly, with M=\lim m(t),

Since h_t\to1 uniformly on any fixed exterior region, V_t=Z_t/h_t has the latter limit as well.

Boundary control on fixed annuli. Choose any sequence t_i\to\infty. We obtain a uniform boundary Hölder estimate away from the collapsing compact core; it will transfer the radial potential to a farthest limiting frontier point. Use the direct end coordinates x=e^{-2t_i/d}y on the end chart. In this coordinate region, denote the rescaled time-t_i exterior by E_i and its rescaled frontier by \Sigma_i. If the original end chart begins at radius R_{\mathrm{chart}}, represent the core by the closed ball of radius R_{\mathrm{chart}}e^{-2t_i/d}. Adjoin that ball to the rescaled boundary portion in the end and call the resulting compact set \mathcal C_i. The radius bound puts these compact sets in a fixed ball. They record the frontier and the collapsing core in Euclidean coordinates; enclosure comparisons will still use the original filled regions. On every fixed compact annulus away from the origin, the rescaled upper and lower conformal bounds give uniform comparison of G_{t_i} with \delta. More explicitly, for an annulus with inner radius a_0>0, fix

If |x|\geq a_0, then |y|=e^{2t_i/d}|x|\geq R(S). Since t_i\geq S+1 and u_t is nonincreasing in time, the radius proof directly gives

Together with H(e^{2t_i/d}x)\to1, these are the required uniform bounds on the total factor F_{t_i} and metric G_{t_i}. The rescaled initial obstacle is eventually disjoint from that annulus. Thus the enclosure property permits both filling and deleting local balls and gives a uniform Euclidean perimeter-quasiminimality ratio. The isoperimetric argument supplies, with constants independent of i, both-phase density

Here p_i lies in a smaller concentric annulus and \rho_0 is chosen so B_{4\rho_0}(p_i) stays in the annulus with the uniform estimates. For example, applying the deleting competitor to a phase of volume b(s) gives b(s)^{(n-1)/n}\leq Cb'(s); integrating this inequality gives the claimed lower volume bound. Apply the same argument to the other phase.

The backgrounds \bar G_{t_i} converge smoothly to \delta on the annulus. Hence the operators for V_i are uniformly elliptic there. Construct the background equilibrium potential in the original ambient manifold by minimizing Dirichlet energy subject to value one on the filled conductor and zero at infinity. Write \mathcal P_i for its pullback to the end chart. The homogeneous energy construction, or smooth conductor exhaustion and weak compactness, extends the potential by one across the conductor. Truncation gives 0\le\mathcal P_i\le1, and the variational inequality makes \mathcal P_i a weak supersolution. Thus q_i=1-\mathcal P_i has a W^{1,2}_{\mathrm{loc}} zero extension, is a weak subsolution, and satisfies 0\le q_i\le1. If M_i=\sup_{B_{4s}(p_i)}q_i, the function M_i-q_i is a nonnegative supersolution and equals M_i on a set of volume at least \eta s^n. The weak Harnack inequality gives

Iteration yields uniform boundary Hölder control

The variational construction of the flow potential gives q_i=-V_i on the exterior portion of the end chart; see Lemma 4.1 and [6, Lemma 3.2]. Thus the estimate controls |V_i| uniformly in i on each fixed annulus, including at singular frontier points.

The largest limiting radius and the area comparison. Take a Hausdorff limit \mathcal C of \mathcal C_i, and let O be the unbounded component of \mathbb R^n\setminus\mathcal C. Every compact subset of O eventually lies in the actual exterior: join its points to infinity by finitely many paths with positive distance from \mathcal C, then use Hausdorff convergence and the absence of a boundary crossing along those paths. The uniform annular bounds and interior elliptic estimates give a common subsequence of F_{t_i} and V_i converging on compact subsets of O. Their far-exterior limits identify both functions throughout O by unique continuation:

where M=\lim_{t\to\infty}m(t). The origin lies in \mathcal C and hence outside the continuation domain O. The boundary Hölder estimate implies V_\infty=0 at every point of \partial O\setminus\{0\}. Set R_* = \max\{|x|:x\in\mathcal C\}. If R_*>0, choose p\in\mathcal C with |p|=R_*. Points on the outward radial ray from p lie in O, so the boundary estimate and the displayed potential give

In particular M>0, and \mathcal C\subset\overline B_{r_\infty}.

