Proof. Let \(T\) be the cycle in a proper \(\operatorname{CAT}(0)\) space \(X\). If \(T=0\), take the zero filling. Otherwise choose a closed ball \(Y=\overline B(o,R)\) whose interior contains \(\operatorname{spt}T\). Properness makes \(Y\) compact. Triangle comparison implies that balls are convex, so the induced metric makes \(Y\) a compact \(\operatorname{CAT}(0)\) space.

For \(x\in X\), let \(P(x)\) be the point of the segment \([o,x]\) at distance \(\min\{R,d(o,x)\}\) from \(o\). Geodesics in a \(\operatorname{CAT}(0)\) space are unique. Comparing two such radial points with their counterparts in the Euclidean comparison triangle, and then using nonexpansion of Euclidean radial projection onto a closed ball, gives \[d(P(x),P(x'))\le d(x,x').\] One can also identify \(P(x)\) as the nearest point of \(Y\): the triangle inequality bounds the distance from \(x\) to \(Y\) below by \(\max\{0,d(o,x)-R\}\), and the radial point attains this bound.

Lipschitz pushforward therefore gives an integral cycle \(P_\#T\) in \(Y\) with \(\mathbf M(P_\#T)\le\mathbf M(T)\). If \(\iota:Y\mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longmapstoP PaperOriginallongmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \let\LongrightarrowP PaperOriginalLongrightarrow\endcsname \let\LongleftrightarrowP PaperOriginalLongleftrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF21AA> >> BDC}% PPaperOriginalhookrightarrow\endcsname\pdfliteral direct{EMC}\endgroup}X\) denotes inclusion, locality gives \(\iota_\#P_\#T=T\), because \(\iota\circ P\) is the identity on a neighborhood of \(\operatorname{spt}T\). Apply the compact-space inequality to \(P_\#T\) and include its filling into \(X\). Its mass does not increase, its boundary is \(T\), and its support is contained in the compact set \(Y\). ◻

Proof. The pushforward and Borel-coefficient properties are part of the basic current calculus of (Ambrosio and Kirchheim 2000). The exact restriction identity also follows from the restriction mass bound and the decomposition \(C=C\mathbin{\vrule height 1.4ex depth -0.3ex width .07ex\vrule height .07ex depth 0ex width .7ex}E+C\mathbin{\vrule height 1.4ex depth -0.3ex width .07ex\vrule height .07ex depth 0ex width .7ex}(Y\setminus E)\), using minimality of the mass measure in both directions. The representation in (iii) is the integer-rectifiable representation (Ambrosio and Kirchheim 2000, Lemma 4.1 and Theorem 4.5); subdivision makes chart domains bounded and disjointification makes their images disjoint. The slicing assertion is (Ambrosio and Kirchheim 2000, Theorems 5.6–5.7). The compactness and closure theorems (Ambrosio and Kirchheim 2000, Theorems 5.2 and 8.5) give (ii): completeness is automatic, the bounded normal masses are assumed, and compactness of \(Y\) supplies uniform tightness. The product statements, including the action convention, are (Wenger 2007, Definition 3.1 and Theorem 3.2). They apply because an integral current is normal and its support in \(Y\) is bounded.

Here is the geometric verification of the filling hypotheses. Choose a point \(o\) in the support of a nonzero cycle \(B\), and let \(H(t,x)\) be the point at fraction \(t\) along \([o,x]\). Triangle comparison gives \[d(H(t,x),H(t,x'))\le t\,d(x,x'),\qquad d(H(t,x),H(s,x))=|t-s|d(o,x).\] Thus \(H\) is jointly Lipschitz on \([0,1]\times Y\). On the support of \(B\), its time speed is bounded by \(D=\mathop{\mathrm{diam}}(\operatorname{spt}B)\), and its spatial Lipschitz constant is at most one. In each of the \(k+1\) terms of (8), a \(1\)-Lipschitz scalar test after composition with \(H\) has time derivative at most \(D\), and the remaining spatial derivatives have Lipschitz bounds at most one. The mass bound therefore gives \[ \mathbf M\bigl(H_\#([0,1]\times B)\bigr) \le (k+1)D\mathbf M(B). \tag{11}\] The boundary is \(B\): the terminal map is the identity, the initial map is constant and hence has zero pushforward in positive dimension, and \(\partial B=0\). The filling is supported on segments from \(o\) to \(\operatorname{spt}B\), all lying within distance \(D\) of \(o\). Consequently the space has local cone inequalities in every positive current dimension. Taking \(D\le\mathop{\mathrm{diam}}Y\) gives the second estimate in (9), for example with \(L_k=(k+1)\mathop{\mathrm{diam}}Y\).

The complete space \(Y\) has the convex geodesic bicombing given by its unique geodesics. Wenger’s Euclidean-exponent isoperimetric theorem (Wenger 2005, Theorem 1.2 and Corollary 1.4), or its cone-inequality form applied successively in dimension, gives the first estimate in (9) for every \(k\ge1\). No sharp value of \(K_k\) is used here.

For a fixed boundary \(B\), the cone gives at least one filling. A minimizing sequence has bounded mass and fixed boundary, so (ii) gives an integral weak limit with that boundary. Lower semicontinuity proves attainment. Finally, \(Y\) is complete and geodesic, hence quasiconvex, and it has the local cone inequalities in dimensions \(1,\ldots,k\) just verified. The cycle assertion in (Wenger 2007, Theorem 1.4) applies to the weakly convergent sequence \(B_j\): its normal masses are bounded because its boundary masses are zero. It gives \(V(B_j-B)\to0\) directly. Finiteness and subadditivity of filling volume imply \[|V(B_j)-V(B)|\le V(B_j-B),\] which proves the second limit. ◻

Proof. The linear bound in (9) implies \[V(B)-c\mathbf M(B)^{q_0}\le L_n\mathbf M(B)-c\mathbf M(B)^{q_0}.\] The right side has finite supremum, and positivity implies \(\mathbf M(B)<(L_n/c)^n\). A positive maximizing sequence therefore has uniformly bounded masses and zero boundaries. By Theorem 4, a subsequence converges weakly to an integral cycle \(A\). Filling continuity and lower semicontinuity of mass give \[V(A)-c\mathbf M(A)^{q_0} \ge\limsup_j\bigl(V(B_j)-c\mathbf M(B_j)^{q_0}\bigr).\] Thus \(A\) attains the positive supremum, and in particular \(\mathbf M(A)>0\). Maximality and subadditivity now give \[c\bigl(\mathbf M(A)^{q_0}-\mathbf M(B)^{q_0}\bigr) \le V(A)-V(B)\le V(A-B),\] which is (12). ◻

Proof. At a density-one point \(z\) of \(E_i\), every fixed finite configuration in \(\mathbb R^k\) can be approximated by points of \((E_i-z)/t\) as \(t\downarrow0\). Indeed, a point of such a configuration that stayed a positive distance from \((E_i-z)/t\) along a sequence of scales would give a missing Euclidean ball of radius comparable to \(t\) at distance \(O(t)\) from \(z\), contradicting density one. Applying this observation to a single direction and using the bi-Lipschitz lower bound for \(\phi_i\) shows that \(N_z(w)\ge\operatorname{Lip}(\phi_i^{-1})^{-1}|w|\). The upper Lipschitz bound gives \(N_z(w)\le\operatorname{Lip}(\phi_i)|w|\). Thus \(N_z\) is a norm.

For any four points \(a,b,c,d\) of a \(\operatorname{CAT}(0)\) space, \[ d(a,c)^2+d(b,d)^2 \le d(a,b)^2+d(b,c)^2+d(c,d)^2+d(d,a)^2. \tag{17}\] To verify this, let \(m\) be the midpoint of \([a,c]\). Midpoint comparison bounds \(d(b,m)^2\) and \(d(d,m)^2\) by the corresponding average squared distances to \(a,c\), minus \(d(a,c)^2/4\). Combine these bounds with \(d(b,d)^2\le2d(b,m)^2+2d(d,m)^2\) to obtain (17).

