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 (Bi and Zhu 2026b, 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 \[\|v_\sigma\|_{C^{1,\alpha}(\overline{E_\sigma}\cap V_0)}\le C, \qquad 0<\alpha<\theta.\] 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 \[\widetilde u_t\ge u_t,\qquad \widetilde u_t=u_t\quad\hbox{on }\overline{E_t},\qquad \|\widetilde u_t\|_{C^{1,\alpha}}\le C.\] 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 6, 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. ◻

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 (Wickramasekera 2014, 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 (Wickramasekera 2014, 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 \(D(y)=\mathop{\mathrm{dist}}_{\mathbb R^n}(y,\Sigma')\), let \(\mu_r\) denote the perimeter measure of \(\Sigma\cap B_r\), and define the fixed median \[ q(r)=\inf\left\{a\ge0: \mu_r(\{D\le a\})\ge\tfrac12\mu_r(B_r)\right\}. \tag{26}\] 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 \[f(r_i)\ge\tfrac12\sup_{0<s\le R_i}f(s).\] 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 \[ \frac{q(\rho r_i)}{q(r_i)}\le2\rho. \tag{27}\] Pass to a simultaneous cone subsequence at these radii, and put \[a_i=\frac{q(r_i)}{r_i}>0, \qquad \Sigma_i=r_i^{-1}\Sigma, \qquad \Sigma'_i=r_i^{-1}\Sigma'.\] 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 (25). 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 \[ \mathop{\mathrm{dist}}_{\mathbb R^n}(y,\Sigma'_i)=(1+o(1))\zeta_i(y) \tag{28}\] 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 (28) 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 \[L_i=A_i^{ab}\partial_{ab}+B_i^a\partial_a+c_i\] 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 \[ \|W_i\|_{L^p(K)}\le C_K \quad\hbox{for every }K\Subset\operatorname{Reg}C. \tag{29}\] 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 \[\widehat L_i\phi_i=-1,\qquad \phi_i|_{\partial B}=0.\] 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 \[\zeta_i\ge c\lambda_i\phi_i.\] Combining this with (29) yields \(\lambda_i/a_i\le C_K\). The assumed upper bound for \(|F_i|\) and a finite cover now give \[ \|F_i/a_i\|_{L^\infty(K)}\le C_K. \tag{30}\] 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 \[ W\ge0, \qquad W\in W^{2,q}_{\rm loc}(\operatorname{Reg}C) \quad(q<\infty), \qquad (\Delta_C+|A_C|^2)W\le0. \tag{31}\] The upper median set also retains positive measure on the fixed regular core. Equation (28) 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 (27), 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 (28) 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 \[ \inf_{\operatorname{Reg}C\cap B_\rho}W\le3\rho. \tag{32}\] 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 (28) 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 \[\int\eta^2|\nabla_C h|^2 \le2\int\eta(1-h)|\nabla_C h||\nabla_C\eta|,\] and hence \[ \int\eta^2|\nabla_C h|^2 \le4\int|\nabla_C\eta|^2. \tag{33}\]

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 \[ \int_C|\nabla_C\chi|^2 \le\sum_j\int_{C\cap B_{2\rho_j}(x_j)}|\nabla_C\theta_j|^2 \le C\sum_j\rho_j^{k-2}\longrightarrow0. \tag{34}\] 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 (33), 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 (34), 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 \[\|\varphi\|_\infty \|\nabla_C h\|_{L^2(\mathop{\mathrm{supp}}\varphi)} \|\nabla_C\chi\|_2\longrightarrow0.\] The other term converges by dominated convergence. Hence \[\int_C\langle\nabla_C h,\nabla_C\varphi\rangle\ge0\] 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 (Bombieri and Giusti 1972); the precise form needed here is stated in (De Silva and Jerison 2011, 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 \[\frac{1}{\mathcal H^k(C\cap B_r)} \int_{C\cap B_r}(h+\epsilon) \le C\operatorname*{ess\,inf}_{C\cap B_r}(h+\epsilon).\] The essential infimum of \(h\) is zero by (32). 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. ◻

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 \[\rho=\frac{a-1}{2}R(T),\qquad \overline{B_\rho(p)}\subset\{|y|>R(T)\}.\] 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 \[Hv_s(y)\leq e^{-s}\left[\left(\frac{R(s)}{|y|}\right)^d-1\right] \qquad (|y|\geq R(s)).\] Using \(v_s\leq0\) on the remaining time interval gives, for \(r=|y|\geq R(T)\), \[H u_{T+1}\leq H-1+e^{-T}+(e^T-1)R_{\max}^d r^{-d}.\] 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 \[e^{-T-1}\leq u_{T+1}\leq K e^{-T},\qquad K=4, \qquad r\geq R(T).\]