If R_*=0, then \mathcal C=\{0\} and the same harmonic limit holds on every fixed annulus. Its sign V_\infty\leq0 forces M=0. For any fixed R>0, the normalized boundaries are eventually inside B_R, while the total factors converge to 1 on S_R by harmonic compactness and unique continuation. The minimizing-enclosure property would give A\leq\omega R^{n-1} for every R>0, contradicting A>0. Thus R_*>0 and M>0. The enclosing coordinate spheres in this argument also enclose the original compact core.

Finally, for any R>r_\infty, outer containment and the minimizing property give, after taking the subsequential limit,

Letting R\downarrow r_\infty yields

□

Proof of Lemma 2.1. We first establish existence and the equality of infima without using frontier regularity. We then use the local obstacle lemma proved below to identify the exterior.

Confinement and existence. On the coordinate end put Y=\nabla r/|\nabla r|_g. Its divergence is the outward mean curvature of a coordinate sphere, so (2.1) gives

for a sufficiently large R_0. For E\in\mathcal A and almost every R>R_0, the BV trace on S_R=\{r=R\} is defined and the perimeter measure gives S_R zero mass. Gauss–Green on the bounded region E\cap\{r>R\} yields

There is no flux at infinity. The first boundary integral is no larger than the removed perimeter, while the second is the area added by clipping E at S_R. Therefore, writing E_R=E\setminus\{r>R\},

The inequality is strict if the removed region has positive volume.

Fix R_1>R_0. For each competitor choose a good slicing radius in (R_0,R_1); the radius need not work for any other competitor. Clipping confines a minimizing sequence to the same compact truncation at R_1. The competitor O gives a finite perimeter bound. BV compactness and lower semicontinuity on a slightly larger compact set now give a minimizer. Containment of O and absence from the specified exterior tail pass to the L^1 limit. Since clipping did not change the infimum, this is a global minimizer in \mathcal A, not a minimizer constrained by an artificial outer wall.

For any global minimizer, the strict part of (13.2) excludes positive volume beyond every good slicing radius above R_0. Choosing one such radius below a fixed R_2\in(R_0,R_1) confines every minimizer to the truncation at R_2. The same BV argument shows that any sequence of minimizers has an L^1-convergent subsequence. Its limit remains admissible, and lower semicontinuity together with minimality makes its perimeter equal to the common minimum. This proves the compactness assertion. The minimum is positive: a zero-perimeter indicator would be constant almost everywhere on connected M, whereas every admissible set contains the nonempty open set O^\circ and misses an end tail.

Recovery by full smooth cuts. A smooth exterior D determines the filled competitor

The identity includes boundary on B. Consequently the perimeter infimum is at most a_g(B).

For the reverse inequality, start with any E\in\mathcal A. Choose a smooth vector field supported in a collar of B and equal there to the normal pointing from O into X. Its flow \Phi_t satisfies \Phi_t(O)\supset O for small positive t, with a positive exterior collar between their boundaries. Hence E_t=\Phi_t(E) contains a neighborhood of O up to null sets. The tangential Jacobian of \Phi_t tends uniformly to one on the compact perimeter support, so P_g(E_t)\to P_g(E).

Fix t>0. Strict BV approximation, with the constant value one retained on a smaller neighborhood of O and zero retained outside a fixed compact set, supplies smooth functions u_j\in[0,1] such that

For completeness, smooth in finitely many coordinate charts only between these two constant regions and use a partition of unity; the usual strict approximation proof is local on precisely that region. Choose \delta_j\downarrow0 slowly enough that \|u_j-\mathbf1_{E_t}\|_{L^1}/\delta_j\to0. Coarea and Sard’s theorem give a regular value s_j\in(\delta_j,1-\delta_j) with

Also \|\mathbf1_{F_j}-\mathbf1_{E_t}\|_{L^1} \le\|u_j-\mathbf1_{E_t}\|_{L^1}/\delta_j. Lower semicontinuity thus gives P_g(F_j)\to P_g(E_t). Each F_j contains a neighborhood of O and has compact smooth boundary.

Retain the component of M\setminus\overline{F_j} containing the distant end, and denote its closure by D_j. This amounts to filling the bounded complementary components. The boundary of a compact smooth domain has finitely many components; the operation selects entire boundary components and introduces none. Thus D_j is a connected smooth exterior, closed in X, and its full intrinsic boundary lies in \operatorname{int}X, with

Let first j\to\infty, then t\downarrow0. This proves a_g(B)\le P_g(E) for every competitor and hence (2.4). Applied to a minimizer, the same argument and the lower bound a_g(B) show convergence of the resulting cut areas to its full perimeter. It also proves independence of the inner metric extension.