Approximate the four Euclidean domain points \(0,u,u+v,v\) by scaled points of \(E_i-z\). Apply (17) to their images, divide by \(t^2\), and use the two-point differential (14). It follows that \[N_z(u+v)^2+N_z(u-v)^2\le2N_z(u)^2+2N_z(v)^2.\] Apply the same inequality to \((u+v)/2\) and \((u-v)/2\), and use homogeneity, to obtain the reverse inequality. The parallelogram law therefore holds. Polarization produces a positive definite inner product and the matrix \(G_i(z)\) in (15). Its entries are measurable, since they are finite linear combinations of squared metric differentials in fixed rational directions. Setting \(w'=0\) proves (16). ◻

Proof. Fix a tuple of \(d\) scalar \(1\)-Lipschitz tests. Almost everywhere on each domain their compositions with \(\psi_i\) are differentiable in the sense of scalar Lipschitz extensions. If \(\ell\) is one of the resulting derivative rows, (20) gives \[|\ell(w)|\le\sqrt{P_i(z)[w,w]}\qquad(w\in\mathbb R^d).\] To see the bound in an arbitrary direction, approximate that direction by rescaled domain points at a density-one point, and then use differentiability and (20). If \(P_i\) is positive definite, transform it to the identity and apply Hadamard’s inequality to the rows. Their absolute determinant is at most \(\sqrt{\det P_i}\). If \(P_i\) is singular, all rows vanish on its kernel; the full determinant and \(\det P_i\) both vanish, giving the same bound. Substitution in the representation yields (5) with the measure \(\eta\).

The exceptional set is allowed to depend on the fixed test tuple. The measure \(\eta\) is independent of that tuple and controls every tuple separately; this is precisely what the definition of mass requires. ◻

Proof. Denote the right side of (22) by \(\eta\), initially allowing \(\eta(Y)=\infty\). We first prove \(\eta\le\|C\|\); this will also establish finiteness before we apply the preceding lemma.

Fix \(0<\varepsilon<1\). Each chart domain, up to a Lebesgue-null set, has a countable Borel partition into pieces \(E\) on which there is a fixed positive definite matrix \(Q\), depending on the piece, satisfying \[ (1-\varepsilon)|z-z'|_Q \le d(\phi_i(z),\phi_i(z')) \le(1+\varepsilon)|z-z'|_Q, \qquad z,z'\in E. \tag{23}\] Here is a construction. Approximate the measurable matrix \(G_i(z)\) by an element of a countable dense family of positive definite matrices, with a sufficiently small relative error. The centered estimate (16) then gives a radius \(1/l\), depending on the point, on which the two inequalities with this fixed matrix and the prescribed \(\varepsilon\) hold against every other point of the original chart domain. Group points according to the matrix and \(l\), subdivide into sets of diameter less than \(1/l\), and disjointify. These sets can be chosen Borel. Indeed, for any fixed matrix and radius, the non-strict distance inequalities against all domain points with \(0<|z-z'|<1/l\) can be tested on a fixed countable dense subset of the domain, using continuity in \(z'\). All metric differentiability points are covered by this countable construction.

At almost every density point of a piece \(E\), differentiation of (23) in domain directions gives \[|w|_{G_i(z)}\le(1+\varepsilon)|w|_Q, \qquad J_i(z)\le(1+\varepsilon)^k\sqrt{\det Q}.\] On \(\phi_i(E)\), every scalar coordinate of \(Q^{1/2}\phi_i^{-1}\) is \((1-\varepsilon)^{-1}\)-Lipschitz by the lower inequality in (23). By McShane’s theorem (Rieffel 2006, Theorem 2.1), extend these coordinates separately to all of \(Y\), preserving that Lipschitz constant, and call the resulting tuple \(\pi\).

For any Borel subset \(D\subset E\), choose the bounded Borel first coefficient supported on \(\phi_i(D)\) and equal there to \(\operatorname{sgn}(\theta_i)\circ\phi_i^{-1}\). At density points of \(E\), the extended coordinates have the same derivatives as \(Q^{1/2}z\), since they agree with them on the piece. The images of all original charts are disjoint. Hence (6), applied to this coefficient and tuple, gives exactly \[\int_D|\theta_i|\sqrt{\det Q}\,dz = C(b,\pi_1,\ldots,\pi_k) \le(1-\varepsilon)^{-k}\|C\|(\phi_i(D)).\] The mass bound is legitimate for this Borel coefficient by its standard extension from the first slot. Combining the last two estimates gives \[\int_D|\theta_i|J_i\,dz \le\left(\frac{1+\varepsilon}{1-\varepsilon}\right)^k \|C\|(\phi_i(D)).\] For a Borel set \(B\subset Y\), take \(D=E\cap\phi_i^{-1}(B)\) and sum over all pieces and all charts. Their images are disjoint, while discarded Lebesgue-null sets make no contribution to \(\eta\). Thus \[\eta(B)\le \left(\frac{1+\varepsilon}{1-\varepsilon}\right)^k\|C\|(B).\] This proves that \(\eta\) is finite; letting \(\varepsilon\downarrow0\) gives \(\eta\le\|C\|\).

Conversely, (16) gives the quadratic majorant \(P_i=G_i\) for every chart. Its measure (21) is exactly the now finite measure \(\eta\). Lemma 7 yields \(\|C\|\le\eta\), which completes the proof of the equality. ◻

Proof. By Proposition 8, it suffices to take \(y=\phi_i(z)\), where \(z\) is a density-one point of \(E_i\), the metric differential exists, \(z\) is a Lebesgue point of \(\rho_i=|\theta_i|J_i\) extended by zero off \(E_i\), and \(|\theta_i(z)|\ge1\). These conditions hold at almost every point carrying chart mass. Fix \(\varepsilon>0\). For all sufficiently small \(\rho\), the domain points in the ellipsoid \[\{z+w:|w|_{G_i(z)}<\rho/(1+\varepsilon)\}\] map into \(B(y,\rho)\), by (16). This ellipsoid has volume \[\frac{\omega_k\rho^k}{(1+\varepsilon)^kJ_i(z)}.\] Lebesgue differentiation on dilates of this fixed ellipsoid shows that the contribution of this chart to \(\rho^{-k}\|C\|(B(y,\rho))\) has lower limit at least \(\omega_k|\theta_i(z)|/(1+\varepsilon)^k\). Other charts contribute nonnegative mass. Letting \(\varepsilon\downarrow0\) and using \(|\theta_i(z)|\ge1\) proves (24). ◻

Proof. Fix a level at which \(U_t\) is integral, and write \(a=a(t)\) and \(P=P(t)\). Its boundary is an integral \((n-1)\)-cycle. By the induction hypothesis there is an integral \(n\)-current \(R\) in \(Y\) satisfying \[\partial R=\partial U_t,\qquad d:=\mathbf M(R)\le \left(\frac{P}{n\omega_n^{1/n}}\right)^{n/(n-1)}.\] Indeed, the coefficient on the right is the sharp filling coefficient in dimension \(n-1\). The currents \(B=A-U_t+R\) and \(A-B=U_t-R\) are integral cycles. Restriction decomposes the mass measure exactly, so \[\mathbf M(B)\le m-a+d,\qquad \mathbf M(A-B)\le a+d.\] No disjointness involving \(R\) is needed for these upper bounds.

Suppose first that \(d<a\le m/2\). The extremizer inequality (12) and the coarse filling estimate in Theorem 4 give \[\begin{align*} cq_0(m/2)^{1/n}(a-d) &\le c\bigl(m^{q_0}-(m-a+d)^{q_0}\bigr)\\ &\le V(A-B) \le K_n(a+d)^{q_0} \le K_n(2a)^{q_0}. \end{align*}\] The first inequality follows by integrating the derivative of \(s^{q_0}\) over \([m-a+d,m]\subset[m/2,m]\). Thus \[ d\ge a-Ca^{1+1/n},\qquad C=\frac{K_n2^{q_0}}{cq_0(m/2)^{1/n}} =nK_n2^{1+2/n}. \tag{43}\] If \(d\ge a\), the same lower bound holds automatically. Choose \(0<\delta<m/2\) so small that \[C\delta^{1/n}\le1-(1-\eta)^{n/(n-1)}.\] For \(0<a\le\delta\), combining (43) with the upper bound for \(d\) yields \[P\ge n\omega_n^{1/n}a^{(n-1)/n} (1-Ca^{1/n})^{(n-1)/n} \ge (1-\eta)n\omega_n^{1/n}a^{(n-1)/n}.\] The case \(a=0\) is immediate. All constants used to choose \(\delta\) depend only on the fixed extremizer, the dimension, and \(\eta\). ◻

Proof. Fix a point \(z_0\in Z\) and normalize each differentiated test by \(\pi_i(z_0)=0\); adding constants does not change its current evaluation. The normalized \(1\)-Lipschitz functions form a compact metric space in the uniform topology, by the Arzelà–Ascoli theorem. Choose a countable dense family of normalized \(k\)-tuples \((\pi^j)_{j\ge1}\).

