Proof. Write \(\nabla=\nabla_\Sigma\) and \(\beta=Nu\). The restriction formula is \[ \Delta_\Sigma u=\mathop{\mathrm{tr}}_{T\Sigma}\mathop{\mathrm{Hess}}_g u-mh\beta. \tag{22}\] This holds both almost everywhere and weakly: in a \(C^{1,1}\) parametrization, the induced divergence expression has Lipschitz coefficients, and the restricted smooth ambient function has Lipschitz first derivatives. The ordinary chain rule therefore gives the same formula as the distributional divergence.

Taking the tangential trace in Lemma 4, and using \(|\nabla_g u|^2=|\nabla u|^2+\beta^2\), gives \[2\Delta_\Sigma u \le -(m-2)|\nabla u|^2-m(b^2+f^2)-m\beta^2-2mh\beta.\] Equivalently, \[m(h^2-b^2)\ge mf^2+m(h+\beta)^2+(m-2)|\nabla u|^2+2\Delta_\Sigma u.\] Multiply by \(v^2\) and integrate by parts. Since \(c\alpha=2\) and \(c\alpha^2=m-2\), the gradient terms become \[c|\nabla v|^2-4v\langle\nabla v,\nabla u\rangle +(m-2)v^2|\nabla u|^2 =c|\nabla v-\alpha v\nabla u|^2.\] This proves the first inequality. The squared differential norm is the sum over an orthonormal tangent frame, so \(|d(\Phi|_\Sigma)|^2\le mf^2\) by Lemma 4, proving the second. Reversing a local normal reverses both \(h\) and \(\beta\); all expressions are therefore globally defined. Finally, all coefficients are bounded near \(\bar U\), so \(H^1_0(U)\) approximation proves the last assertion. ◻

Proof. The sharp Euclidean Sobolev inequality of Aubin and Talenti (Aubin 1976; Talenti 1976), in our normalization, is \[ c\int_{\mathbb R^m}|D\varphi|^2\,dx \ge Y_m\left(\int_{\mathbb R^m}|\varphi|^q\,dx\right)^{2/q}, \qquad \varphi\in\mathop{\mathrm{Lip}}_c(\mathbb R^m). \tag{24}\] Before multiplying by \(c\), the sharp constant is \(m(m-2)|\mathbb S^m|^{2/m}/4\). Normalize the metric at a chart center. In a sufficiently small chart, its matrix \(G\) satisfies \((1-\delta)\mathrm{Id}\le G\le(1+\delta)\mathrm{Id}\). Comparing inverse metrics and volume densities in (24) gives \[ c\int_\Sigma|\nabla_\Sigma\varphi|^2\,d\sigma \ge Y_m\left(\frac{1-\delta}{1+\delta}\right)^{m/2} \|\varphi\|_q^2 \tag{25}\] for chart-supported \(\varphi\). The gradient integral is at least \((1-\delta)^{m/2}(1+\delta)^{-1}\) times its Euclidean value, whereas the squared critical norm is at most \((1+\delta)^{m/q}\) times its Euclidean value, and \(m/q=(m-2)/2\).

Choose \(\delta\) so that the coefficient in (25) is at least \(Y_m-\epsilon\). Cover \(\bar U\) by finitely many such charts and choose chart-supported Lipschitz functions \(\chi_i\) with \(\sum_i\chi_i^2=1\) near \(\bar U\). They are obtained by normalizing a finite family of cutoffs by the square root of the sum of their squares near \(\bar U\), and then cutting off outside that neighborhood. The triangle inequality in \(L^{q/2}\) and the product rule give \[\begin{align*} \|v\|_q^2 &=\left\|\sum_i(\chi_i v)^2\right\|_{q/2} \le\sum_i\|\chi_i v\|_q^2,\tag{26}\\ \sum_i|\nabla_\Sigma(\chi_i v)|^2 &=|\nabla_\Sigma v|^2 +v^2\sum_i|\nabla_\Sigma\chi_i|^2. \tag{27}\end{align*}\] Summing (25) proves (23) first for compactly supported Lipschitz functions. Taking, for example, \(\epsilon=Y_m/2\) gives the continuous inclusion into \(L^q\), so approximation proves it for every \(v\in H^1_0(U)\).

For compactness, extend a bounded \(H^1_0(U)\) sequence by zero. Each \(\chi_i v_j\) is a bounded Euclidean \(H^1\) sequence in its chart, supported in a fixed compact subset. Rellich compactness gives an \(L^2\)-convergent subsequence in each of the finitely many charts. The identity \(v_j=\sum_i\chi_i(\chi_i v_j)\) then gives convergence in \(L^2(U)\). No boundary regularity of \(U\) is used. ◻

Proof. Write \(Y=Y(U)\ge0\), the last inequality following from Lemma 8. For a normalized minimizing sequence, Hölder’s inequality and \(h^2-b^2\ge-b^2\) give \[ \|v_j\|_2^2\le\sigma(U)^{2/m},\qquad c\|\nabla_\Sigma v_j\|_2^2 \le Q_\Sigma(v_j)+mb^2\sigma(U)^{2/m}. \tag{28}\] Thus, after a subsequence, \[v_j\rightharpoonup\psi\text{ in }H^1_0(U),\qquad v_j\mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF27F6> >> BDC}% PPaperOriginallongrightarrow\endcsname\pdfliteral direct{EMC}\endgroup}\psi\text{ in }L^2(U) \text{ and almost everywhere}.\] Set \(t=\int_U|\psi|^q\,d\sigma\in[0,1]\) and \(z_j=v_j-\psi\). The Brézis–Lieb lemma (Brezis and Lieb 1983, Theorem 1) gives \[ \int_U|z_j|^q\,d\sigma\mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF27F6> >> BDC}% PPaperOriginallongrightarrow\endcsname\pdfliteral direct{EMC}\endgroup}1-t. \tag{29}\] Weak convergence splits the quadratic form: \(Q_\Sigma(v_j)=Q_\Sigma(\psi)+Q_\Sigma(z_j)+o(1)\). The bounded potential and (23) give \[Q_\Sigma(z_j)\ge (Y_m-\epsilon)\|z_j\|_q^2 -(C_{\epsilon,U}+mb^2)\|z_j\|_2^2.\] Letting \(j\to\infty\) and then \(\epsilon\downarrow0\), we obtain \[ Y\ge Q_\Sigma(\psi)+Y_m(1-t)^{2/q} \ge Yt^{2/q}+Y_m(1-t)^{2/q}. \tag{30}\] If \(t=0\), this contradicts \(Y<Y_m\). If \(Y=0\), it forces \(t=1\). If \(Y>0\) and \(0<t<1\), strict concavity gives \(t^{2/q}+(1-t)^{2/q}>1\), and the right side is strictly greater than \(Y\). Thus \(t=1\) in every case. Lower semicontinuity of the gradient energy and strong \(L^2\) convergence of the potential term give \(Q_\Sigma(\psi)\le Y\); normalization gives the reverse inequality. ◻

Proof of Theorem 7. First let \(\Sigma\) be connected and boundaryless, and fix a nonempty open \(U\Subset\Sigma\) with \(\Sigma\setminus\bar U\ne\varnothing\). Suppose \(Y(U)<Y_m\), and take the normalized minimizer \(\psi\) supplied by Lemma 10. Write \(Y=Q_\Sigma(\psi)\).

Push the probability measure \(|\psi|^q\,d\sigma\), extended by zero, to \(M\) by the immersion. Its support is compact and it has no atoms. Indeed, an immersion is locally an embedding, so a fiber in the compact support meets each member of a finite embedding-chart cover in at most one point; every such fiber has zero area measure. Lemma 6 supplies fixed parameters \(p,a\) such that the coordinates \(w_1,\ldots,w_{n+1}\) of \(\Phi_{p,a}|_\Sigma\) satisfy \[ \int_\Sigma|\psi|^q w_j\,d\sigma=0,\qquad \sum_{j=1}^{n+1}w_j^2=1. \tag{31}\] These coordinates and \(u=u_{p,a}\) have bounded first derivatives near \(\bar U\). Multiplication by \(w_j\) or \(w_j^2\) preserves \(H^1_0(U)\).