The connected exterior. Lemma 2.2, proved in the next subsection, identifies a closed representative with frontier T, whose singular part has zero \mathcal H^{n-1}_g measure and whose regular charts have exactly one filled side and one exterior side. The complement has one component containing the distant end. Any other component U is bounded, with \partial U\subset T. Since \mathcal H^{n-1}_g(T)=P_g(E)<\infty, the finite-boundary-measure criterion for perimeter gives U finite perimeter. On its regular boundary charts its outer normal is opposite to \nu_E. The singular part has zero perimeter measure, so the perimeter formula for unions gives

The open set U has positive volume and is bounded; its perimeter is positive, for otherwise its indicator would be constant on connected M. Filling it contradicts minimality. Thus the exterior is connected. This argument does not require the singular frontier to be a smooth manifold.

If n\le7, Lemma 2.2 makes T a compact embedded C^{1,1} hypersurface. A smooth field transverse to its continuous outward normal has a flow collar of T. A small positive flow displaces T into \operatorname{int}X and enlarges its filled side. In a smaller collar, smooth a C^1 defining function in C^1 while retaining positive derivative along the transverse field. Its smooth zero set is a small graph over the displaced frontier. The graph displacement extends to a collar isotopy, fixed on O and on the distant end. The exterior remains connected, so these are full cuts. Letting the smoothing error and then the flow time tend to zero gives the asserted C^1 and area convergence.

Finally every bounded finite-perimeter superset of a minimizer still contains O, so global minimality gives the claimed outer minimization directly.□

Proof of Lemma 2.2. The obstacle first converts constrained minimization to an ordinary almost-minimization estimate. Extend the outward unit normal of O to a smooth field Z with |Z|_g\le1 and bounded divergence. It suffices to work in a collar and to cut off farther away. For a local replacement F, the set F\cup O is admissible. Perimeter submodularity and Gauss–Green give, after canceling unchanged perimeter outside the replacement region V,

where \Lambda bounds |\operatorname{div}_gZ| on the region. To check the second inequality without assuming that O has finite total perimeter, take Z=\nu_O on \partial O\cap V and cut it off in a larger neighborhood. Since F\cap O and O agree outside a compact subset of V, Gauss–Green for their indicator difference gives

All the perimeters in this calculation are relative to V.

Insertion and deletion of small balls in (13.3), followed by the relative isoperimetric inequality, give the usual two-sided volume-density bounds and upper and lower perimeter-density bounds. They identify the frontier of the closed representative with the support of its perimeter measure. The codimension-one almost-minimizer regularity theorem gives a C^{1,\alpha} regular part and a singular set of dimension at most n-8; the Riemannian version uses the smooth area integrand in coordinate charts. The error in a radius-r ball is O(r^n), hence vanishes after rescaling perimeter by r^{1-n}. Blowups are therefore genuine Euclidean perimeter-minimizing cones. These are the density and dimension-reduction conclusions of [24, Chapters 21, 26, and 28]; for the smooth minimizing case see [34, Chapter 7, Theorem 5.8].

To see why no contact point is singular, take a tangent filled set at a point of \partial O. It contains the tangent half-space of O, and its boundary cone is supported in the complementary closed half-space. That boundary is a stationary multiplicity-one minimizing cone. Write k=n-1. On its spherical link the nonnegative coordinate height z satisfies, weakly, \Delta z=-(k-1)z. This identity follows by restricting linear coordinate functions to a stationary cone. The link is compact and stationarity includes its singular set, so testing the identity by the constant function gives 0=-(k-1)\int z. Since k-1>0, the height vanishes. Thus the boundary cone is the hyperplane bounding the half-space. Constancy and the fact that it is the boundary of a set give multiplicity one; the two density bounds exclude an empty boundary. Density-one regularity for almost-minimizers now gives a single contact graph.

Apply (13.3) to flows of both signs in such a graph cylinder. The resulting first-variation functional has a bounded weak mean curvature. Its graph function v solves a uniformly elliptic prescribed-mean-curvature equation with bounded right-hand side. The coefficients are smooth functions of position, v, and Dv, on the bounded gradient range supplied by C^{1,\alpha} regularity. Difference quotients first yield local W^{2,2} regularity. In nondivergence form the principal coefficients are C^{0,\alpha} and the right-hand side is bounded; the local linear W^{2,p} estimates give v\in W^{2,p} for every finite p. Sobolev embedding then gives C^{1,\alpha} for each \alpha<1. See [16, Chapter 9] for these elliptic estimates. Off the obstacle, all local replacements are admissible, so the weak mean curvature is zero; interior elliptic regularity gives smoothness and minimality on every regular patch.