Write \(\lambda=\|C\|\). For each tuple \(\pi\), the first-slot extension to bounded Borel functions gives a finite signed measure \(b\mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longmapstoP PaperOriginallongmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \let\LongrightarrowP PaperOriginalLongrightarrow\endcsname \let\LongleftrightarrowP PaperOriginalLongleftrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF21A6> >> BDC}% PPaperOriginalmapsto\endcsname\pdfliteral direct{EMC}\endgroup}C(b,\pi)\) with density \(\rho_\pi\) relative to \(\lambda\), where \(|\rho_\pi|\le1\) almost everywhere. Choose measurable versions for the countable family and put \[\rho(z)=\sup_{j\ge1}|\rho_{\pi^j}(z)|.\] This satisfies \(0\le\rho\le1\). If a normalized tuple \(\pi\) is approximated uniformly by tuples in the family, current continuity, with their common Lipschitz bound, shows for every Lipschitz \(b\) that \[|C(b,\pi)|\le\int_Z|b|\rho\,d\lambda.\] Thus \(\rho\lambda\) is a mass-controlling measure for all normalized \(1\)-Lipschitz tuples and hence, by rescaling the differentiated slots, for all tests. Minimality of the mass measure gives \(\lambda\le\rho\lambda\). It follows that \(\rho=1\) almost everywhere.

For any finite set of the tuples, select at each point the first index where the largest \(|\rho_{\pi^j}|\) is attained and select its sign. This produces bounded Borel functions \(b_j\) with \(\sum_j|b_j|\le1\) and \[\sum_j C(b_j,\pi^j) =\int_Z\max_j|\rho_{\pi^j}|\,d\lambda.\] These coefficients can be replaced, with an arbitrarily small error in the displayed sum, by Lipschitz coefficients satisfying the same constraint. To see this, regularity of the finite Borel measure \(\lambda\) gives Lipschitz approximations to each \(b_j\) in \(L^1(\lambda)\). For example, an indicator is approximated between a compact set and an open neighborhood by a Lipschitz function obtained from distances to those sets, and simple functions suffice for the general case. Apply to the vector of approximations the Lipschitz map \[(v_1,\ldots,v_N)\mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longmapstoP PaperOriginallongmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \let\LongrightarrowP PaperOriginalLongrightarrow\endcsname \let\LongleftrightarrowP PaperOriginalLongleftrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF27FC> >> BDC}% PPaperOriginallongmapsto\endcsname\pdfliteral direct{EMC}\endgroup} \frac{(v_1,\ldots,v_N)}{\max\{1,\sum_j|v_j|\}}.\] This retains \(L^1\) convergence and enforces the constraint. Monotone convergence of the finite maxima to \(\rho=1\) now proves that the supremum over all admissible finite Lipschitz lists equals \(\mathbf M(C)\); the reverse inequality follows directly from the mass bound.

It remains to make the lists countable without depending on \(C\). For each fixed list length, the admissible lists of coefficients and normalized tuples form a separable metric space in the uniform topology: they are a subspace of a finite product of \(C(Z)\), which is separable because \(Z\) is compact metric. Choose a countable dense subset within that admissible space and take the union over list lengths. An admissible list can be approximated by these lists with its constraint and differentiated Lipschitz bounds preserved. The first-slot errors are controlled by their uniform norms times \(\mathbf M(C)\), and convergence in the differentiated slots follows from current continuity. Hence the countable collection has the same supremum. A common Lipschitz bound on the first coefficients is neither asserted nor needed. ◻

Proof. For a bounded open interval \(I\), define the bounded Lipschitz function \[H(s)=\int_I1_{\{t<s\}}\,dt.\] Let \(b\) be Lipschitz and let \(\pi\) be an \((n-1)\)-tuple of \(1\)-Lipschitz functions. Restriction, the chart representation, and Fubini give \[ \int_I\partial U_t(b,\pi)\,dt =A(H(u),b,\pi)=-A(b,H(u),\pi). \tag{46}\] For the first equality, the boundary evaluation has the measurable representative \(U_t(1,b,\pi)\), obtained by integrating the chart determinants with the factor \(1_{\{u>t\}}\). It is integrable on bounded intervals. The second equality is the product rule applied to the cycle identity \(\partial A=0\).

The chain rule in the charts gives \[D(H(u))=1_{\{u\in I\}}Du\qquad\mu\text{-almost everywhere}.\] There is no ambiguity at the two endpoints of \(I\): the differential of a Lipschitz scalar function vanishes almost everywhere on each fixed level set. This follows by applying Euclidean differentiation at density points of that level set in each chart. At points of the charts the other differentiated covectors have norm at most one, so the Euclidean determinant bound in Proposition 8 applies.

In particular, for an admissible finite list \(\mathcal T\) as in Lemma 13, (46) yields \[ \left|\int_I\sum_{(b,\pi)\in\mathcal T} \partial U_t(b,\pi)\,dt\right| \le\int_{\{u\in I\}}|Du|\,d\mu. \tag{47}\] Let \(\gamma=u_\#(|Du|\mu)\). Lebesgue differentiation of (47), simultaneously for the countable collection of lists on \(Y\), gives \[P(t)\le\frac{d\gamma_{\mathrm{ac}}}{dt}(t) \qquad\text{for almost every }t.\] Here we used (44) for each integral slice. Thus the passage from a signed integrated evaluation to total slice mass excludes only a countable union of null sets.

Set \(\lambda=u_\#\mu\) and \(\zeta=u_\#(|Du|^2\mu)\). For every interval \(I\), Cauchy–Schwarz on \(u^{-1}(I)\) and the bound \(|Du|\le\operatorname{Lip}(u)\) imply \[\gamma(I)^2\le\lambda(I)\zeta(I),\qquad \gamma(I)\le\operatorname{Lip}(u)\lambda(I).\] Divide by the appropriate powers of the interval length and let symmetric intervals shrink to an almost every point. The densities of \(\lambda_{\mathrm{ac}}\) and \(\zeta_{\mathrm{ac}}\) are respectively \(-a'\) and \(-e'\). Combining the resulting inequalities with the bound for \(P\) proves (45). ◻

Proof. If \(u=0\) almost everywhere, take \(u^*=0\). Otherwise let \(L=\operatorname{Lip}(u)\) and \(U=\operatorname*{ess\,sup}_\mu u\). Then \(U>0\) and \(L>0\): a function with Lipschitz constant zero is constant, and a positive constant would have positivity set of mass \(m>\delta\). For \(0\le t\le U\), define \[a(t)=\mu\{u>t\},\qquad R(t)=\left(\frac{a(t)}{\omega_n}\right)^{1/n}.\] The function \(R\) is nonincreasing and right-continuous, \(R(t)>0\) for \(t<U\), and \(R(U)=0\). For almost every \(t\in(0,U)\), Proposition 12 and Lemma 14 give \[ -R'(t)=\frac{-a'(t)}{n\omega_n R(t)^{n-1}} \ge c_0,\qquad c_0=\frac{1-\eta}{L}>0. \tag{49}\] This estimate also controls finite differences, even if \(R\) has a singular part. Indeed, on every compact subinterval of \((0,U)\), the nonnegative measure \(-dR\) has absolutely continuous density \(-R'\ge c_0\) and a nonnegative singular part. Therefore \[ R(s)-R(t)\ge c_0(t-s) \qquad(0<s<t<U). \tag{50}\]

Define the generalized inverse for \(\rho\ge0\) by \[ h(\rho)=\sup\{0\le t<U:\rho<R(t)\}, \qquad \sup\varnothing:=0. \tag{51}\] It is nonincreasing, \(h(0)=U\), and \(h(\rho)=0\) for \(\rho\ge R(0)\). It is also \(c_0^{-1}\)-Lipschitz. To check the latter claim, let \(\rho_1<\rho_2\) with \(h(\rho_1)>h(\rho_2)\) and choose \[h(\rho_2)<s<t<h(\rho_1).\] The definition and monotonicity imply \(R(s)\le\rho_2\) and \(R(t)>\rho_1\), whence \(c_0(t-s)\le R(s)-R(t)\le\rho_2-\rho_1\). Letting \(s\) and \(t\) approach the two inverse values proves the claim.

For \(0<t<U\), right continuity of \(R\) implies \[ h(\rho)>t\quad\mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longmapstoP PaperOriginallongmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \let\LongrightarrowP PaperOriginalLongrightarrow\endcsname \let\LongleftrightarrowP PaperOriginalLongleftrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF27FA> >> BDC}% PPaperOriginalLongleftrightarrow\endcsname\pdfliteral direct{EMC}\endgroup}\quad \rho<R(t). \tag{52}\] In the reverse implication, if \(\rho<R(t)\), right continuity supplies some \(s>t\) with \(\rho<R(s)\); the other implication follows from monotonicity. Consequently \(u^*(x)=h(|x|)\) is Lipschitz and compactly supported, and its superlevel sets have Euclidean volume \(a(t)\). The layer-cake formula proves the asserted equality of all positive power integrals.