Let \(B_Q\) be the symmetric bilinear form associated with \(Q_\Sigma\). For a bounded Lipschitz multiplier \(\zeta\) near \(\bar U\), differentiate the quotient along \(\psi(1+s\zeta)\) to obtain \[ B_Q(\psi,\psi\zeta) =Y\int_\Sigma|\psi|^q\zeta\,d\sigma. \tag{32}\] For small \(|s|\), the factor \(1+s\zeta\) is bounded away from zero, and differentiated mass integrands are dominated by a constant times \(|\psi|^q\). No regularity or positivity of the minimizer is needed.

Fix a coordinate \(w=w_j\) and put \(B_w=\int_\Sigma|\psi|^q w^2\,d\sigma\). Since \(|w|\le1\) and its weighted mean is zero, \[\|\psi(1+sw)\|_q^2=1+(q-1)B_ws^2+o(s^2).\] The numerator has zero linear term by (32); its second variation at the minimum therefore gives \[ Q_\Sigma(\psi w)\ge(q-1)YB_w. \tag{33}\] The product rule and the first variation with \(\zeta=w^2\) give \[ \begin{aligned} Q_\Sigma(\psi w) &=B_Q(\psi,\psi w^2) +c\int_\Sigma\psi^2|\nabla_\Sigma w|^2\,d\sigma\\ &=YB_w+c\int_\Sigma\psi^2|\nabla_\Sigma w|^2\,d\sigma. \end{aligned} \tag{34}\] Subtract and sum over the coordinates. Since \(c=q-2\) and \(\sum_jw_j^2=1\), the result is \[ \int_\Sigma\psi^2|d(\Phi_{p,a}|_\Sigma)|^2\,d\sigma\ge Y. \tag{35}\] Combining this with Lemma 8 forces \[ \nabla_\Sigma\psi=\alpha\psi\nabla_\Sigma u \quad\text{almost everywhere on }U. \tag{36}\]

Choose a compactly supported Lipschitz cutoff \(\chi\) on \(\Sigma\) equal to one near \(\bar U\). By (20), \[F=(\chi e^{-\alpha u})E_0\psi\in H^1(\Sigma)\] is the zero extension of \(e^{-\alpha u}\psi\), and (36) gives \(\nabla_\Sigma F=0\) almost everywhere on all of \(\Sigma\). The local Poincaré inequality in coordinate balls makes \(F\) constant almost everywhere on each ball. Overlapping balls have the same constant, so connectedness makes \(F\) constant on \(\Sigma\). It vanishes on the nonempty open set \(\Sigma\setminus\bar U\) and therefore vanishes everywhere, contradicting \(\|\psi\|_q=1\). Thus \(Y(U)\ge Y_m\).

Every proper compact support in a connected boundaryless \(\Sigma\) is contained in such a \(U\): choose a coordinate ball outside the support and finitely many relatively compact neighborhoods covering the support that miss a smaller ball. This proves the inequality unless the support is the entire compact component. In that case, let \(D_\delta\) be the image of a Euclidean ball of radius \(\delta\) in a fixed chart about a point. Remove that point with a cutoff \(\eta_\delta\) equal to zero in \(D_\delta\), equal to one outside \(D_{2\delta}\), and satisfying \(|\nabla_\Sigma\eta_\delta|\le C/\delta\). For Lipschitz \(v\), metric comparability gives \[\int_\Sigma|\nabla_\Sigma((1-\eta_\delta)v)|^2\,d\sigma \le 2\int_{D_{2\delta}}|\nabla_\Sigma v|^2\,d\sigma +C\|v\|_\infty^2\delta^{m-2}\mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF27F6> >> BDC}% PPaperOriginallongrightarrow\endcsname\pdfliteral direct{EMC}\endgroup}0.\] Also \(\eta_\delta v\to v\) in \(L^2\) and \(L^q\). The potential is bounded on the compact component, so the inequality for \(\eta_\delta v\) passes to \(v\). This is where \(m>2\) is needed.

Finally, pass to \(\Sigma^\circ\) if \(\Sigma\) has boundary. A compact support meets only finitely many connected components, because the components are open. If \(a_j=\int_{\Sigma_j}|v|^q\,d\sigma\), the connected result gives \[Q_\Sigma(v)\ge Y_m\sum_j a_j^{2/q} \ge Y_m\left(\sum_j a_j\right)^{2/q}.\] For \(v\in H^1_0(U)\), use its defining approximation: Lemma 9 gives convergence in \(L^q\), and boundedness of the potential near \(\bar U\) gives convergence of the quadratic form. This proves the final assertion. ◻

Proof. Choose a finite smooth partition on a neighborhood of \(K\), with each partition function compactly supported in a coordinate chart. The smooth multiplier formula follows by inserting a multiplier in the distributional test field. Smooth positivity of the metric and volume density on the compact chart supports therefore bounds the Euclidean BV norms of the localized functions. Extend each of them by zero within its chart.

For a Euclidean BV function \(v\), convolution satisfies \(D(v*\rho_\varepsilon)=Dv*\rho_\varepsilon\) and \(\|D(v*\rho_\varepsilon)\|_1\le |Dv|(\mathbb R^n)\). Apply the fundamental theorem of calculus to the convolution and let its scale tend to zero in \(L^1\). This gives \[\|v(\cdot+h)-v\|_1\le |h|\,|Dv|(\mathbb R^n),\qquad \|v*\rho_\varepsilon-v\|_1 \le\varepsilon C_\rho |Dv|(\mathbb R^n).\] For fixed \(\varepsilon\), the convolutions have common compact support and uniformly bounded sup norms and first derivatives, by the uniform \(L^1\) bound on \(v\). Approximation on a finite mesh makes this family precompact in \(L^1\). Taking scales to zero and a diagonal subsequence gives compactness in each chart, hence for the original finite sum. Total variation is a supremum of continuous divergence integrals, so is lower semicontinuous. A further almost-everywhere convergent subsequence of characteristic functions has values in \(\{0,1\}\). ◻

Proof. Write \(J_F\) for the positive ambient volume Jacobian. The Piola pullback of a target test field \(X\) is \[Q_X(x)=J_F(x)(dF_x)^{-1}X(Fx).\] Ordinary smooth change of variables, tested against smooth scalar functions, gives \(\mathop{\mathrm{div}}Q_X=J_F(\mathop{\mathrm{div}}X)\circ F\). Gauss–Green for \(E\) implies \[\int_{FE}\mathop{\mathrm{div}}X\,d\mathop{\mathrm{Vol}}_g =\int_{\partial^*E} \langle X(Fx),J_F(x)(dF_x)^{-*}N_E(x)\rangle_g \,d\mathcal H_g^{n-1}(x).\] Thus the outward vector measure \(-D\chi_{FE}\) is the pushforward of the vector density \(a(x)=J_F(x)(dF_x)^{-*}N_E(x)\) against the perimeter measure of \(E\). Since \(F\) is injective and \(a\ne0\), its polar direction at \(y=F(x)\) is \(a(x)/|a(x)|\). Its total variation is therefore the pushforward of \(|a|\,d|D\chi_E|\), with no cancellation. Linear algebra in orthonormal bases gives \(|a|=J_{n-1}(dF|_{N_E^\perp})\), proving (38). Differentiation of the Gram determinant at \(t=0\) gives tangential divergence. The Jacobians and their time derivatives are uniformly bounded over all tangent planes above the fixed compact set. This justifies differentiation under the integral and proves the final estimate. ◻

Proof. Concentric balls provide finite-perimeter competitors at every volume in \((0,|B|)\). A minimizing sequence has uniformly bounded perimeter and is supported, up to null sets, in the compact set \(\overline B\). Lemma 11 gives an \(L^1(M)\) limit \(\chi_E\) of the same volume with \(P(E)\le I(v)\). The limit vanishes almost everywhere outside \(\overline B\); since \(\partial B\) has ambient volume zero, it is admissible in (37). Thus it attains the infimum.