For the stronger low-dimensional contact conclusion we apply the smooth-obstacle theorem directly: a pure-perimeter minimizer with a C^2 obstacle in ambient dimension below eight has C^{1,1} frontier, smooth away from contact [19, Regularity Theorem 1.3(iii), pp. 368–369]. The smooth metric and obstacle satisfy precisely those local hypotheses. Its graph functions are consequently in W^{2,\infty}. Finally the dimension estimate makes the singular set \mathcal H^{n-1}-null, while the regular frontier agrees with the reduced boundary up to such a null set. This proves the area identity.□

Proof of Lemma 2.3. Compactness and strict negativity give a signed-distance collar 0\le t\le2\varepsilon outside B on which Y=\partial_t obeys

For a good slicing level s\in(0,2\varepsilon) set O_s=O\cup\{0<t<s\} and W_s=O_s\setminus E. Gauss–Green, using a smooth extension of Y across B, gives

The second integral includes any original frontier at t=0. The perimeter added by adjoining O_s is the first boundary term; the removed perimeter is at least the second. Therefore

The enlarged set is admissible. Minimality forces W_s to have zero volume. For each minimizer choose one good level s\in(\varepsilon,2\varepsilon). The common smaller collar 0\le t<\varepsilon is then filled almost everywhere. The density bounds for the closed representative exclude any frontier point in that collar, so it lies in the interior of E.

The frontier is now separated from the obstacle. Every local perimeter replacement supported sufficiently near it is admissible, so it is locally perimeter minimizing there.

To express this local minimality in the language of integral currents, choose a coordinate ball U with compact closure disjoint from the obstacle, and orient that chart. Since E has finite perimeter, P_g(E;U)<\infty. Write C=\partial[\![E]\!] as a current in U. If Z is an integral (n-1)-cycle compactly supported in U, coning in the ball coordinates fills it by an integral n-current S compactly supported in U. A top-dimensional integral current has the form S=[\![k]\!], where k is integer-valued and belongs to BV(U); here k has compact support and \partial S=Z. Thus C+Z=\partial[\![\mathbf 1_E+k]\!]. The function c(t)=\min\{1,\max\{0,t\}\} is 1-Lipschitz and maps integers to \{0,1\}. Consequently c(\mathbf 1_E+k)=\mathbf 1_F for a finite-perimeter set F agreeing with E outside a compact subset of U. The contraction property of BV variation and set minimality give

Since P_g(E;U)=\mathbf M_g(C), this is the local integral-current minimizing property used by interior boundary regularity. The orientation is needed only in this chart; reversing it changes all current signs and leaves the mass comparison unchanged.

Applying ambient flows of both signs gives

for each smooth vector field V supported near T. Thus stationarity holds across the entire frontier, with no extra first-variation term at the singular set. Lemma 2.2 and interior minimizing-boundary regularity give the stated dimensional conclusions; see [34, Chapter 7, Theorem 5.8]. For the integral-current formulation and its singular-set dimension reduction, see [13, 14]. Detachment removes contact with the obstacle; the dimension restriction is what removes interior singularities.

Lemma 2.1 supplies connectedness and boundedness of the removed region, so the exterior retains exactly the original end. It is closed in the complete ambient manifold. To check completeness also for intrinsic Cauchy sequences, choose a subsequence whose successive intrinsic distances are less than 2^{-j} and join its successive terms by curves in D of lengths less than 2^{1-j}. Their concatenation has finite length and therefore an ambient endpoint. Closedness puts the endpoint in D. Appending it to the concatenated curve gives intrinsic convergence, since the remaining tail lengths tend to zero. The original Cauchy sequence then converges as well. When n\le7, the exterior is in addition a smooth manifold with boundary T.

Every bounded finite-perimeter enlargement of E remains a competitor containing O, proving outer minimization. In the smooth case any full cut of D determines precisely such an enlargement; all its components and portions on T contribute to its perimeter. The full-area identity follows from (2.4) and the zero (n-1)-dimensional measure of the singular set.□

Proof. We cut off the metric on a distant annulus and remove the resulting scalar-curvature error by a Dirichlet conformal correction. The annular flux then recovers the original ADM mass.