We give the exact energy substitution, including possible atoms and singular continuous parts of \(a\). Strict decrease in (50) implies \[ h(R(t))=t\qquad(0<t<U). \tag{53}\] Indeed, all levels \(s<t\) satisfy \(R(s)>R(t)\), while no \(s\ge t\) is eligible in (51) with \(\rho=R(t)\). If \(R\) is continuous at \(t\), (52) and this continuity show that the entire fiber \(h^{-1}(\{t\})\) is the singleton \(\{R(t)\}\).

The monotone function \(R\) is continuous except at countably many points and differentiable with finite derivative almost everywhere. Its derivative, wherever it is relevant to (49), is nonzero. Let \(N\) be the null set where the Lipschitz function \(h\) is not differentiable. The set \(h(N)\) is null because Lipschitz functions map Lebesgue null sets to null sets. By (53), every level \(t\) with \(R(t)\in N\) lies in \(h(N)\). It follows that for almost every \(t\in(0,U)\) both derivatives needed below exist, and differentiating the inverse identity gives \[ h'(R(t))=\frac1{R'(t)}. \tag{54}\] For example, this follows directly by letting \(s\to t\) in \(h(R(s))-h(R(t))=s-t\) at a continuity and differentiability point of \(R\).

Since \(h\) is absolutely continuous and nonincreasing on \([0,R(0)]\), the measure \(|h'(\rho)|\,d\rho\) pushes forward under \(h\) to Lebesgue measure on \((0,U)\). One may verify this first on an interval \((a,b)\subset(0,U)\): integration of \(-h'\) over its inverse image equals \(b-a\) by the fundamental theorem of calculus; constant portions have zero derivative. Intervals determine the measure. At almost every target level the fiber consists of the single point \(R(t)\), so this substitution and (54) give \[\begin{align*} \int_{\mathbb R^n}|\nabla u^*|^2\,dx &=n\omega_n\int_0^{R(0)}\rho^{n-1}|h'(\rho)|^2\,d\rho \\ &=\int_0^U\frac{n\omega_n R(t)^{n-1}}{-R'(t)}\,dt \\ &=\int_0^U \frac{\bigl(n\omega_n R(t)^{n-1}\bigr)^2}{-a'(t)}\,dt. \tag{55}\end{align*}\] The denominators are positive and finite almost everywhere by (49). The preceding substitution also justifies the equality for a nonnegative integrand before its finiteness is known.

For clarity, a jump of \(R\) produces an interval on which \(h\) is constant. In particular, an atom at \(U\) produces a central constant part of \(u^*\). Such intervals contribute no energy. A singular continuous decrease causes no additional term either: the exact substitution uses the absolutely continuous measure \(|h'|\,d\rho\), and the inverse image of any null set of levels has zero measure for this measure. Thus no absolute-continuity assumption on \(a\) has entered (55).

Finally, (42) gives \[\bigl(n\omega_n R(t)^{n-1}\bigr)^2 \le(1-\eta)^{-2}P(t)^2.\] Using (45) in (55) therefore yields \[\int_{\mathbb R^n}|\nabla u^*|^2\,dx \le(1-\eta)^{-2}\int_0^U(-e'(t))\,dt \le(1-\eta)^{-2}E(u).\] The last inequality only uses monotonicity of \(e\) and remains true if its distributional derivative has a singular part. ◻

Proof. The assertion is immediate for \(f=0\). By homogeneity, assume \(\int f^p\,dx=1\). Write \(f=f(r)\), \(r=|x|\). Continuity and radial monotonicity give a finite \(r_0>0\) such that \(f>0\) on \([0,r_0)\) and \(f=0\) on \([r_0,\infty)\). For \(D>0\) define \[g_D(r)=A_D(1+r^2)^{-(n-2)/2}\quad(0\le r<D), \qquad g_D(r)=0\quad(r\ge D),\] where \(A_D>0\) is chosen so that \(\int g_D^p\,dx=1\). For \(0<r<r_0\) match the cumulative radial masses by \[ n\omega_n\int_0^r s^{n-1}f(s)^p\,ds =n\omega_n\int_0^{t(r)}s^{n-1}g_D(s)^p\,ds. \tag{57}\] Both cumulative functions are \(C^1\) with strictly positive derivative on their open radial intervals. Their inverse composition \(t\) is therefore \(C^1\) on \((0,r_0)\), strictly increasing, and satisfies \(t(0+)=0\) and \(t(r_0-)=D\). Differentiation gives \[ d_T:=t'(r)\left(\frac{t(r)}r\right)^{n-1} =\frac{f(r)^p}{g_D(t(r))^p}. \tag{58}\] The radial map \(T(x)=t(|x|)x/|x|\) has positive eigenvalues \(t'\) and \(n-1\) copies of \(t/r\) and transports \(f^p\,dx\) to \(g_D^p\,dx\). This follows either from (57) by radial integration, or from the change of variables and (58). The jump of \(g_D\) at \(D\) is harmless: the calculation uses its positive continuous interior profile and never differentiates the cutoff.

Let \[\ell=p\left(1-\frac1n\right)=\frac{2(n-1)}{n-2}, \qquad \ell-1=\frac p2.\] Change of variables, the arithmetic–geometric mean inequality for the positive eigenvalues of \(DT\), and radial integration by parts give \[\begin{align*} n\int_{\mathbb R^n}g_D^\ell\,dx &=n\int_{|x|<r_0}f^\ell d_T^{1/n}\,dx \\ &\le\int_{|x|<r_0}f^\ell \left(t'+(n-1)\frac tr\right)\,dx \\ &=-\ell\int_{|x|<r_0}f^{p/2}f't\,dx \\ &\le\ell\left(\int_{\mathbb R^n}|\nabla f|^2\,dx\right)^{1/2} \left(\int_{\mathbb R^n}|x|^2g_D^p\,dx\right)^{1/2}. \tag{59}\end{align*}\] The last equality needed for Cauchy–Schwarz is the transport identity \(\int f^pt^2\,dx=\int|x|^2g_D^p\,dx\). To justify the integration by parts, first integrate over \(\varepsilon<r<R<r_0\). The boundary term is \(n\omega_n[r^{n-1}tf^\ell]_{\varepsilon}^R\). It tends to zero at \(r_0\) because \(f(r_0)=0\) and \(t\le D\), and at zero because \(n>1\) and \(t,f\) are bounded. The resulting derivative integral is absolutely integrable: \(f'\) is essentially bounded, \(t\) is bounded, and the source ball is bounded. The source profile is Lipschitz, so its radial integration by parts is justified by absolute continuity.

Now put \[I_0=\int_{\mathbb R^n}(1+|x|^2)^{-n}\,dx, \qquad A_\infty=I_0^{-1/p},\qquad g(r)=A_\infty(1+r^2)^{-(n-2)/2}.\] The normalizing constants \(A_D\) converge to \(A_\infty\) as \(D\to\infty\). Moreover, \[\int g_D^\ell\,dx\mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longmapstoP PaperOriginallongmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \let\LongrightarrowP PaperOriginalLongrightarrow\endcsname \let\LongleftrightarrowP PaperOriginalLongleftrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF27F6> >> BDC}% PPaperOriginallongrightarrow\endcsname\pdfliteral direct{EMC}\endgroup}\int g^\ell\,dx, \qquad \int|x|^2g_D^p\,dx\mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longmapstoP PaperOriginallongmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \let\LongrightarrowP PaperOriginalLongrightarrow\endcsname \let\LongleftrightarrowP PaperOriginalLongleftrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF27F6> >> BDC}% PPaperOriginallongrightarrow\endcsname\pdfliteral direct{EMC}\endgroup}\int|x|^2g^p\,dx.\] For both integrals the unnormalized radial integrand is of order \(r^{1-n}\) at infinity. Thus both are finite for every \(n>2\), and the convergence follows by monotone convergence for the unnormalized integrals and convergence of \(A_D\). In particular, no limit of the transport maps is required.