For the regularity conclusions, choose a minimizing region supplied by Ghomi–Spruck in the cited arXiv version (Ghomi and Spruck 2022, Lemma 7.2(i),(ii), p. 35), and denote it by \(E\). Set \(K=\mathop{\mathrm{supp}}|D\chi_E|\). These unconditional parts of the Lemma give interior smoothness outside the stated singular set and \(C^{1,1}\) regularity in a neighborhood of the confining sphere in a Cartan–Hadamard manifold; see also Morgan (Morgan 2003, sec. 2 and Corollary 3.8) for the interior theory. For the boundary part, Ghomi and Spruck use the Euclidean obstacle regularity of Stredulinsky and Ziemer (Stredulinsky and Ziemer 1997) and its local graph extension to the Riemannian setting. The graph and singular-set description gives \(|K|=0\). Off \(K\), \(\chi_E\) is locally constant almost everywhere. Take \(E\) to be the union of the components of \(M\setminus K\) on which \(\chi_E=1\) almost everywhere. This changes only a null set. Every ball centered on \(K\) has positive perimeter, hence contains both phases in positive volume, so \(\partial E=K=\Gamma\). At regular points the perimeter measure is the induced area measure. The singular set has zero \(m\)-dimensional Hausdorff measure, giving (40).

The free portion in the \(C^{1,1}\) collar is smooth by interior regularity for the constant-mean-curvature equation. Hence the actual interior singular set avoids this collar. It is closed away from the collar and contained in the compact set \(\Gamma\), so it is a compact subset of \(B\). Finally, the \(C^{1,1}\) regularity is that of the whole hypersurface near \(\partial B\); the transition between its free and contact portions does not create a boundary of \(\Sigma\). ◻

Proof. Positivity was just established. Fix a compact interval \([v_-,v_+]\subset(0,|B|)\). The radius of a concentric ball varies continuously with its volume, and its boundary area is continuous in its radius. Consequently there is \(C_0<\infty\) such that \(I(v)\le C_0\) for all \(v\in[v_-,v_+]\).

Consider the family \[\mathcal K=\bigl\{\chi_F:\ |F\setminus B|=0, \ v_-\le |F|\le v_+,\ P(F)\le C_0\bigr\}.\] BV compactness on a compact neighborhood of \(\overline B\), together with lower semicontinuity and \(L^1\) continuity of volume, shows that \(\mathcal K\) is compact in \(L^1(M)\). For each \(F\in\mathcal K\) there is a smooth vector field \(Z_F\) compactly supported in \(B\) for which \[A_{Z_F}(F):=\int_F\mathop{\mathrm{div}}Z_F\,d\mathop{\mathrm{Vol}}_g\ne0.\] Otherwise \(D\chi_F\) would vanish in \(B\), contradicting \(v_-\le |F|\le v_+\). Reverse the sign of \(Z_F\) if necessary so that \(A_{Z_F}(F)>0\). For each fixed \(Z\), the functional \(A_Z\) is \(L^1\)-continuous. A finite subcover of \(\mathcal K\) therefore gives smooth fields \(Z_1,\ldots,Z_k\), compactly supported in \(B\), and a number \(\delta>0\) such that \[ \max_{1\le j\le k} A_{Z_j}(F)\ge 2\delta \qquad\text{for every }F\in\mathcal K. \tag{42}\]

Let \(\Psi_j^t\) be the flow of \(Z_j\). These flows preserve \(B\) in both time directions. If \(J_j(t,x)\) is the ambient volume Jacobian, then \[G_j(t,F):=|\Psi_j^t(F)|=\int_F J_j(t,x)\,d\mathop{\mathrm{Vol}}_g(x), \qquad \partial_tG_j(0,F)=A_{Z_j}(F).\] Smoothness of the finitely many flows gives uniform bounds on \(\partial_t^2J_j\) on a fixed compact set for small \(|t|\). Since \(|F|\le |B|\), there is a common \(\tau>0\) such that \[|\partial_tG_j(t,F)-A_{Z_j}(F)|\le\delta \quad (|t|\le\tau,\ F\in\mathcal K,\ 1\le j\le k).\] For an index satisfying (42), the function \(t\mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF21A6> >> BDC}% PPaperOriginalmapsto\endcsname\pdfliteral direct{EMC}\endgroup}G_j(t,F)\) thus has derivative at least \(\delta\) on \([-\tau,\tau]\). It follows that every \(w\) with \(|w-|F||<\delta\tau\) is attained for some \(t\) satisfying \[ |\Psi_j^t(F)|=w,\qquad |t|\le \delta^{-1}|w-|F||. \tag{43}\]

By Lemma 12, perimeter transforms by the tangential Jacobian of a smooth diffeomorphism on the reduced boundary. Applied to these finitely many flows, this gives \[\begin{align*} P(\Psi_j^t(F)) &=\int_{\partial^*F} J_m\bigl(D\Psi_j^t(x)|_{T_x\partial^*F}\bigr) \,d\mathcal H_g^m(x),\\ |P(\Psi_j^t(F))-P(F)|&\le C|t|P(F) \qquad (|t|\le\tau). \end{align*}\] Here \(C\) is uniform over the finitely many flows and all \(m\)-planes above the compact set \(\overline B\). Equation (43) and the bound \(P(F)\le C_0\) show that changing the volume by \(\Delta v\) costs at most \(C C_0\delta^{-1}|\Delta v|\) in perimeter.

Apply this construction first to a minimizer at \(v\) and then to a minimizer at \(w\), where \(v,w\in[v_-,v_+]\) and \(|v-w|<\delta\tau\). Both are in \(\mathcal K\), and the resulting competitors give \[|I(v)-I(w)|\le C C_0\delta^{-1}|v-w|.\] This proves local Lipschitz continuity. All constants here may depend on the fixed ball and interval; no uniform regularity of the minimizers is required. ◻

Proof. For a smooth flow \(\Psi^t\) with \(E_t=\Psi^t(E)\), set \(P(t)=P(E_t)\) and \(V(t)=|E_t|\). Whenever the flow is admissible, \[ P(t)\ge I(V(t)),\qquad P(0)=I(v),\qquad V(0)=v. \tag{44}\] For a field supported on a free regular patch, admissibility holds for both signs of \(t\). Differentiating at the minimum in (44) gives \(P'(0)=I'(v)V'(0)\). The first variation formulas are \[V'(0)=\int_\Sigma\langle Z,N\rangle\,d\sigma, \qquad P'(0)=m\int_\Sigma h\langle Z,N\rangle\,d\sigma.\] Arbitrary smooth compactly supported normal speeds on the patch therefore give \(mh=I'(v)\). This holds on every free regular patch, so \(h=h_0\) throughout \(\Sigma\cap B\). Such patches exist by (41).

To determine the sign of \(h_0\), put \(\rho(x)=d(o,x)\) and \(F(x)=\rho(x)^2/2\). The function \(F\) is smooth on \(M\), including at \(o\). Hessian comparison gives \[\mathop{\mathrm{Hess}}F=d\rho\otimes d\rho+\rho\mathop{\mathrm{Hess}}\rho \ge d\rho\otimes d\rho +(g-d\rho\otimes d\rho) =g,\] with the inequality at \(o\) understood by smooth extension. The flow of \(-\nabla F\) is \[\Psi^t(x)=\exp_o\bigl(e^{-t}\exp_o^{-1}(x)\bigr).\] It sends \(B\) into itself for \(t\ge0\). The volume and perimeter Jacobian formulas give \[\begin{align*} V'(0)&=-\int_E\Delta F\,d\mathop{\mathrm{Vol}}_g\le -nv<0,\tag{45}\\ P'(0)&=-\int_{\partial^*E} \mathop{\mathrm{tr}}_{T_x\partial^*E}(\mathop{\mathrm{Hess}}F)\,d\mathcal H_g^m(x) \le -mP(E)<0. \tag{46}\end{align*}\] These formulas apply to the reduced boundary as a rectifiable set, so they include its contact portion and do not require smoothness at singular points. The integrands and their flow derivatives are bounded on a fixed compact neighborhood of \(\overline B\). The one-sided derivative of (44) now yields \[P'(0)-I'(v)V'(0)\ge0.\] Since \(V'(0)<0\), division reverses the inequality and gives \[ I'(v)\ge\frac{P'(0)}{V'(0)}>0. \tag{47}\]