Step 1: the cutoff and Dirichlet correction. Choose a smooth radial cutoff \chi_R=\chi(r/R), with \chi=1 on (-\infty,1], \chi=0 on [2,\infty), and 0\le\chi\le1. On the end put

and set h_R=h on the compact core. For large R this is a smooth positive metric, equal to h on r\le R and Euclidean on r\ge2R. Extend \chi_R=1 on the compact core and define

Thus Q_R is supported in the annulus \mathcal A_R=\{R<r<2R\}. The differentiated metric bounds and the scalar-curvature formula give

The metrics h_R are uniformly comparable with h, so the Sobolev inequality

has a constant independent of large R on the homogeneous completion of C_c^\infty(\operatorname{int}N). This is the Euclidean-end Sobolev inequality combined with a compact-core Poincaré inequality with trace on one fixed end annulus. The completion imposes zero trace at \Gamma.

Since \|Q_R\|_{n/2}\to0, the bilinear form

is coercive, with \mathcal B_R(v,v)\ge (a_n/2)\int|dv|^2. Solve

Then

All integrals in these estimates use h_R. Local elliptic regularity makes v_R smooth up to \Gamma. On the Euclidean end it is harmonic and belongs to L^{2^*}; the scaled mean-value estimate implies v_R\to0 at infinity.

Step 2: uniform and compact convergence. We first prove a uniform supremum estimate by annular rescaling. Cover \mathcal A_R by a fixed number of balls of radius comparable to R in \{R/2<r<4R\}, and rescale each by R. The rescaled metrics have uniform ellipticity and bounded coefficients through order two; the rescaled zeroth-order term R^2Q_R is O(R^{-\tau}). The scalar interior boundedness estimate for this equation gives

The same bound holds on the closed source annulus by using these slightly larger balls. Off that annulus v_R is harmonic, has zero trace on \Gamma, and tends to zero on the end. The maximum principle on the inner and outer complementary regions therefore gives \|v_R\|_\infty\le CR^{-\tau} on all of N. In particular z_R>0 for large R. On each fixed compact region the equation for v_R is \Delta_hv_R=0 once R is large. Interior and fixed smooth boundary estimates, with v_R=0 on \Gamma, give v_R\to0 in C^2 there, and in every fixed higher norm.

Put \widehat h_R=z_R^{4/d}h_R. The conformal scalar formula now gives exactly

The global supremum estimate and \sup_N|h_R-h|_h=O(R^{-\tau}) give the stated metric comparison and completeness. The end is z_R^{4/d}\delta, with z_R positive Euclidean harmonic and tending to one, hence is harmonically flat to every order.

Step 3: the ADM limit and the boundary flux. Write

on that Euclidean end. Let \nu denote the h-unit normal at \Gamma pointing into N, and put I_R=\int_\Gamma\partial_\nu z_R\,dA_h. The boundary C^2 convergence gives I_R=o(1). Since h_R=h near \Gamma, integrating a_n\Delta_{h_R}z_R=Q_Rz_R over a large truncation and then letting its radius tend to infinity gives, with the outward normal of the exterior domain equal to -\nu at \Gamma,

The nonlinear factor is harmless:

For the remaining integral, write b=h-\delta and

On \mathcal A_R,

Indeed all nonlinear scalar terms and the change of volume density are bounded by C(|b_R||D^2b_R|+|Db_R|^2), where b_R=\chi_Rb. Their annular integral is O(R^{d-2\tau}). The outer linear flux is zero since b_R=0 there; its inner flux is the negative of the original ADM flux on S_R. Consequently

The last term tends to zero, by the scalar tail assumption (or by integrability and uniform metric comparison). Thus

As a_nd=4k, the flux identity yields

Finally H_\nu(\Gamma,\widehat h_R)\to H_\nu(\Gamma,h) uniformly by the compact C^2 convergence. Strict negativity therefore persists on the compact boundary.□

Proof of Theorem 1.1. The case 3\le n\le7 is the numerical Bray–Lee theorem described in Section 1, with the classical positive mass theorem when the frontier is empty. Assume n\ge8. Proposition 12.1 has proved the nonempty-frontier case for harmonically flat ends. We now let (E,h) have the general asymptotics of the theorem and full perimeter A>0.