The ratio on the limiting left side of (59) can be evaluated from \(g\) itself. Direct differentiation shows that \[-g'(r)=Krg(r)^{p/2},\qquad K=(n-2)A_\infty^{-2/(n-2)}>0.\] If \(J=\int g^\ell\,dx\) and \(M_2=\int|x|^2g^p\,dx\), radial integration by parts with the vector field \(x\) gives \[nJ=-\ell\int g^{p/2}g'r\,dx=\ell K M_2, \qquad \int|\nabla g|^2\,dx=K^2M_2.\] Its boundary term \(r^ng(r)^\ell\) vanishes at zero and is \(O(r^{2-n})\) at infinity. Therefore \[ \frac{nJ}{\ell M_2^{1/2}} =\left(\int|\nabla g|^2\,dx\right)^{1/2}. \tag{60}\] Taking \(D\to\infty\) in (59) proves \(\int|\nabla f|^2\,dx\ge\int|\nabla g|^2\,dx\).

The transport argument has reduced the lower energy bound to the single comparison profile \(g\). It remains to compute this energy and thus fix the threshold used in the current’s critical minimization. The inverse stereographic map \[x\mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longmapstoP PaperOriginallongmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \let\LongrightarrowP PaperOriginalLongrightarrow\endcsname \let\LongleftrightarrowP PaperOriginalLongleftrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF27FC> >> BDC}% PPaperOriginallongmapsto\endcsname\pdfliteral direct{EMC}\endgroup}\frac{(2x,|x|^2-1)}{1+|x|^2}\in\mathbb S^n\] pulls the sphere metric back to \(4(1+|x|^2)^{-2}\) times the Euclidean metric. Its volume Jacobian is \(2^n(1+|x|^2)^{-n}\), so \[ I_0=2^{-n}s_n. \tag{61}\] For \(w(r)=(1+r^2)^{-(n-2)/2}\), direct differentiation gives \[-\Delta w=n(n-2)(1+r^2)^{-(n+2)/2}.\] Integration by parts consequently yields \[\begin{align*} \int_{\mathbb R^n}|\nabla g|^2\,dx &=n(n-2)A_\infty^2I_0\\ &=n(n-2)I_0^{1-2/p} =\frac{n(n-2)}4s_n^{2/n}. \end{align*}\] Here the boundary term \(r^{n-1}g(r)g'(r)\) vanishes at zero and is \(O(r^{2-n})\) at infinity; also \(1-2/p=2/n\). Multiplying the energy lower bound by \(4/(n-2)\) proves (56) for the normalized \(f\), and homogeneity proves the general statement. All moments and boundary limits used above remain valid when \(n=3\) or \(n=4\); no \(L^2\) integrability of the full profile \(g\) is needed.

The coefficient is also optimal within the class in the statement. Indeed, choose a nonnegative nonincreasing Lipschitz cutoff \(\chi\) equal to one on \([0,1]\) and zero on \([2,\infty)\), and set \(g_R(x)=g(x)\chi(|x|/R)\). Then \(g_R\to g\) in \(L^p\), and \(\nabla g_R\to\nabla g\) in \(L^2\): the tail of \(\nabla g\) tends to zero, while the extra cutoff energy is bounded by \[\frac{\operatorname{Lip}(\chi)^2}{R^2} \int_{R<|x|<2R}g(x)^2\,dx=O(R^{2-n})\mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longmapstoP PaperOriginallongmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \let\LongrightarrowP PaperOriginalLongrightarrow\endcsname \let\LongleftrightarrowP PaperOriginalLongleftrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF27F6> >> BDC}% PPaperOriginallongrightarrow\endcsname\pdfliteral direct{EMC}\endgroup}0.\] Each \(g_R\) is admissible, and its Sobolev quotient tends to the computed value for \(g\). ◻

Proof. Fix temporarily \(\eta\in(0,1)\) and its corresponding \(\delta\). Combining Lemmas 15 and 16 gives \[ S_*(1-\eta)^2\|z\|_p^2\le4\beta E(z) \quad\text{if }z\ge0\text{ is Lipschitz and }\mu\{z>0\}\le\delta. \tag{63}\] This also holds when \(z=0\) almost everywhere.

For a general Lipschitz \(u\) with \(\|u\|_2>0\), set \[\tau=\frac{\|u\|_2}{\sqrt\delta},\qquad z=(|u|-\tau)_+.\] Chebyshev’s inequality gives \(\mu\{z>0\}\le\delta\). Scalar Lipschitz composition in the charts gives \(E(z)\le E(u)\); at the finitely many corner levels the gradient vanishes almost everywhere on the corresponding level sets, so the usual almost-everywhere chain rule applies. In addition, \(0\le|u|-z\le\tau\), whence \[\||u|-z\|_p\le m^{1/p}\tau.\] For every \(\xi>0\), the triangle inequality followed by \((b+d)^2\le(1+\xi)b^2+(1+\xi^{-1})d^2\) now yields \[\begin{align*} \frac{S_*(1-\eta)^2}{1+\xi}\|u\|_p^2 &\le S_*(1-\eta)^2\|z\|_p^2 +\frac{S_*(1-\eta)^2}{\xi}m^{2/p}\tau^2\\ &\le4\beta E(u) +\frac{S_*(1-\eta)^2m^{2/p}}{\xi\delta}\|u\|_2^2. \end{align*}\] Choose \(\eta>0\) and \(\xi>0\) sufficiently small that \(S_*(1-\eta)^2/(1+\xi)\ge S_*-\epsilon\), and then fix the associated \(\delta\). The last display proves the assertion with, for example, \[C_\epsilon=\frac{S_*(1-\eta)^2m^{2/p}}{\xi\delta}.\] If \(\|u\|_2=0\), then \(\|u\|_p=0\) and the asserted inequality is immediate. ◻

Proof. Let \(\mathcal L^2\) denote the Hilbert space of chart covector fields \(L\) with \(\int|L|^2\,d\mu<\infty\). The Lipschitz level-set property from Section 2, applied to the difference of two representatives, shows that their gradients agree whenever the functions agree \(\mu\)-almost everywhere. We first prove that the graph of the Lipschitz gradient in \(L^2(\mu)\times\mathcal L^2\) is closable. Suppose that \(u_j\) are Lipschitz, \[u_j\mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longmapstoP PaperOriginallongmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \let\LongrightarrowP PaperOriginalLongrightarrow\endcsname \let\LongleftrightarrowP PaperOriginalLongleftrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF27F6> >> BDC}% PPaperOriginallongrightarrow\endcsname\pdfliteral direct{EMC}\endgroup}0\quad\hbox{in }L^2(\mu),\qquad Du_j\mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longmapstoP PaperOriginallongmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \let\LongrightarrowP PaperOriginalLongrightarrow\endcsname \let\LongleftrightarrowP PaperOriginalLongleftrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF27F6> >> BDC}% PPaperOriginallongrightarrow\endcsname\pdfliteral direct{EMC}\endgroup}L\quad\hbox{in }\mathcal L^2.\] For a Lipschitz scalar function \(b\) and a Lipschitz \((n-1)\)-tuple \(\pi\), the identity \(\partial A=0\) gives \[ A(b,u_j,\pi)=-A(u_j,b,\pi). \tag{64}\] The right-hand side tends to zero by the mass bound and \(\|u_j\|_1\le m^{1/2}\|u_j\|_2\). For the fixed tuple \(\pi\), the limit of the left-hand side is integration of \(b\) against the signed measure \[\nu_\pi(B)=\sum_i \int_{\phi_i^{-1}(B\cap Z_i)} \theta_i\det\bigl(L_i,D(\pi\circ\phi_i)\bigr)\,dz, \qquad B\subset Y\ \hbox{Borel}.\] Here \(L_i\) is the row of coordinate components of \(L\) in the \(i\)th chart. The determinant estimate of Proposition 8 shows that this is a finite signed measure and that \[\|\nu_\pi\|_{\mathrm{TV}} \le \left(\prod_{a=1}^{n-1}\operatorname{Lip}(\pi_a)\right) \int|L|\,d\mu.\] The same estimate applied to \(Du_j-L\) justifies the asserted limit. Thus \(\int b\,d\nu_\pi=0\) for every Lipschitz \(b\) on \(Y\). Lipschitz functions are uniformly dense in \(C(Y)\), so they determine finite signed measures on the compact metric space \(Y\). It follows that \(\nu_\pi=0\) as a measure.

Fix a chart \(i\). Extend each coordinate of \(\phi_i^{-1}:Z_i\to\mathbb R^n\) to a Lipschitz scalar function \(F^a\) on \(Y\). At almost every density point of \(E_i\), \[D(F^a\circ\phi_i)=e_a^*.\] Take for \(\pi\) each of the \(n\) tuples obtained by omitting one member of \((F^1,\ldots,F^n)\). Restricting the corresponding zero measures \(\nu_\pi\) to \(Z_i\) shows, component by component, that \(\theta_i L_i=0\) almost everywhere on \(E_i\). The multiplicity is nonzero, so \(L_i=0\). There are only countably many charts and finitely many tuples per chart; hence \(L=0\) in \(\mathcal L^2\). Notice the order of this argument: we first obtained zero signed measures by testing on the whole cycle, and then localized those measures. No boundary or normality property of a chart restriction is used.