It remains to control the contact set \(C=\Gamma\cap\partial B\). At every point of \(C\), the \(C^1\) boundary of \(E\) is tangent to the sphere and has the same outward normal \(N=\nabla\rho\): the region lies on the inner side of the sphere. At almost every such point \(x\), its local \(C^{1,1}\) graph has a second-order expansion. In normal coordinates at \(x\), with the common outward normal pointing upward, write the boundary of \(E\) and the sphere as graphs \(u\) and \(\beta\). They have equal values and first derivatives at \(x\), and \(u\le\beta\). Thus \(D^2u(x)\le D^2\beta(x)\). In these coordinates their second fundamental forms for the outward normal are \(-D^2u(x)\) and \(-D^2\beta(x)\), respectively. Hessian comparison on the sphere therefore gives \[ \mathrm{II}_{\Gamma}(x)\ge\mathrm{II}_{\partial B}(x), \qquad h(x)\ge R_B^{-1}>0 \quad\text{for almost every }x\in C. \tag{48}\]

Choose an open collar \(U\) of \(\partial B\) whose closure avoids \(S\) and \(o\) and in which \(\Gamma\) is \(C^{1,1}\). For arbitrary \(\varphi\in C_c^\infty(U)\) with \(\varphi\ge0\), the smooth field \(Z=-\varphi\nabla\rho\), extended by zero, has an admissible flow for \(t\ge0\), because \(\rho\) is nonincreasing along its trajectories. The one-sided variation inequality gives \[ m\int_\Sigma(h-h_0)\langle Z,N\rangle\,d\sigma\ge0. \tag{49}\] For completeness, the perimeter first variation used here is obtained by integrating \(\mathop{\mathrm{div}}_\Sigma Z=\mathop{\mathrm{div}}_\Sigma Z^{\mathsf T} +mh\langle Z,N\rangle\) on the whole \(C^{1,1}\) hypersurface in the collar. The tangential divergence integrates to zero, since its support is compact in \(\Sigma\). Thus there is no additional term on the transition between the free and contact portions.

The integrand in (49) vanishes on the free portion, where \(h=h_0\), while \(\langle Z,N\rangle=-\varphi\) on \(C\). Consequently \[\int_C(h-h_0)\varphi\,d\sigma\le0 \qquad (\varphi\in C_c^\infty(U),\ \varphi\ge0).\] Nonnegative smooth ambient tests determine the sign of a Radon measure, so \(h\le h_0\) almost everywhere on \(C\), irrespective of whether \(C\) has relative interior. Combining this with (48), \(h=h_0\) on the free portion, and (47), gives \[ 0\le h\le h_0,\qquad h^2\le h_0^2 \quad\text{almost everywhere on }\Sigma. \tag{50}\] ◻

Proof. For a bounded Lipschitz scalar \(\varphi\), distributional testing gives \[ D(\varphi\chi_E)=\varphi D\chi_E+ \chi_E\nabla\varphi\,d\mathop{\mathrm{Vol}}_g. \tag{53}\] First prove this for smooth multipliers. On a compact neighborhood of a test field’s support, approximate \(\varphi\) smoothly, uniformly and in \(W^{1,1}\). Uniform convergence controls the measure term and \(W^{1,1}\) convergence controls the second term, since \(|\chi_E|\le1\). This proves the displayed distributional identity.

Take \[\varphi_\varepsilon(y)= \min\{1,\max\{0,(d(x,y)-r)/\varepsilon\}\}, \qquad r+\varepsilon<\operatorname{inj}(x).\] Its derivative is supported in a compact annulus. The products \(\varphi_\varepsilon\chi_E\) converge in \(L^1\) to \(\chi_{E\setminus B_r(x)}\), because the smooth distance sphere has ambient volume zero. Lower semicontinuity and (53) give \[\begin{split} P(E\setminus B_r(x)) &\le\liminf_{\varepsilon\downarrow0} \left(\int\varphi_\varepsilon\,d|D\chi_E| +\frac{|E\cap(B_{r+\varepsilon}(x)\setminus B_r(x))|} {\varepsilon}\right)\\ &\le P(E;M\setminus\overline B_r(x)) +\left.\frac d{ds}\right|_{s=r}|B_s(x)|. \end{split}\] Polar coordinates identify the last derivative with the area of the distance sphere. Finally, disjoint spheres can carry positive mass for a finite measure at only countably many radii. ◻

Proof. Suppose \(S\ne\varnothing\), since otherwise the assertion is empty. By (41) and (40), there is a free smooth boundary point. Choose a neighborhood \(W\) of that point with \(\overline W\subset B\setminus S\) so small that \(\Gamma\cap W\) is smooth. Extend a nonnegative, nonzero normal variation of a smaller patch to a smooth field \(Z\) compactly supported in \(W\). With \(N\) the outward normal, this gives \[a:=\int_{\Sigma\cap W}\langle Z,N\rangle\,d\sigma>0.\]

Let \(\phi_t\) be its flow and put \(G(t)=\mathop{\mathrm{Vol}}(\phi_t(E))-\mathop{\mathrm{Vol}}(E)\). The volume Jacobian formula gives \(G'(0)=a\). Choose \(t_0>0\) so that \(G'(t)\ge a/2\) for \(0\le t\le t_0\). The flow preserves \(W\) and \(B\) and fixes \(M\setminus W\). Whenever a finite-perimeter set \(F\) agrees with \(E\) almost everywhere on \(W\), the volume Jacobian minus one is supported in \(W\), so \[ \mathop{\mathrm{Vol}}(\phi_t(F))-\mathop{\mathrm{Vol}}(F)=G(t). \tag{55}\] The tangential Jacobian formula and locality of the reduced boundary likewise give \[P(\phi_t(F))-P(F) =\int_{\partial^*E\cap W} \left(J_m\bigl(D\phi_t|_{T_x\partial^*E}\bigr)-1\right) \,d\mathcal H_g^m(x).\] On the compact support of the flow, the tangential Jacobians satisfy \(|J_m-1|\le Lt\) uniformly over all tangent \(m\)-planes for \(0\le t\le t_0\). Hence \[ \bigl|P(\phi_t(F))-P(F)\bigr|\le L P(E;W)t. \tag{56}\] Every volume deficit \(0\le\delta\le at_0/2\) can therefore be repaired by a time \(0\le t\le2\delta/a\), at perimeter cost at most \(C_R\delta\), where \(C_R=2LP(E;W)/a\). The bound is independent of \(F\) as long as \(F=E\) almost everywhere on \(W\).

Compactness of \(S\) and its disjointness from \(\overline W\cup\partial B\) give \(\rho_*>0\) such that \[B_{2\rho_*}(x)\subset B\setminus\overline W \qquad(x\in S).\] After decreasing \(\rho_*\), smoothness of the ambient metric on a compact neighborhood of \(S\) gives uniform constants \(C_V,C_A\) with \[ \mathop{\mathrm{Vol}}(B_r(x))\le C_Vr^n, \qquad \mathcal H_g^m(\partial B_r(x))\le C_Ar^m \quad(x\in S,\ 0<r<\rho_*). \tag{57}\] Decrease \(\rho_*\) again so that \(C_V\rho_*^n\le at_0/2\).

Fix \(x\in S\). For almost every \(r\in(0,\rho_*)\), one has \(P(E;\partial B_r(x))=0\). At such radii set \(F=E\setminus B_r(x)\); Lemma 16, applied to \(E\), gives \[ P(F)\le P(E;M\setminus\overline{B_r(x)}) +\mathcal H_g^m(\partial B_r(x)). \tag{58}\] The volume deficit is \(\delta=\mathop{\mathrm{Vol}}(E\cap B_r(x))\le C_Vr^n\), and \(F=E\) on \(W\). Use (55) to repair this deficit by a flow time \(t\le2\delta/a\). The repaired set remains in \(B\) and has volume \(v\). Minimality, (56), and (58) imply \[\begin{split} P(E;B_r(x)) &\le\mathcal H_g^m(\partial B_r(x))+C_R\delta\\ &\le C_Ar^m+C_RC_Vr^n\le C_0r^m, \end{split}\] where \(C_0=C_A+C_RC_V\rho_*\) because \(n=m+1\). The constants are uniform in \(x\in S\).