The capacitary coefficient at infinity. Let w be its capacitary potential and c its capacity. The local minimizing density bounds and the weak Harnack contraction for the equilibrium potential, as in the proof of Lemma 4.1, give its constant extension the continuous boundary value one on the full frontier. For any 0<\alpha<\min\{\tau,1\}, the positive function r^{-d}(1-r^{-\alpha}) is h-superharmonic for all sufficiently large r: its Euclidean Laplacian is -\alpha(d+\alpha)r^{-n-\alpha}, while the metric error is O(r^{-n-\tau}). Comparison with a fixed positive multiple on a large coordinate sphere, followed by the maximum principle on the end, gives w=O(r^{-d}). Scaled interior estimates give Dw=O(r^{-d-1}), D^2w=O(r^{-d-2}). Consequently

Extend a cutoff of w, supported in its end chart, smoothly by zero to \mathbb R^n. Its Euclidean Laplacian has a finite \alpha-moment. The Newton potential, split into inner, comparable and outer annuli, is therefore a monopole plus O_1(r^{-d-\alpha}). Its difference from the extended function is a decaying harmonic function on all of \mathbb R^n, hence zero. The cutoff has reduced this uniqueness statement to Euclidean space. If the monopole coefficient is c_0, flux integration on \{w<s\}, for regular values s\uparrow1, gives

The flux at infinity is -d\omega c_0; the other boundary is the regular level w=s, and the buried boundary is absent. Monotone convergence and the capacity normalization identify c_0=c. Thus the finite-energy construction and this end argument give

In particular the conformal mass formula applies.

A strict smooth inner boundary. Fix a small \lambda>0 and set

Then h_\lambda\ge h, its asymptotically flat rate is \min\{\tau,d\}>d/2, and

Since 1\le p\le1+\lambda, scalar integrability and the stated scalar decay persist. The end expansion gives

To apply the static part of Corollary 6.2, write this perturbation as

The normalized harmonic factor equals one on the frontier and lies strictly between zero and one on the connected open exterior. Constant scaling preserves local perimeter minimality of the original filled set. Its smooth ambient metric supplies the minimizing-boundary density and regular-patch estimates, and the harmonic factor has the constant boundary data required by the separation argument. The asymptotically flat end supplies the positive-mean-curvature coordinate spheres used to confine the prescribed-curvature minimizer. Thus the static detachment conclusion applies to the original conductor. Proposition 7.2 then gives a smooth compact enclosing boundary \Gamma with H_\nu(\Gamma,h_\lambda)<0. Retain its connected end exterior. The metric h_\lambda is smooth on a neighborhood of its closure. All parameters producing \Gamma are fixed before approximation.

Comparison of all enclosing areas and passage to the limit. Apply Lemma 14.1 on this smooth exterior. Let k_j be its approximating metrics, and choose \epsilon_j\downarrow0 so that

Take a minimizing enclosure of the region removed by \Gamma in k_j. Lemma 2.1 gives compact existence, and Lemma 2.3 puts the entire minimizing frontier a positive distance from the strict smooth obstacle. The resulting frontier \Sigma_j is detached and locally perimeter minimizing, and is outer minimizing with all components counted. Lemma 2.1 gives its connected end exterior. It has precisely the boundary class of Theorem 1.1, a harmonically flat end and scalar curvature zero on that distant end.

Every such enclosure contains the original conductor. Its full perimeter in h is therefore at least A, by the original outer-minimizing property. This comparison applies directly to finite-perimeter enclosures, or follows from strict smooth approximation keeping a fixed neighborhood of the conductor. Tangent-plane metric comparison gives

Proposition 12.1 consequently gives

First let j\to\infty at fixed \lambda, and then \lambda\downarrow0. The mass formula proves

The strict boundary and elliptic constants may depend on the fixed \lambda and on \Gamma; this is consistent with the stated order of limits. The area estimate came from metric comparison and conductor inclusion.

The empty frontier. A nonempty locally perimeter-minimizing frontier has positive perimeter by the density lower bound. If A=0, the frontier is therefore empty. On that complete boundaryless manifold apply the same cutoff-density proof with \Gamma=\varnothing; there is then no compact-boundary flux. Applying Lemma 8.1 to the harmonic-flat approximants and passing their masses to the limit gives m_{\rm ADM}(h)\ge0.□

Proof. The proof has three parts. Absorbing the density into the metric gives bounded generalized mean curvature and strict blow-up compactness. Compactness and low-dimensional regularity then exclude cones with too many translation symmetries at every small scale. The quantitative stratification theorem converts this exclusion into the tube estimate.

The conformal metric absorbs the positive perimeter density: d\mathcal H_{g_*}^{n-1}=\widehat\rho\, d\mathcal H_g^{n-1}. Also f\,dV_{g_*}=\Phi\widehat\rho\,dV_g, so the displayed functional is exactly the corresponding weighted perimeter minus weighted volume functional. All estimates below are first made in g_*. Choose neighborhoods

Smoothness and positivity make their metric and forcing bounds uniform after working on a slightly larger compact neighborhood of \overline U_1.