The closure of the graph is therefore the graph of an operator \(D\), and, with its graph norm, it is the asserted Hilbert space \(\mathcal H\). In particular, functions that agree almost everywhere have the same gradient. Applying (62) to differences of Lipschitz approximants shows that a graph-norm Cauchy sequence is Cauchy in \(L^p\). Its \(L^p\) limit agrees with its \(L^2\) limit, and passing to the limit proves both the continuous inclusion and the extension of (62).

To prove the first calculus assertion, choose Lipschitz approximants \(u_{a,j}\to u_a\) in \(\mathcal H\), and pass to a common subsequence converging almost everywhere. Write \(U_j=(u_{1,j},\ldots,u_{k,j})\) and \(U=(u_1,\ldots,u_k)\). Bounded first derivatives give \(\Phi(U_j)\to\Phi(U)\) in \(L^2\). The constant \(\Phi(0)\) is harmless because \(\mu\) is finite. The chart chain rule gives \[D\Phi(U_j)=\sum_a\partial_a\Phi(U_j)Du_{a,j}.\] For each summand, subtract the proposed limit by writing \[\partial_a\Phi(U_j)(Du_{a,j}-Du_a) +\bigl(\partial_a\Phi(U_j)-\partial_a\Phi(U)\bigr)Du_a.\] The first term tends to zero in \(\mathcal L^2\) by the derivative bound; the second does so by dominated convergence. Closedness proves assertion (i).

For assertion (ii), fix a level \(t\) and choose \(\eta\in C_c^\infty((-1,1))\) with \(0\le\eta\le1\) and \(\eta(0)=1\). The functions \[\Phi_\epsilon(s)=\int_{-\infty}^s \eta\bigl((a-t)/\epsilon\bigr)\,da\] converge uniformly to zero, their derivatives are bounded by one, and \(\Phi_\epsilon'(s)\to\mathbf1_{\{t\}}(s)\). Assertion (i) and dominated convergence give \[\Phi_\epsilon(u)\mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longmapstoP PaperOriginallongmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \let\LongrightarrowP PaperOriginalLongrightarrow\endcsname \let\LongleftrightarrowP PaperOriginalLongleftrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF27F6> >> BDC}% PPaperOriginallongrightarrow\endcsname\pdfliteral direct{EMC}\endgroup}0\quad\hbox{in }L^2, \qquad D\Phi_\epsilon(u)\mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longmapstoP PaperOriginallongmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \let\LongrightarrowP PaperOriginalLongrightarrow\endcsname \let\LongleftrightarrowP PaperOriginalLongleftrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF27F6> >> BDC}% PPaperOriginallongrightarrow\endcsname\pdfliteral direct{EMC}\endgroup}\mathbf1_{\{u=t\}}Du \quad\hbox{in }\mathcal L^2.\] Closedness forces the latter limit to be zero. A Lipschitz, piecewise \(C^1\) scalar function with finitely many corners can now be smoothed by convolution. The smoothed functions converge uniformly, their derivatives have a common bound, and the derivatives converge at every point other than the corners. The level-set conclusion removes the contribution of those corners, so dominated convergence and closedness give the claimed chain rule.

In particular, \(D(u_+)=\mathbf1_{\{u>0\}}Du\). If \(u_j\to u\) in \(\mathcal H\), the positive parts converge in \(L^2\), and their gradient difference is \[\mathbf1_{\{u_j>0\}}(Du_j-Du) +\bigl(\mathbf1_{\{u_j>0\}}-\mathbf1_{\{u>0\}}\bigr)Du.\] Along any subsequence with almost-everywhere convergence, the second term tends to zero in \(\mathcal L^2\) by the level-set conclusion at zero and dominated convergence. The first term is bounded in norm by \(\|Du_j-Du\|_2\). Applying this observation to subsequences proves continuity for the original sequence. Taking positive parts of Lipschitz approximants proves assertion (iii).

Finally fix \(y\in Y\). The distance \(r=d(y,\cdot)\) is bounded, \(|Dr|\le1\) almost everywhere, and \(\mu\) is finite. Each term of (26) is therefore continuous as a linear functional of \(g\) in the graph norm. Approximate a nonnegative \(g\in\mathcal H\) by the nonnegative Lipschitz functions just constructed and pass to the limit. This proves assertion (iv). ◻

Proof. Projecting the whole weighted current. First consider Lipschitz functions \(u_j\) with \(\sup_j\|u_j\|_{\mathcal H}\le M_0<\infty\). Fix a chart \(i\), extend its inverse coordinates to a Lipschitz map \(F=(F^1,\ldots,F^n):Y\to\mathbb R^n\), and fix a Lipschitz function \(\chi\) on \(Y\). Define finite signed measures \(\lambda_{j,\chi}\) on \(\mathbb R^n\) by \[ \int h\,d\lambda_{j,\chi} =A\bigl((h\circ F)\chi u_j,F^1,\ldots,F^n\bigr) \tag{65}\] for bounded Borel \(h\). The chart representation defines these measures directly. They are supported in the fixed compact set \(F(Y)\), and the current mass bound gives \[\sup_j\|\lambda_{j,\chi}\|_{\mathrm{TV}} \le \left(\prod_{a=1}^n\operatorname{Lip}(F^a)\right) \|\chi\|_\infty m^{1/2}M_0.\]

For \(h\in C_c^\infty(\mathbb R^n)\), expanding the row \(D(h\circ F)=\sum_a(\partial_a h\circ F)DF^a\) in a determinant shows that \[\int\partial_l h\,d\lambda_{j,\chi} =A\bigl(\chi u_j,F^1,\ldots,F^{l-1},h\circ F, F^{l+1},\ldots,F^n\bigr).\] All terms except \(a=l\) in the row expansion vanish because they have a repeated row. Apply the cycle identity (64), with a permutation of rows if necessary, to move \(h\circ F\) to the first coefficient. The intrinsic determinant bound then gives \[ \left|\int\partial_lh\,d\lambda_{j,\chi}\right| \le C_F\|h\|_\infty\int|D(\chi u_j)|\,d\mu \le C_\chi\|h\|_\infty, \tag{66}\] uniformly in \(j\) and \(l\). Indeed, \[\int|D(\chi u_j)|\,d\mu \le m^{1/2}\bigl(\|\chi\|_\infty\|Du_j\|_2 +\operatorname{Lip}(\chi)\|u_j\|_2\bigr).\] Constants here may depend on the fixed chart, \(\chi\), and \(M_0\).

Translation compactness. We give the compactness consequence of (66) in detail. This is the usual translation and mollification proof of Euclidean BV compactness (Simon 2014, chap. 2, Theorem 2.6), expressed directly for the projected signed measures. Let \(\tau_w(x)=x+w\). For a smooth compactly supported test \(h\), the fundamental theorem of calculus along \(x+tw\) and (66) imply \[\left|\int h\,d\bigl((\tau_w)_\#\lambda_{j,\chi} -\lambda_{j,\chi}\bigr)\right| \le C_\chi\|h\|_\infty\sum_{l=1}^n|w_l|.\] Smooth compactly supported tests recover the total variation of finite signed Radon measures, by regularity and approximation of continuous tests. After changing the constant, we obtain \[ \| (\tau_w)_\#\lambda_{j,\chi}-\lambda_{j,\chi}\|_{\mathrm{TV}} \le C_\chi|w|. \tag{67}\] Choose a nonnegative smooth convolution kernel \(\rho_\epsilon\) of integral one supported in the ball of radius \(\epsilon\). Averaging (67) gives \[\|\lambda_{j,\chi}-(\rho_\epsilon*\lambda_{j,\chi})\,dx\|_{\mathrm{TV}} \le C_\chi\epsilon.\] For fixed \(\epsilon\), the densities \(\rho_\epsilon*\lambda_{j,\chi}\) have a common compact support and uniform bounds on their values and first derivatives. The Arzelà–Ascoli theorem makes them precompact in the uniform norm on that support and hence in \(L^1(\mathbb R^n)\). Approximation by these families, first fixing \(\epsilon\) and then letting it decrease to zero, proves that \(\{\lambda_{j,\chi}:j\ge1\}\) is totally bounded in total variation. The space of finite signed measures is complete in that norm, so this family is precompact.