For an arbitrary \(0<r<\rho_*/2\), choose a good radius \(\rho\in(r,2r)\). Monotonicity of the perimeter measure gives \[P(E;B_r(x))\le P(E;B_\rho(x))\le2^m C_0r^m.\] Together with (51), this proves the claim with \(r_0=\rho_*/2\). ◻

Proof. If \(S=\varnothing\), then \(\Sigma=\Gamma\) is compact and we take \(\eta_j=1\), including when \(\Sigma\) has several components. Otherwise, use \(\mathcal H^{m-2}(S)=0\) as follows. For each positive integer \(j\), choose a finite covering \[ \begin{gathered} S\subset\bigcup_{i=1}^{N_j}B_{r_{ji}}(x_{ji}), \qquad x_{ji}\in S,\\ 0<r_{ji}<\min(2^{-j},r_0/4), \qquad\sum_{i=1}^{N_j}r_{ji}^{m-2}<2^{-j}. \end{gathered} \tag{60}\] Indeed, Hausdorff nullity first gives a countable open-ball cover with arbitrarily small radii and sum. Recenter the balls at points of \(S\) with a fixed enlargement, starting with tolerances small enough to absorb this enlargement, and take a finite subcover by compactness.

Put \(\theta(t)=\min\{1,\max\{0,t-1\}\}\) for \(t\ge0\), and define \[\eta_{ji}(x)=\theta\!\left(\frac{d(x,x_{ji})}{r_{ji}}\right), \qquad \eta_j=\min_{1\le i\le N_j}\eta_{ji}\big|_\Sigma.\] Each ambient cutoff vanishes on \(B_{r_{ji}}(x_{ji})\), is one outside \(B_{2r_{ji}}(x_{ji})\), and has Lipschitz constant at most \(r_{ji}^{-1}\). A finite minimum is Lipschitz, and almost everywhere \[|\nabla_\Sigma\eta_j|^2 \le\sum_{i=1}^{N_j}|\nabla_\Sigma\eta_{ji}|^2.\] There is no restriction on the overlap of the balls. Applying the assumed area bound at their doubled radii gives \[ \begin{split} \int_\Sigma|\nabla_\Sigma\eta_j|^2\,d\sigma &\le\sum_i r_{ji}^{-2} \sigma(\Sigma\cap B_{2r_{ji}}(x_{ji}))\\ &\le2^m C\sum_i r_{ji}^{m-2}<2^m C\,2^{-j}. \end{split} \tag{61}\]

The open balls in (60) form a neighborhood of \(S\) on which \(\eta_j\) vanishes. Its support is therefore contained in a compact subset of \(\Gamma\setminus S\). The embedded graph charts identify the subspace and manifold topologies on \(\Sigma\), so this support is compact in \(\Sigma\). For each fixed \(x\in\Sigma\), closedness of \(S\) gives \(d(x,S)>0\). As the maximum radius tends to zero, the doubled balls eventually miss \(x\), and \(\eta_j(x)=1\). This proves (59), without assuming completeness of \(\Sigma\) or finiteness of its components. ◻

Proof. Lemma 15 gives \(0\le h\le h_0:=I'(v)/m<\infty\) almost everywhere on \(\Sigma\). If \(S\ne\varnothing\), the singular-dimension bound gives \[\dim_{\mathcal H}S\le n-8=m-7<m-2, \qquad \mathcal H^{m-2}(S)=0.\] Lemma 17 and the compact embedded boundary structure therefore verify the hypotheses of Lemma 18. That lemma supplies cutoffs with (59); when \(S=\varnothing\) take \(\eta_j=1\).

Apply Theorem 7 to these cutoffs: \[c\int_\Sigma|\nabla_\Sigma\eta_j|^2\,d\sigma +m\int_\Sigma(h^2-b^2)\eta_j^2\,d\sigma \ge Y_m\left(\int_\Sigma\eta_j^q\,d\sigma\right)^{2/q}.\] The area is \(I(v)<\infty\), and \(|h^2-b^2|\le h_0^2+b^2\) almost everywhere. Dominated convergence and (59) prove (62), since \(2/q=(m-2)/m\). The same construction applies in every dimension under consideration: when \(4\le n\le7\) the singular set is empty, and when \(n=8\) its possible infinite zero-dimensional compact set still admits the covers in (60). ◻

Proof. The profile is locally Lipschitz by Lemma 14, hence differentiable almost everywhere. At a differentiability volume choose a minimizer. Lemma 15 gives \(h_0=I'(v)/m>0\) and \(h^2\le h_0^2\) almost everywhere on its regular boundary. Lemma 19 therefore gives \[m(h_0^2-b^2)I(v)\ge Y_m I(v)^{(m-2)/m}.\] Since \(I(v)>0\) and \(Y_m=m s_m^{2/m}\), division yields \[h_0^2\ge b^2+\left(\frac{s_m}{I(v)}\right)^{2/m}.\] Taking the positive square root proves (63). ◻

Proof. Choose radii \(0<R_0<R\) such that \(E\subset B_{R_0}(o)\) up to a null set, and put \(B=B_R(o)\). The annulus \(B_R(o)\setminus\overline{B_{R_0}(o)}\) has positive volume, so \(\mathop{\mathrm{Vol}}(B)>\mathop{\mathrm{Vol}}(E)\). Recall the model area and volume functions \[\mathcal A_b(r)=s_m\mathop{\mathrm{sn}}_b(r)^m, \qquad \mathcal V_b(r)=s_m\int_0^r\mathop{\mathrm{sn}}_b(t)^m\,dt.\] Both are strictly increasing from zero to infinity. For \(A>0\) put \[ R_b(A)=\mathcal A_b^{-1}(A), \qquad \mathcal W_b(A)=\mathcal V_b(R_b(A)). \tag{64}\] Thus \(\mathcal W_b\) is strictly increasing. The identities \(\mathop{\mathrm{sn}}_b'=\mathop{\mathrm{cs}}_b\) and \(\mathop{\mathrm{cs}}_b^2-b^2\mathop{\mathrm{sn}}_b^2=1\) give \[ \begin{split} \mathcal W_b'(A) &=\frac{\mathop{\mathrm{sn}}_b(R_b(A))}{m\mathop{\mathrm{cs}}_b(R_b(A))}\\ &=\frac{1}{m\sqrt{b^2+(s_m/A)^{2/m}}}. \end{split} \tag{65}\]

On every interval \([\varepsilon,v]\subset(0,\mathop{\mathrm{Vol}}(B))\), the positive Lipschitz function \(I\) has image in a compact subset of \((0,\infty)\), where \(\mathcal W_b\) is continuously differentiable. Thus \(\mathcal W_b\circ I\) is absolutely continuous there. Proposition 20, the chain rule, and (65) imply \[(\mathcal W_b\circ I)'(s) =\mathcal W_b'(I(s))I'(s)\ge1 \quad\text{for almost every }s\in[\varepsilon,v].\] Integrating yields \[\mathcal W_b(I(v)) \ge\mathcal W_b(I(\varepsilon))+v-\varepsilon \ge v-\varepsilon.\] Letting \(\varepsilon\downarrow0\), we obtain \[ \mathcal W_b(I(v))\ge v \qquad(0<v<\mathop{\mathrm{Vol}}(B)). \tag{66}\] This endpoint passage uses only nonnegativity of \(\mathcal W_b(I(\varepsilon))\).

Set \(V=\mathop{\mathrm{Vol}}(E)\). The choice of \(B\) gives \(0<V<\mathop{\mathrm{Vol}}(B)\), and \(E\) is an admissible competitor in (37). Consequently \[P(E)\ge I(V).\] This step uses the full ambient perimeter of \(E\), with no regularity assumption on its boundary. If \(r>0\) is determined by \(\mathcal V_b(r)=V\), then \(\mathcal W_b(\mathcal A_b(r))=V\). Equation (66) and strict monotonicity of \(\mathcal W_b\) give \(I(V)\ge\mathcal A_b(r)=\mathcal I_b(V)\), which proves the assertion. ◻

Proof. Strict smooth approximation on complete manifolds (Güneysu and Pallara 2015, Theorem 2.12 in the author preprint) provides smooth compactly supported functions \(f_j\) with \[\|f_j-\chi_E\|_{L^1(M)}\mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF27F6> >> BDC}% PPaperOriginallongrightarrow\endcsname\pdfliteral direct{EMC}\endgroup}0, \qquad \int_M|\nabla f_j|\,d\mathop{\mathrm{Vol}}_g\mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF27F6> >> BDC}% PPaperOriginallongrightarrow\endcsname\pdfliteral direct{EMC}\endgroup}P(E).\] The cited result assumes completeness, smoothness, connectedness, and absence of boundary; it imposes no curvature condition. We may take real parts, since doing so cannot increase the \(L^1\) error or gradient integral, and lower semicontinuity gives the reverse limiting bound.