First variation and local estimates. Let V=|\partial^*E| be the multiplicity-one boundary varifold, computed in g_*. Domain variation by any compactly supported smooth vector field Y gives

Here the divergence theorem for finite-perimeter sets is applied to fY. Thus V has bounded generalized mean curvature, with no additional first-variation measure on its singular set. In particular

The minimizing comparison also gives

for changes supported in W\Subset U_1. The usual ball comparisons and relative isoperimetric inequality for this bounded-volume-forcing functional give uniform local perimeter upper bounds and two-phase density lower bounds. In particular, for x in the perimeter support in \overline U_0 and all sufficiently small s, both phases occupy at least c s^n of B_s^{g_*}(x), and

These are the local estimates for almost-minimizing boundaries; they do not say that the boundary is Euclidean area minimizing in a fixed coordinate chart.

Strict blow-up compactness. Consider singular points x_j\in\overline U_0 and radii r_j\downarrow0. Choose g_*-orthonormal frames at x_j and write

The rescaled sets E_j=\Psi_j^{-1}(E) minimize P_{g_j}-\int f_j\,dV_{g_j} on domains exhausting \R^n. On every fixed ball g_j\to\delta and f_j\to0 smoothly. The preceding perimeter bounds give, after passage to a subsequence,

We need strict perimeter convergence as well, since L^1 convergence by itself could lose cancelling sheets.

Fix 0<a<b in a bounded coordinate ball. Coarea permits radii s_j\in(a,b) for which the traces exist, neither perimeter measure charges \partial B_{s_j}, and

Glue E_\infty inside B_{s_j} to E_j outside. The modification is compactly supported in a larger fixed ball. The BV gluing formula and minimality give

The last two terms tend to zero. Uniform convergence of the metrics compares their perimeters with Euclidean perimeter by factors 1+o(1), so

Let b\downarrow a at a radius with |D\chi_{E_\infty}|(\partial B_a)=0, and use lower semicontinuity. It follows that

The same argument on other relatively compact balls gives local perimeter-measure convergence. Together with D\chi_{E_j}\rightharpoonup D\chi_{E_\infty}, Reshetnyak continuity now gives boundary-varifold convergence: the unoriented tangent plane is a continuous even function of the measure-theoretic normal. Changing between the g_j and Euclidean varifold conventions introduces only vanishing metric errors. In particular there is no residual varifold mass from disappearing or cancelling sheets.

The limit is a Euclidean locally perimeter-minimizing boundary. Indeed, for F\triangle E_\infty\Subset B_a, repeat the same gluing with F inside B_{s_j}. On the gluing annulus F=E_\infty, so the trace error is unchanged. Letting j\to\infty and then b\downarrow a proves P(E_\infty;B_a)\le P(F;B_a) at continuity radii. Such a ball can contain any compact modification. Finally, the two-phase density bounds at x_j pass to every fixed rescaled ball centered at zero. Both phases of E_\infty have positive density there; in particular its boundary varifold is nonzero.

Uniform exclusion of highly symmetric cones. We claim that there are single constants \epsilon,R_0>0 such that, at every singular point in \overline U_0 and every physical scale 0<s<R_0, the rescaled varifold is not \epsilon-close to an (n-7)-symmetric comparison cone. Comparison is in the weak-varifold metric on B_1 used in quantitative stratification, with balls contained in the buffered neighborhood. Such a cone is dilation invariant about zero and translation invariant along an (n-7)-plane; it is not assumed to be a minimizing boundary.

If the claim fails, choose x_j,r_j as above and comparison cones C_j with distance less than 1/j on B_1. Denote their invariant (n-7)-planes by K_j and pass to K_j\to K in the Grassmannian. Closeness on the open ball gives a uniform mass bound for C_j on B_{1/2}, by using a cutoff equal to one there and supported in B_{3/4}. No mass bound at the boundary of the comparison ball is needed. Cone homogeneity propagates this interior bound to every fixed B_R. Radon compactness on the Grassmann bundle therefore gives a global varifold limit C, homogeneous about zero and translation invariant along K. The strict compactness just proved identifies it on the comparison ball:

There was no first-variation bound for the arbitrary C_j. Consequently it is this identification with the minimizing-boundary limit, not their mass bounds alone, that gives rectifiability, integrality and multiplicity one in B_1. Homogeneity extends these properties to C globally.