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 \[\gamma=2^{n-1}(eK)^q.\] 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 \[\mathcal H_\delta^{n-1}(\Sigma_{T+1}\cap B_\rho(p)) \geq c(n,\gamma)\rho^{n-1}.\] It follows that the fixed area satisfies \[\begin{split} A&\geq 2^{-(n-1)/2}e^{-q(T+1)} c(n,\gamma)\rho^{n-1}\\ &=2^{-(n-1)/2}c(n,\gamma)e^{-q} \left(\frac{e^{2/d}-1}{2}\right)^{n-1}R_{\max}^{n-1}. \end{split}\] 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 (43) at every real time. ◻

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, \[Z_2\chi=b_2r^{-d}+O_j(r^{-d-1}),\qquad b_2>0.\] 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 \[|y|^{-d}(Z_2\chi)(y/|y|^2)\] 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 \[ 0\le M-2A\le M,\qquad |Z_1-Z_1\chi|\le C(n)M r^{-d}\quad(r\ge2R), \tag{46}\] where the first inequality uses Lemma 16 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 \[0\le c_t\le m(t)\le m(0),\qquad \mu(t)\longrightarrow0.\] 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 \[|F_tp-1|+r|D(F_tp)|\le C(R_c/r)^d\quad(r\ge2R_c).\] 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 \[\mathcal F(L)=P_{h_{t,\lambda}}(L)+ \theta\int_{L\setminus K_t}h_0\,dV_{h_{t,\lambda}}, \qquad h_0\ge0 .\] 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 \[P(L\cap\{r<s\}) \le P(L)-\int_{L\cap\{r>s\}}\operatorname{div}X\,dV.\] 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 \[ w_{\Sigma,\lambda}=\frac{(1+\lambda)w}{p},\qquad c_{\Sigma,\lambda}=(1+\lambda)c,\qquad m_\lambda=m+2\lambda c. \tag{47}\] 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 17 give \[0\le D_\Gamma:=c_\Gamma-c_{\Sigma,\lambda} \le m_\lambda-c_{\Sigma,\lambda}=\mu+\lambda c.\] 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 17 has retained end factor \[B=Fp\left(1-\frac{1-a}{2}w_\Gamma\right)\] and retained end mass \[m_B=m_\lambda-(1-a)c_\Gamma \le \mu+\lambda c+a m_\lambda .\] 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 \[e_\delta:=B(z_\delta-1)\ge0\] 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 \[B-J= \frac{\lambda}{2}Fw -\frac12Fp(w_\Gamma-w_{\Sigma,\lambda}) +\frac a2Fp w_\Gamma .\] 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 (45) therefore yields \[ |Bz_\delta-J| \le C(n)\bigl(\mu+\lambda c+a m_\lambda+A_\delta\bigr)r^{-d}, \qquad r\ge2R_1 . \tag{48}\] 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 \[0\le M_{\delta_t}\longrightarrow0,\qquad \sup_{r\ge2R_1}r^d|B_tz_{\delta_t}-J_t|\longrightarrow0.\] Apply the one-end reduction (46) 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 (46). Lemma 21, followed by the end-factor comparison, now gives \[J_t\longrightarrow1 \quad\hbox{uniformly on }\{r>R_2\}\] 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 \[\bar G_t=h_t^{4/d}\delta,\qquad F_t=h_tU_t,\qquad J_t=h_tW_t,\qquad W_t=\tfrac12(U_t-V_t).\] 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 \[\sup_{r\ge s}r^d|J_t-1| \le s^d\sup_{S_s}|J_t-1|\longrightarrow0.\] The same conclusion holds for \(W_t\), since \(h_t\to1\) with the stated decay. This proves Equation (44). ◻