Choose \(\eta\in C_c^\infty(U)\) with \(0\le\eta\le1\) and \(\eta=1\) near \(K\), and put \(u_j=\eta f_j\). Then \(u_j\to\chi_E\) in \(L^1(M)\), and \(\chi_E\nabla\eta=0\) almost everywhere. Hence \[\int_M|\nabla u_j|\,d\mathop{\mathrm{Vol}}_g \le\int_M|\nabla f_j|\,d\mathop{\mathrm{Vol}}_g +\|\nabla\eta\|_\infty\|f_j-\chi_E\|_1.\] Lower semicontinuity shows that the left side also converges to \(P(E)\). Thus all approximating functions now have support in the same compact subset of \(U\).

Write \(e_j=\|u_j-\chi_E\|_1\), and choose \(0<\delta_j<1/2\) with \(\delta_j\to0\) and \(e_j/\delta_j\to0\). For \(\delta_j<t<1-\delta_j\), pointwise comparison with \(\chi_E\) gives \[ \bigl|\{u_j>t\}\mathbin{\triangle}E\bigr| \le\delta_j^{-1}e_j. \tag{67}\] The ordinary coarea formula for the smooth function \(u_j\) gives \[\int_{\delta_j}^{1-\delta_j}P(\{u_j>t\})\,dt \le\int_M|\nabla u_j|\,d\mathop{\mathrm{Vol}}_g.\] By Sard’s Theorem, almost every \(t\) is a regular value. We can therefore choose such a \(t_j\in(\delta_j,1-\delta_j)\) with \[P(\{u_j>t_j\}) \le\frac{1}{1-2\delta_j}\int_M|\nabla u_j|\,d\mathop{\mathrm{Vol}}_g+\frac1j.\] Set \(E_j=\{u_j>t_j\}\). Its boundary is smooth and compact in \(U\). Equation (67) gives \(L^1\) convergence of the characteristic functions. The displayed perimeter bound and lower semicontinuity give the claimed perimeter convergence. ◻

Proof. We use two elementary BV identities. For a compactly supported Lipschitz function \(\varphi\) and bounded \(u\in BV(M)\), the product rule is \[D(\varphi u)=\varphi\,Du+u\nabla\varphi\,d\mathop{\mathrm{Vol}}_g.\] It follows from the distributional product rule for smooth multipliers by smoothing \(\varphi\) locally in charts: the approximations converge uniformly, and their gradients converge in \(L^1\) on the fixed compact support. The boundedness of \(u\) permits passage in the second term. For \(0\le u\le1\), the coarea identity is \[ |Du|(M)=\int_0^1P(\{u>t\})\,dt. \tag{68}\] The integrand is measurable: the defining perimeter supremum can be taken over a countable dense family of compactly supported test fields. For completeness, apply the strict smooth approximation theorem (Güneysu and Pallara 2015, Theorem 2.12 in the author preprint) to \(u\in BV(M)\). It applies to arbitrary integrable functions. The ordinary coarea formula then proves the inequality “\(\ge\)”: take a subsequence whose level sets converge in \(L^1\) for almost every \(t\), and use lower semicontinuity and Fatou’s Lemma. Such a subsequence exists because the integral over \(t\in\mathbb R\) of the \(L^1\) distance between the two level indicators equals the \(L^1\) distance between the functions. For the reverse inequality, write \(u=\int_0^1\chi_{\{u>t\}}\,dt\), test against \(\mathop{\mathrm{div}}X\) for \(X\in C_c^\infty(TM)\) with \(|X|\le1\), and take the supremum in the definition of total variation. This also shows that almost every level in (68) has finite ambient perimeter.

Fix \(o\in M\) and, for \(R>0\), define \[\varphi_R(x)=\min\{1,\max\{0,R+1-d(o,x)\}\}.\] Completeness makes its support compact. It equals one on \(B_R(o)\), vanishes outside \(B_{R+1}(o)\), and has Lipschitz constant at most one. The product rule and (68) give \[ \begin{split} \int_0^1P(\{\varphi_R\chi_E>t\})\,dt &=|D(\varphi_R\chi_E)|(M)\\ &\le\int_M\varphi_R\,d|D\chi_E| +\int_E|\nabla\varphi_R|\,d\mathop{\mathrm{Vol}}_g\\ &\le P(E)+|E\setminus B_R(o)|. \end{split} \tag{69}\] Almost every level is a bounded finite-perimeter set, and for every \(0<t<1\) its volume is at least \(v_R=|E\cap B_R(o)|\). The hypothesis and monotonicity of \(J\) therefore bound the left side of (69) below by \(J(v_R)\). Since \(v_R\to|E|\) and \(|E\setminus B_R(o)|\to0\), continuity of \(J\) proves the assertion. ◻

Proof of Theorem 1. Null sets have zero perimeter and \(\mathcal I_b(0)=0\). The preceding bounded-set comparison and Lemma 23, with \(J=\mathcal I_b\), prove the assertion for all \(n\ge2\) and normalized curvature parameters \(b\in\{0,1\}\).

For an arbitrary negative curvature bound, put \(b=\sqrt{-\kappa}>0\) and \(\widetilde g=b^2g\). Then \[\mathop{\mathrm{sec}}_{\widetilde g}=b^{-2}\mathop{\mathrm{sec}}_g\le-1, \qquad |E|_{\widetilde g}=b^n|E|_g, \qquad P_{\widetilde g}(E)=b^{n-1}P_g(E).\] The last identity holds for ambient BV perimeter: constant metric rescaling leaves divergence unchanged, while \(d\mathop{\mathrm{Vol}}_{\widetilde g}=b^n d\mathop{\mathrm{Vol}}_g\) and \(|X|_{\widetilde g}=b|X|_g\). In the dual definition of perimeter, the substitution \(Y=bX\) therefore multiplies the supremum by \(b^{n-1}\). The model functions have the matching relations \[\mathcal V_b(r)=b^{-n}\mathcal V_1(br),\qquad \mathcal A_b(r)=b^{-(n-1)}\mathcal A_1(br),\qquad \mathcal I_b(V)=b^{-(n-1)}\mathcal I_1(b^nV).\] Applying the normalized result to \(\widetilde g\) and dividing by \(b^{n-1}\) gives the claimed inequality for \(g\) and \(\kappa\). The case \(\kappa=0\) was proved directly, without a curvature limit. ◻

Proof. Almost-minimality and the regular part. Write \(\theta=(n-1)/n\) and \(\Lambda=A/V\). For every bounded finite-perimeter set \(F\), 1 gives \[ \begin{split} P(E)-P(F) &\le A\left[1-\left(\frac{|F|}{V}\right)^\theta\right] \le\Lambda|E\triangle F|. \end{split} \tag{72}\] Indeed \(z^\theta\ge\min\{z,1\}\) for \(z\ge0\), and the difference of the volumes is bounded by the volume of the symmetric difference. If \(E\triangle F\) is contained, up to a null set, in a compact subset of a bounded open set \(O\), locality of perimeter cancels the part outside \(O\). Thus \[P(E;O)\le P(F;O)+\Lambda|E\triangle F|.\] It suffices to consider competitors with \(P(F;O)<\infty\); agreement with \(E\) near the complement of \(O\) then gives finite global perimeter as well.