Isolating a chart in total variation. We now isolate the chart. Formula (65) also defines \(\lambda_{j,\mathbf1_{Z_i}}\), using the bounded Borel coefficient \(\mathbf1_{Z_i}\). By regularity of \(\mu\) and distance cutoffs on the compact metric space \(Y\), choose Lipschitz \(\chi_k\) with \[\|\chi_k-\mathbf1_{Z_i}\|_2\mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longmapstoP PaperOriginallongmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \let\LongrightarrowP PaperOriginalLongrightarrow\endcsname \let\LongleftrightarrowP PaperOriginalLongleftrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF27F6> >> BDC}% PPaperOriginallongrightarrow\endcsname\pdfliteral direct{EMC}\endgroup}0.\] The mass bound and Cauchy–Schwarz give, uniformly in \(j\), \[\begin{align*} \|\lambda_{j,\chi_k}-\lambda_{j,\mathbf1_{Z_i}}\|_{\mathrm{TV}} &\le \left(\prod_{a=1}^n\operatorname{Lip}(F^a)\right) \int|\chi_k-\mathbf1_{Z_i}|\,|u_j|\,d\mu \\ &\le C_FM_0\|\chi_k-\mathbf1_{Z_i}\|_2. \tag{68}\end{align*}\] For each fixed \(k\) the family on the left with coefficient \(\chi_k\) is precompact by the preceding argument. Uniform approximation (68) therefore makes \(\{\lambda_{j,\mathbf1_{Z_i}}:j\ge1\}\) precompact as well. There is no need for the Lipschitz constants of \(\chi_k\) to remain bounded.

Since \(F\circ\phi_i\) is the identity on \(E_i\), the density of the isolated measure is exactly \[d\lambda_{j,\mathbf1_{Z_i}}(z) =\mathbf1_{E_i}(z)\theta_i(z)(u_j\circ\phi_i)(z)\,dz.\] Consequently, for any \(j,k\), \[\begin{align*} \|u_j-u_k\|_{L^1(Z_i,\mu)} &=\int_{E_i}|\theta_i|J_i |(u_j-u_k)\circ\phi_i|\,dz\\ &\le\operatorname{Lip}(\phi_i)^n \|\lambda_{j,\mathbf1_{Z_i}} -\lambda_{k,\mathbf1_{Z_i}}\|_{\mathrm{TV}}. \end{align*}\] Here \(J_i\le\operatorname{Lip}(\phi_i)^n\). This calculation allows arbitrary signed, unbounded integer multiplicities: the fixed \(\theta_i\) appears as \(|\theta_i|\) in the total variation of a difference. Likewise, (68) already uses the full mass measure \(|\theta_i|J_i\,dz\), so it requires no bound or regularity of the multiplicities. All integrations by parts above concerned the whole cycle \(A\) with a Lipschitz coefficient. None concerned a restriction to one chart.

Assembling the charts. We may now take a diagonal subsequence that is Cauchy in \(L^1(Z_i,\mu)\) for every \(i\). The omitted charts have uniformly small contributions, because \[\int_{\bigcup_{i>N}Z_i}|u_j|\,d\mu \le M_0\,\mu\left(\bigcup_{i>N}Z_i\right)^{1/2} \mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longmapstoP PaperOriginallongmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \let\LongrightarrowP PaperOriginalLongrightarrow\endcsname \let\LongleftrightarrowP PaperOriginalLongleftrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF27F6> >> BDC}% PPaperOriginallongrightarrow\endcsname\pdfliteral direct{EMC}\endgroup}0\] uniformly in \(j\). The chart images cover \(\mu\) up to a null set, and \(\mu\) is finite. The subsequence is thus Cauchy in \(L^1(\mu)\). Proposition 18 supplies a uniform \(L^p\) bound. Since \(p>2\), interpolation with \(\alpha=(p-2)/(2(p-1))>0\) gives \[\|u_j-u_k\|_2 \le\|u_j-u_k\|_1^\alpha\|u_j-u_k\|_p^{1-\alpha} \mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longmapstoP PaperOriginallongmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \let\LongrightarrowP PaperOriginalLongrightarrow\endcsname \let\LongleftrightarrowP PaperOriginalLongleftrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF27F6> >> BDC}% PPaperOriginallongrightarrow\endcsname\pdfliteral direct{EMC}\endgroup}0.\] Finally, approximate the \(j\)th member of an arbitrary bounded sequence in \(\mathcal H\) by a Lipschitz function with graph-norm error at most \(1/j\). The result just proved for those approximants proves compactness for the original sequence. ◻

Proof. Taking \(\epsilon=S_*/2\) in (62) shows that \[\frac{S_*}{2}\|u\|_p^2 \le4\beta E(u)+C_{S_*/2}\|u\|_2^2 \le\max\{1,C_{S_*/2}/n\}\,Q(u).\] Thus \(\Lambda>0\). The constant function \(m^{-1/p}\) has \(L^p\) norm one, so \[ \Lambda\le n m^{1-2/p}=n m^{2/n}<ns_n^{2/n}=S_*. \tag{72}\]

Let \(u_j\) be an \(L^p\)-normalized minimizing sequence. It is bounded in \(\mathcal H\), since \(Q\) is a positive definite quadratic form equivalent to the squared graph norm. By weak compactness in a Hilbert space and Proposition 19, after taking a subsequence we have \[u_j\rightharpoonup u\quad\hbox{in }\mathcal H, \qquad u_j\to u\quad\hbox{in }L^2(\mu) \quad\hbox{and almost everywhere}.\] Put \(z_j=u_j-u\) and \(t=\int|u|^p\,d\mu\). Fatou’s lemma gives \(0\le t\le1\). We give the Brézis–Lieb splitting (Brezis and Lieb 1983, Theorem 1) in the form needed here: \[ \int|z_j|^p\,d\mu\mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longmapstoP PaperOriginallongmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \let\LongrightarrowP PaperOriginalLongrightarrow\endcsname \let\LongleftrightarrowP PaperOriginalLongleftrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF27F6> >> BDC}% PPaperOriginallongrightarrow\endcsname\pdfliteral direct{EMC}\endgroup}1-t. \tag{73}\] For every \(\epsilon>0\), the mean-value theorem and Young’s inequality give a finite \(C_\epsilon\) such that, for real \(a,b\), \[\bigl||a+b|^p-|a|^p\bigr| \le\epsilon|a|^p+C_\epsilon|b|^p.\] Apply this with \(a=z_j\) and \(b=u\), and set \(R_j=|u_j|^p-|z_j|^p-|u|^p\). The functions \[\bigl(|R_j|-\epsilon|z_j|^p\bigr)_+\] converge to zero almost everywhere and are bounded by \((C_\epsilon+1)|u|^p\). Dominated convergence, followed by \(\epsilon\downarrow0\), shows that \(R_j\to0\) in \(L^1\), since the \(L^p\) norms of \(z_j\) are bounded. This proves (73).

Weak convergence in \(\mathcal H\) also gives the quadratic splitting \[Q(u_j)=Q(u)+Q(z_j)+o(1).\] The definition of \(\Lambda\) gives \(Q(u)\ge\Lambda t^{2/p}\), with the same conclusion if \(u=0\). For fixed \(\epsilon>0\), apply (62) to \(z_j\) and use \(\|z_j\|_2\to0\) to obtain \[\liminf_j Q(z_j)\ge(S_*-\epsilon)(1-t)^{2/p}.\] Letting \(\epsilon\downarrow0\) therefore yields \[ \Lambda\ge\Lambda t^{2/p}+S_*(1-t)^{2/p}. \tag{74}\] If \(t<1\), the inequalities \(a^{2/p}\ge a\) for \(0\le a\le1\) and \(S_*>\Lambda\) make the right-hand side strictly larger than \(\Lambda\): indeed it is at least \(\Lambda t+S_*(1-t)>\Lambda\). Thus \(t=1\). Weak lower semicontinuity now gives \(Q(u)\le\Lambda\), so \(u\) is a minimizer with \(\|u\|_p=1\). Replacing it by \(|u|\) preserves both its \(L^p\) norm and its energy, by Proposition 18. We take \(u\ge0\).