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 \[f_t=F_t-1-\frac{m(t)}2r^{-d},\qquad j_t=J_t-1-\frac{\mu(t)}2r^{-d}.\] The evolution and mass identities give \[\partial_t f_t=2(f_t-j_t)+\frac2d r\partial_r f_t.\] For \(y_s=e^{2(t-s)/d}x\), the chain rule therefore gives \[\frac{d}{ds}\bigl(e^{-2s}f_s(y_s)\bigr) =-2e^{-2s}j_s(y_s).\] Here is the required quantitative bound for \(j_t\). Equation (44) 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 \[K_t(y)=|y|^{-d}(J_t(y/|y|^2)-1)\] 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 \(r\ge2s_0\), \[|j_s(x)|\leq C\varepsilon r^{-d-\beta},\qquad |f_{t_0}(x)|\leq C_0r^{-d-\beta}.\] Integrating the characteristic equation gives the explicit estimate \[|f_t(x)|\leq \left[C_0e^{-2\beta(t-t_0)/d} +\frac{dC}{\beta}\varepsilon \bigl(1-e^{-2\beta(t-t_0)/d}\bigr)\right]r^{-d-\beta}.\] 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 \(M=\lim m(t)\), \[F_t\longrightarrow1+\tfrac M2r^{-d},\qquad Z_t=F_t-2J_t\longrightarrow-1+\tfrac M2r^{-d}.\] 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 \[L\geq\max\left\{1,\frac d2\log\frac{R_{\max}}{a_0}\right\}, \qquad S=t_i-L\geq0.\] 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 \[e^{-t_i}\leq u_{t_i}(y)\leq u_{S+1}(y)\leq4e^{-S}, \qquad 1\leq U_{t_i}(x)\leq4e^L.\] 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 \[|B_s(p_i)\cap E_i|\geq\eta s^n, \qquad |B_s(p_i)\setminus E_i|\geq\eta s^n, \qquad p_i\in\Sigma_i,\quad 0<s<\rho_0.\] 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 \[\sup_{B_s(p_i)}q_i\leq(1-\theta) \sup_{B_{4s}(p_i)}q_i, \qquad 0<\theta<1.\] Iteration yields uniform boundary Hölder control \[|q_i(x)|\leq C\left(\frac{|x-p_i|}{\rho_0}\right)^\kappa \quad (|x-p_i|<\rho_0),\qquad \kappa>0.\] The variational construction of the flow potential gives \(q_i=-V_i\) on the exterior portion of the end chart; see Lemma 6 and (Bi and Zhu 2026c, 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: \[F_\infty(x)=1+\frac M2|x|^{-d},\qquad V_\infty(x)=-1+\frac M2|x|^{-d},\] 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 \[0=-1+\frac{M}{2R_*^d},\qquad R_*=r_\infty:=(M/2)^{1/d}.\] 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, \[A\leq\omega R^{n-1} \left(1+\frac{M}{2R^d}\right)^q.\] Letting \(R\downarrow r_\infty\) yields \[A\leq\omega(2M)^{(n-1)/d},\qquad m(0)\geq M\geq\frac12 \left(\frac{A}{\omega}\right)^{d/(n-1)}.\] ◻

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 6, 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 \[\Delta_\delta w =(\delta^{ij}-h^{ij})\partial_{ij}w +h^{ij}\Gamma_{ij}^\ell\partial_\ell w =O(r^{-n-\tau}).\] 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 \[\int_{\{w<s\}}|dw|_h^2\,dV_h=s\,d\omega c_0.\] 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 \[0<w<1\ \text{in the open exterior},\qquad w=c r^{-d}+O_1(r^{-d-\alpha}),\qquad 0<\alpha<\min\{\tau,1\}.\] In particular the conformal mass formula applies.

A strict smooth inner boundary.

Fix a small \(\lambda>0\) and set \[p=1+\lambda w,\qquad h_\lambda=p^{4/d}h.\] Then \(h_\lambda\ge h\), its asymptotically flat rate is \(\min\{\tau,d\}>d/2\), and \[R_{h_\lambda}=p^{-4/d}R_h\ge0,\qquad R_{h_\lambda}\,dV_{h_\lambda}=p^2R_h\,dV_h.\] Since \(1\le p\le1+\lambda\), scalar integrability and the stated scalar decay persist. The end expansion gives \[m_{\rm ADM}(h_\lambda)=m_{\rm ADM}(h)+2\lambda c.\] To apply the static part of Corollary 13, write this perturbation as \[h^{\rm base}_\lambda=(1+\lambda)^{4/d}h,\qquad \widehat u_\lambda=\frac{1+\lambda w}{1+\lambda},\qquad h_\lambda=\widehat u_\lambda^{4/d}h^{\rm base}_\lambda.\] 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 15 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 23 on this smooth exterior. Let \(k_j\) be its approximating metrics, and choose \(\epsilon_j\downarrow0\) so that \[k_j\ge(1-\epsilon_j)h_\lambda,\qquad H_\nu(\Gamma,k_j)<0,\qquad m_{\rm ADM}(k_j)\longrightarrow m_{\rm ADM}(h_\lambda).\] Take a minimizing enclosure of the region removed by \(\Gamma\) in \(k_j\). Lemma 2 gives compact existence, and Lemma 4 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 gives its connected end exterior. It has precisely the boundary class of Theorem 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 \[\mathcal H_{k_j}^{k}(\Sigma_j) \ge(1-\epsilon_j)^{k/2} \mathcal H_{h_\lambda}^{k}(\Sigma_j) \ge(1-\epsilon_j)^{k/2} A .\] Proposition 22 consequently gives \[m_{\rm ADM}(k_j)\ge \frac12\left( \frac{(1-\epsilon_j)^{k/2}A}{\omega} \right)^{d/k}.\] First let \(j\to\infty\) at fixed \(\lambda\), and then \(\lambda\downarrow0\). The mass formula proves \[m_{\rm ADM}(h)\ge \frac12(A/\omega)^{d/k}.\] 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 16 to the harmonic-flat approximants and passing their masses to the limit gives \(m_{\rm ADM}(h)\ge0\). ◻