We apply the local smooth-manifold conclusion of (Antonelli, Pasqualetto, et al. 2022, Corollary 1.6). To check its quasi-perimeter hypothesis, on measurable subsets of \(O\) define \[G(F)=\Lambda|F\triangle(E\cap O)|.\] This functional is finite at the empty set and satisfies \[|G(F_1)-G(F_2)|\le\Lambda|F_1\triangle F_2|.\] Its continuity exponent is therefore \(1>1-1/n\), as required in Condition (1.3) of that reference. The localized inequality says that \(E\) minimizes \(P+G\) under compact modifications; in particular it does so under the volume-preserving modifications of Definition 1.2. For a ball \(O\) centered on \(\Gamma\), the remaining hypothesis \(P(E;O)>0\) follows from the definition of support. The cited local conclusion applies in every dimension \(n\ge2\) and requires no global lower Ricci bound.

It gives a \(C^{1,\alpha}_{\mathrm{loc}}\) regular part of full perimeter, a relatively closed singular part of Hausdorff dimension at most \(n-8\), and local upper area growth. Here and below the local exponent may be decreased so that \(0<\alpha<1\). These are statements about the perimeter support: one may take the density-one representative supplied by the theorem, whose boundary equals that support. For completeness, the perimeter measure is concentrated on the reduced boundary, giving one inclusion. Conversely, a boundary point of this representative is an essential boundary point; if the perimeter vanished in a neighborhood, the characteristic function would be constant almost everywhere there, a contradiction.

The support is unchanged by modifying \(E\) on a null set. It is a closed subset of a compact set containing \(E\), since completeness makes closed bounded sets compact. Hence \(\Gamma\) and its closed singular subset \(Z\) are compact. The local area estimates become [eq:equality-area-growth] with uniform constants by a finite cover of \(\Gamma\).

Near a regular point, the whole support is a graph dividing a coordinate neighborhood into two connected sides. On either side \(D\chi_E=0\), so \(\chi_E\) is constant almost everywhere. One way to see this last fact is to express the distributional derivatives in coordinates, compensating for the volume density, and mollify on smaller coordinate balls. Overlapping balls give the same constant. The two side values must differ: otherwise the characteristic function would be constant across the neighborhood, since the graph has zero ambient volume. The occupied side consequently defines a consistent outward normal. The divergence formula on a graph identifies its perimeter with its induced area. Together with the full-perimeter conclusion, this proves [eq:equality-perimeter-area]. In particular, every neighborhood of a support point meets the regular part, so \(\Sigma\) is dense in \(\Gamma\).

The graph equation. Choose graph coordinates \((y,z)\in\mathbb R^m\times\mathbb R\) with \(z=u(y)\) on \(\Sigma\) and \(E\) below the graph, up to a null set. Let \(G(y,z)\) be the ambient metric matrix and \(W(y,z)=\sqrt{\det G(y,z)}\). The graph area integrand is \[L_g(y,z,p) =W(y,z)\left[(-p,1)^T G(y,z)^{-1}(-p,1)\right]^{1/2}.\] For a smooth compactly supported function \(\zeta\) in the graph domain, the graph \(u+t\zeta\) gives admissible sets for both signs of small \(t\). The sharp deficit \(P(F)-n\omega_n^{1/n}|F|^\theta\) is nonnegative and vanishes at \(F=E\). Its volume derivative has coefficient \(n\omega_n^{1/n}\theta V^{\theta-1}=H_0\). Define \[\mathcal B(y,z,p)=\partial_zL_g(y,z,p)-H_0W(y,z).\] Differentiating area and the volume between the two graphs gives \[ \int\left[ \partial_pL_g(y,u,Du)\cdot D\zeta +\mathcal B(y,u,Du)\zeta\right]dy=0. \tag{73}\] The derivative \(D\) here is Euclidean differentiation in \(y\). The matrix \(\partial_{pp}L_g\) is positive definite. Indeed, the Hessian of a positive definite norm is positive in directions transverse to its radial kernel, and a nonzero slope variation \((-v,0)\) is not radial at \((-p,1)\). On compact sets of arguments the equation is therefore uniformly elliptic.

The local graph-regularity result in 25 applies to this weak equation: a \(C^{1,\alpha}\) graph stationary for area minus \(H_0\) times volume is locally \(C^{1,1}\). Thus \(Du\) is locally Lipschitz. The proof of that result follows in the next subsection; it obtains this conclusion without an advance bound on second derivatives.

The signed curvature. We can now interpret the graph equation geometrically. On a \(C^{1,1}\) graph the induced metric and its volume density are locally Lipschitz. For a tangential field \(X=X^iF_i\) in a parametrization \(F\), with induced metric \((a_{ij})\), metric compatibility gives, almost everywhere, \[\mathop{\mathrm{div}}_\Sigma X =\partial_iX^i+\tfrac12X^i a^{jk}\partial_i a_{jk} =\frac{1}{\sqrt{\det a}}\, \partial_i\bigl(\sqrt{\det a}\,X^i\bigr).\] The identities are also distributional: Lipschitz coefficients obey the product rule, and the weak mixed derivatives of \(F\) commute. Consequently compactly supported tangential Lipschitz fields have zero integral divergence. Splitting a field into its tangential and normal parts now gives \[ \mathop{\mathrm{div}}_{T\Sigma}Y =\mathop{\mathrm{div}}_\Sigma(Y^{\mathsf T})+H\langle Y,N\rangle, \qquad H=\mathop{\mathrm{div}}_{T\Sigma}N, \tag{74}\] almost everywhere and weakly on each regular chart.

For the graph variation used in [eq:equality-graph-equation], take \(Y=\zeta\,\partial_z\). The area derivative is \(\int\mathop{\mathrm{div}}_{T\Sigma}Y\,d\sigma\), and the volume derivative is its outward flux. By [eq:equality-tangential-divergence], the graph equation becomes \[\int_\Sigma (H-H_0)\langle\partial_z,N\rangle \zeta\,d\sigma=0.\] The factor \(\langle\partial_z,N\rangle\) is strictly positive, because \(E\) occupies the lower side; equivalently its product with the graph area density is \(W\). Arbitrary compactly supported \(\zeta\) therefore give \(H=H_0\) almost everywhere. The coefficient \(H_0\) came from the same global deficit on every patch, so this is the same signed outward curvature on every component. This completes the proof. ◻

Proof. Difference quotients and an energy bound. The positivity of \(\partial_{pp}L_g\) was verified after [eq:equality-graph-equation]. Shrink the graph chart so that its arguments and the segments between nearby arguments lie in one compact ellipticity region. Choose a Euclidean ball \(B_{4\rho}\) compactly contained in its parameter domain. For a coordinate unit vector \(e\) and \(0<\delta<\rho\), put \[v(y)=\frac{u(y+\delta e)-u(y)}{\delta}, \qquad y\in B_{3\rho}.\] Each \(v\) is \(C^1\), though no uniform bound on \(Dv\) is yet known. The identity \[v(y)=\int_0^1\partial_eu(y+s\delta e)\,ds\] does give a uniform \(C^{0,\alpha}\) bound for \(v\).

Subtract the translated equations and divide by \(\delta\). To describe the coefficients explicitly, set \[\begin{aligned} T_s(y)&=(1-s)(y,u(y),Du(y))\\ &\quad+s(y+\delta e,u(y+\delta e),Du(y+\delta e)). \end{aligned}\] The resulting weak equation is \[ \int\left[(aDv+f)\cdot D\zeta +(b\cdot Dv+d)\zeta\right]dy=0, \tag{75}\] where \[\begin{align*} a&=\int_0^1\partial_{pp}L_g(T_s)\,ds, \qquad b=\int_0^1\partial_p\mathcal B(T_s)\,ds,\\ f&=\int_0^1 \bigl(\partial_e\partial_pL_g(T_s) +\partial_z\partial_pL_g(T_s)v\bigr)\,ds,\\ d&=\int_0^1 \bigl(\partial_e\mathcal B(T_s) +\partial_z\mathcal B(T_s)v\bigr)\,ds. \end{align*}\] In these expressions \(\partial_e\) differentiates only the explicit \(y\) argument. The known \(C^{1,\alpha}\) norm of \(u\) bounds the \(C^{0,\alpha}\) norms of all \(T_s\), uniformly in \(s\) and \(\delta\). Smooth composition on the compact argument set, together with the bound for \(v\), shows that \(a,f\) have uniform \(C^{0,\alpha}\) bounds and \(b,d\) are uniformly bounded. The matrices \(a\) are symmetric and uniformly elliptic.