We next construct a boundary-set cone; we do not assert that the moving-center limit E_\infty is conical outside B_1. The radial direction is tangent almost everywhere to a rectifiable cone. Equation (A.4) therefore implies, on B_1\setminus\{0\},

The second expression includes the dimensional term. In polar coordinates the first identity says that the phase is constant along almost every radial segment. Extend those phase values radially to a conical set E_0\subset\R^n. It agrees with E_\infty almost everywhere in B_1. The homogeneous perimeter bound gives locally finite perimeter also at the origin: cutting out a ball of radius a introduces at most O(a^{n-1}) additional perimeter, which vanishes as a\downarrow0. Hence

first by agreement inside B_1, and then by homogeneity.

The conical set E_0 is globally locally perimeter minimizing. Given a compact modification, homothetically contract a ball containing its support into B_1. In that ball E_0 equals E_\infty, so the minimizing comparison is available. Undoing the contraction and using the (n-1)-degree scaling of perimeter proves the comparison at the original scale. Only the phase extension E_0, not the earlier limit outside B_1, is used in this scaling argument.

Splitting as a minimizing boundary. Translation invariance of the rectifiable varifold C along K makes every v\in K tangent almost everywhere. Thus

Distributional constancy in those directions and Fubini yield E_0=K\times F up to a null set, for a locally finite-perimeter conical set F\subset K^\perp. This preserves the phase information that would be absent from a statement about splitting stationary varifolds alone.

The factor F minimizes perimeter. To see this directly, let \ell=\dim K and G\triangle F\Subset D\subset K^\perp be a bounded compact modification. Over a cube Q_L\subset K of side L, replace F by G, retaining E_0 elsewhere. The product gluing formula and the minimality of E_0 give

There is no interface contribution at \partial D because the phases agree near it. The displayed lateral cost at \partial Q_L is essential. Divide by |Q_L| and let L\to\infty; its coefficient tends to zero. Thus P(F;D)\le P(G;D). Here \ell=n-7\ge1; if there are more invariant directions the residual dimension only decreases.

The residual ambient dimension is at most seven. The classical regularity theorem for minimizing boundaries therefore makes the minimizing cone F regular at its vertex [34]. A nontrivial boundary cone smooth at its vertex is a hyperplane, so F is a halfspace. Nontriviality follows from the two-phase density bounds. Consequently C is a multiplicity-one hyperplane through zero.

The strict boundary-varifold convergence to this plane, g_j\to\delta smoothly, and f_j\to0 now give flat-boundary regularity in a fixed smaller ball. One can use the metric-uniform almost-minimizing-boundary theorem or Riemannian Allard regularity; the prescribed-mean-curvature equation then bootstraps the graph to smoothness. This contradicts singularity of E_j at zero. The argument uses Riemannian regularity with converging smooth metrics, not an unsupported bound for Euclidean mean curvature obtained by changing coordinates. It proves the claimed uniform \epsilon,R_0, at every singular center in the patch and every scale below R_0.

The complete quantitative scale interval. Choose finitely many balls B_{\rho_i/2}^{g_*}(p_i) covering \mathcal S, with

They may be shrunk so that the rescaled ambient geometry has the sectional-curvature and injectivity-radius bounds in the cited Riemannian quantitative theorem. The first-variation bound and local perimeter bound already give its bounded generalized mean curvature and finite mass assumptions. The constants are finite on this fixed finite cover; no uniformity over unrelated metrics is asserted.

Apply [27, Theorem 1.3] to B_{2\rho_i}^{g_*}(p_i) after rescaling lengths by \rho_i^{-1}. Its target ball B_1 is B_{\rho_i}^{g_*}(p_i). For every singular point in that target ball and every rescaled scale 0<s<1, the comparison ball lies in B_{2\rho_i}^{g_*}(p_i) and has physical radius \rho_i s<R_0. Thus the same symmetry exclusion holds on the entire interval [r,1), for each 0<r<1. In the notation of quantitative stratification, the rescaled singular set in the target ball lies in

The stratum index is n-8 because the excluded cones have n-7 invariant directions. In particular this statement is stronger than a statement merely about tangent cones or a sequence of scales.

The quantitative theorem gives rescaled tube volume at most C_i r^8. Returning to physical variables, for 0<t<\rho_i/2 we obtain

The entire displayed tube lies in B_{\rho_i}^{g_*}(p_i), which is the target ball where the theorem controls its volume. For t<\min_i\rho_i/2, these finitely many tubes cover B_t^{g_*}(\mathcal S). Summing gives its Ct^8 bound. Finally, smooth metric equivalence on the buffered compact neighborhood compares small tubes and volume elements for g and g_*, proving Equation (A.1). Ambient dimension eight corresponds to stratum index zero and is included.□