Test [eq:equality-quotient-equation] with \(\chi^2v\), where \(\chi\) is supported in \(B_{3\rho}\) and equals one on \(B_{2\rho}\). Such tests are justified by smooth approximation. Ellipticity gives a multiple of \(\int\chi^2|Dv|^2\) on the left. All other terms are bounded by Young’s inequality and the uniform bounds on \(v,f,b,d\), absorbing the terms linear in \(Dv\) into that left side. In particular, the first-order term involving \(b\cdot Dv\) is absorbed in this way. The remaining bound is \(C\int(\chi^2+|D\chi|^2)\), so \[ \int_{B_{2\rho}}|Dv|^2\,dy\le C, \tag{76}\] uniformly in the difference step. No bound on a second derivative of \(u\) has been used.

Harmonic comparison and the gradient bound. Fix \(x\in B_\rho\) and \(0<r\le\min\{\rho,1\}\). On \(B_r(x)\), let \(w-v\in H^1_0(B_r(x))\) solve \[\mathop{\mathrm{div}}(a(x)Dw)=0.\] Existence and uniqueness follow by minimizing the positive quadratic Dirichlet energy in \(v+H^1_0(B_r(x))\). Put \(z=v-w\). Subtract the equations and test with \(z\); the constant vector \(f(x)\) integrates to zero against \(Dz\). With all norms on \(B_r(x)\), ellipticity and the coefficient bounds give \[\begin{split} \lambda\|Dz\|_2^2 &\le Cr^\alpha \bigl(\|Dv\|_2+|B_r|^{1/2}\bigr)\|Dz\|_2\\ &\quad+C\bigl(\|Dv\|_2+|B_r|^{1/2}\bigr)\|z\|_2. \end{split}\] The second line includes the first-order term \(b\cdot Dv\). Zero-boundary Poincaré gives \(\|z\|_2\le Cr\|Dz\|_2\). Since \(r\le r^\alpha\) for \(r\le1\), it follows that \[ \left(\mathchoice{{\setbox 0=\hbox{$\displaystyle\int$}\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\textstyle\int$}\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\scriptstyle\int$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\scriptscriptstyle\int$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}\!\int_{B_r(x)}|Dv-Dw|^2\right)^{1/2} \le Cr^\alpha\left( 1+\left(\mathchoice{{\setbox 0=\hbox{$\displaystyle\int$}\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\textstyle\int$}\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\scriptstyle\int$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\scriptscriptstyle\int$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}\!\int_{B_r(x)}|Dv|^2\right)^{1/2}\right). \tag{77}\] Energy minimization after subtracting an affine function also gives, for every constant vector \(p\), \[ \left(\mathchoice{{\setbox 0=\hbox{$\displaystyle\int$}\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\textstyle\int$}\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\scriptstyle\int$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\scriptscriptstyle\int$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}\!\int_{B_r(x)}|Dw-p|^2\right)^{1/2} \le C\left(\mathchoice{{\setbox 0=\hbox{$\displaystyle\int$}\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\textstyle\int$}\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\scriptstyle\int$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\scriptscriptstyle\int$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}\!\int_{B_r(x)}|Dv-p|^2\right)^{1/2}. \tag{78}\]

We use the following elementary interior estimate for this constant coefficient equation. For \(0<\gamma\le1/4\), \[ \left(\mathchoice{{\setbox 0=\hbox{$\displaystyle\int$}\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\textstyle\int$}\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\scriptstyle\int$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\scriptscriptstyle\int$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}\!\int_{B_{\gamma r}(x)} |Dw-(Dw)_{B_{\gamma r}(x)}|^2\right)^{1/2} \le C\gamma \left(\mathchoice{{\setbox 0=\hbox{$\displaystyle\int$}\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\textstyle\int$}\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\scriptstyle\int$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\scriptscriptstyle\int$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}\!\int_{B_r(x)}|Dw-p|^2\right)^{1/2}. \tag{79}\] Here subscripts on a function denote its average. To justify the estimate at the weak regularity in use, mollify \(Dw-p\) in an interior ball. Its components solve the same constant coefficient equation. A linear coordinate change, with distortion controlled by ellipticity, makes them harmonic. The mean value formula reproduces each such function by convolution with a smooth radial averaging kernel of radius comparable to \(r\). Differentiating that kernel and applying Cauchy–Schwarz bounds its gradient on \(B_{r/2}(x)\) by \[Cr^{-1}\left(\mathchoice{{\setbox 0=\hbox{$\displaystyle\int$}\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\textstyle\int$}\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\scriptstyle\int$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\scriptscriptstyle\int$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}\!\int_{B_r(x)}|Dw-p|^2\right)^{1/2}.\] The radius of the averaging kernel is chosen small enough that its support remains in the transformed interior ball. Letting the mollification scale tend to zero proves the same estimate for a representative of \(Dw-p\), and its oscillation on \(B_{\gamma r}(x)\) is at most this bound times \(2\gamma r\). This proves [eq:equality-harmonic-oscillation].

We control the mean gradient and its oscillation together: harmonic comparison contracts the oscillation, while the coefficient errors are summable on geometric scales. Define \[p_r=\mathchoice{{\setbox 0=\hbox{$\displaystyle\int$}\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\textstyle\int$}\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\scriptstyle\int$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\scriptscriptstyle\int$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}\!\int_{B_r(x)}Dv, \qquad J(r)=\left(\mathchoice{{\setbox 0=\hbox{$\displaystyle\int$}\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\textstyle\int$}\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\scriptstyle\int$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}{{\setbox 0=\hbox{$\scriptscriptstyle\int$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd 0}}\!\int_{B_r(x)}|Dv-p_r|^2\right)^{1/2}.\] Using \(p=p_r\) in the preceding estimates and comparing averages on the two balls gives \[\begin{align*} J(\gamma r) &\le C\gamma J(r) +C\gamma^{-m/2}r^\alpha\bigl(1+|p_r|+J(r)\bigr),\tag{80}\\ |p_{\gamma r}-p_r|&\le\gamma^{-m/2}J(r). \tag{81}\end{align*}\] Fix \(\gamma\) so that \(C\gamma\le1/2\), and choose a fixed \(\ell\ge1\) with \(\ell/2\ge\gamma^{-m/2}\). Then \[S(r)=1+|p_r|+\ell J(r) \quad\hbox{satisfies}\quad S(\gamma r)\le(1+C'r^\alpha)S(r).\] Choose one starting radius \(r_*\le\min\{\rho,1\}\). Its starting averages are bounded uniformly for every \(x\in B_\rho\) by [eq:equality-quotient-energy]. On the geometric scales \(r_j=\gamma^jr_*\), the product of \(1+C'r_j^\alpha\) is finite, because \(\sum_jr_j^\alpha<\infty\). Thus \(|p_{r_j}|\) is uniformly bounded. Each fixed-step quotient is \(C^1\), so these averages tend to \(Dv(x)\) as \(j\to\infty\). We obtain a bound for \(Dv(x)\) uniform in \(x\in B_\rho\), in \(e\), and in the sufficiently small step \(\delta\).

Finally, \[Dv(y)=\frac{Du(y+\delta e)-Du(y)}{\delta}\] holds classically. The uniform bound just proved bounds increments of \(Du\) along every coordinate direction. On a smaller coordinate cube, join two points by coordinate segments and add the increment bounds. The sum of their lengths is at most \(\sqrt m\) times the Euclidean distance between the points. Therefore \(Du\) is locally Lipschitz and \(u\in C^{1,1}_{\mathrm{loc}}(\Omega)\). ◻

Proof. If \(Z=\varnothing\), then \(\Sigma=\Gamma\) is compact, and we take \(\eta_j=1\). Otherwise \(n\ge8\), \(m\ge7\), and \[\dim_{\mathcal H}Z\le m-7<m-2, \qquad \mathcal H^{m-2}(Z)=0.\] The compactness, finite area, embedding and uniform growth hypotheses of 18 follow from 24. That lemma supplies the stated cutoffs. In particular their supports are compact in the manifold topology of \(\Sigma\), and the construction does not assume that \(\Sigma\) is complete or has finitely many components. ◻