For each \(\psi\in\mathcal H\), differentiate the quotient \(Q(u+s\psi)/\|u+s\psi\|_p^2\) at \(s=0\). Its denominator is nonzero for sufficiently small \(s\), and Hölder’s inequality justifies the derivative of the \(L^p\) integral: for \(|s|\le1\) the derivative integrand is bounded by \(p(|u|+|\psi|)^{p-1}|\psi|\in L^1\). Since \(u\) is a minimizer and \(\|u\|_p=1\), this gives \[4\beta\int Du\cdot D\psi\,d\mu+n\int u\psi\,d\mu =\Lambda\int u^{p-1}\psi\,d\mu.\] Set \[v=\left(\frac\Lambda n\right)^{1/(p-2)}u.\] Homogeneity preserves the quotient minimum, and \(\Lambda(\Lambda/n)^{-1}=n\) gives (71). Since \(p/(p-2)=n/2\), its \(p\)th moment is \(W=(\Lambda/n)^{n/2}<s_n\) by (72). Testing (71) with \(\psi=v\) gives \(Q(v)=nW\), completing (70). ◻

Proof. We first prove the power assertion under the additional assumption \(v\in L^{2\gamma}\). It is immediate for \(\gamma=1\), so suppose \(\gamma>1\). For \(N\ge1\) define \[w_N=v\min(v,N)^{\gamma-1},\qquad \psi_N=v\min(v,N)^{2\gamma-2},\qquad d_\gamma=\frac{2\gamma-1}{\gamma^2}.\] The scalar functions in these definitions, extended by zero for negative arguments, are Lipschitz and piecewise \(C^1\). Thus both compositions belong to \(\mathcal H\). Their derivatives show that \[Dv\cdot D\psi_N\ge d_\gamma|Dw_N|^2.\] Indeed there is equality below \(N\), while above \(N\) the ratio of the coefficients is one and \(0<d_\gamma\le1\). The gradient vanishes on the level set \(\{v=N\}\), so no value at the corner matters. Testing (71) with \(\psi_N\) is legitimate; for example, \(v^{p-1}\psi_N\le N^{2\gamma-2}v^p\) is integrable. Dropping the nonnegative term \(n\int v\psi_N\,d\mu\) gives \[ 4\beta d_\gamma E(w_N) \le n\int v^{p-2}w_N^2\,d\mu. \tag{76}\]

Apply (62) to \(w_N\) with \(\epsilon=S_*/2\) and multiply by \(d_\gamma\). By (76), \[\frac{d_\gamma S_*}{2}\|w_N\|_p^2 \le n\int v^{p-2}w_N^2\,d\mu +d_\gamma C_{S_*/2}\|w_N\|_2^2.\] Hölder’s inequality with conjugate exponents \(p/(p-2)\) and \(p/2\) gives \[n\int_{\{v>K\}}v^{p-2}w_N^2\,d\mu \le n\left(\int_{\{v>K\}}v^p\,d\mu\right)^{(p-2)/p} \|w_N\|_p^2.\] Since \(v\in L^p\), choose \(K\ge1\), depending on \(\gamma\) but not \(N\), so that the coefficient on the right is at most \(d_\gamma S_*/4\). The contribution of \(\{v\le K\}\) is at most \(nK^{p-2}\|w_N\|_2^2\). Absorbing the tail yields \[\frac{d_\gamma S_*}{4}\|w_N\|_p^2 \le\bigl(nK^{p-2}+d_\gamma C_{S_*/2}\bigr)\|w_N\|_2^2.\] The assumed \(L^{2\gamma}\) moment bounds the right-hand side uniformly in \(N\), because \(w_N\le v^\gamma\). Thus \(\|w_N\|_p\) is uniformly bounded. Equation (76), with another application of Hölder, then uniformly bounds \(E(w_N)\).

On \(\{v<N\}\) the gradient of \(w_N\) is \(\gamma v^{\gamma-1}Dv\). Monotone convergence in these sets shows that \(\gamma v^{\gamma-1}Dv\) belongs to \(\mathcal L^2\). At every truncation level the formulas and the level-set property give \[|w_N|\le v^\gamma,\qquad |Dw_N|\le\gamma v^{\gamma-1}|Dv|.\] The values and gradients converge almost everywhere to \(v^\gamma\) and \(\gamma v^{\gamma-1}Dv\), respectively. The displayed bounds give strong \(L^2\) convergence of both by dominated convergence. Closedness proves \(v^\gamma\in\mathcal H\) with the asserted gradient, and the continuous inclusion into \(L^p\) gives \(v^\gamma\in L^p\).

Starting with \(v\in L^p\), set \(\ell_j=p(p/2)^j\) for \(j\ge0\). The conditional assertion just proved, with \(\gamma=\ell_j/2\), shows \[v\in L^{\ell_j}\quad\mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longmapstoP PaperOriginallongmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \let\LongrightarrowP PaperOriginalLongrightarrow\endcsname \let\LongleftrightarrowP PaperOriginalLongleftrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF27F9> >> BDC}% PPaperOriginalLongrightarrow\endcsname\pdfliteral direct{EMC}\endgroup}\quad v\in L^{\ell_{j+1}}.\] Since \(p/2>1\), these exponents tend to infinity. Finiteness of \(\mu\) then gives every finite positive moment of \(v\). The conditional assertion therefore applies to every \(\gamma\ge1\). Finally, for \(q\ge0\) take \(\gamma=1+q/2\) to obtain \[\int v^q|Dv|^2\,d\mu=\gamma^{-2}E(v^\gamma)<\infty,\] which is (75). ◻

Proof. Let \(t_N=\min(v,N)\). The scalar calculus in Proposition 18 gives \(t_N\in\mathcal H\) and \(Dt_N=\mathbf1_{\{v<N\}}Dv\), where the level-set property handles \(v=N\). For each \(N\), choose a smooth, compactly supported cutoff in the given open neighborhood and equal to one on a neighborhood of the compact rectangle \([0,R]\times[0,N]\). Multiply \(F\) by this cutoff and extend by zero to obtain a global \(C^1\) function with bounded first derivatives. It agrees with \(F\) and its first derivatives on the rectangle. The bounded-derivative calculus consequently gives \(F(r,t_N)\in\mathcal H\) and \[D(F(r,t_N)) =\partial_1F(r,t_N)Dr +\partial_2F(r,t_N)\mathbf1_{\{v<N\}}Dv.\] Writing \(L=\|\,|Dr|\,\|_\infty\), the growth assumption gives bounds independent of \(N\), \[|F(r,t_N)|\le C(1+v)^M,\qquad |D(F(r,t_N))|\le C(1+v)^M(L+|Dv|).\] The squares of these bounds are integrable by Lemma 21: they are bounded by a constant times \((1+v^{2M})(1+|Dv|^2)\). Since \(v\) is finite almost everywhere, the values and the displayed gradient formulas converge almost everywhere to the values and right-hand side in (77). Dominated convergence gives strong convergence in the two \(L^2\) spaces, and closedness proves the result. A distance function has bounded values and \(|Dr|\le1\), so it satisfies the hypotheses. ◻

Proof. We verify the hypotheses of Lemma 22 for the actual functions of \((r,v)\) that occur here. This is necessary when \(\beta<1\), since the scalar function \(v^\beta\) itself is not \(C^1\) at zero.

For \(j=1,2\) and \(z>0\), define \[F_j(r,z)=z\,w(\kappa r z^\beta)^j, \qquad t=\kappa r z^\beta.\] On this half-plane, \[\partial_zF_j=w(t)^j+j\beta t\,w(t)^{j-1}w'(t),\qquad \partial_rF_j=j\kappa z^{1+\beta}w(t)^{j-1}w'(t).\] As \(z\downarrow0\), uniformly for \(r\) in bounded intervals, \(w(t)=1+O(t^2)\) and \(w'(t)=O(t)\). Thus \[F_j(r,z)=z+O(z^{1+2\beta}),\qquad \partial_zF_j(r,z)\mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longmapstoP PaperOriginallongmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \let\LongrightarrowP PaperOriginalLongrightarrow\endcsname \let\LongleftrightarrowP PaperOriginalLongleftrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF27F6> >> BDC}% PPaperOriginallongrightarrow\endcsname\pdfliteral direct{EMC}\endgroup}1, \qquad \partial_rF_j(r,z)\mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longmapstoP PaperOriginallongmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \let\LongrightarrowP PaperOriginalLongrightarrow\endcsname \let\LongleftrightarrowP PaperOriginalLongleftrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF27F6> >> BDC}% PPaperOriginallongrightarrow\endcsname\pdfliteral direct{EMC}\endgroup}0.\] Extension by \(F_j(r,z)=z\) on \(z\le0\) is therefore jointly \(C^1\). The functions and their first derivatives have polynomial growth in \(z\ge0\) on bounded \(r\) intervals.

For the other tests, write \[F_g(r,z)=\kappa^2z^{2+2\beta}f(\kappa r z^\beta),\qquad F_\psi(r,z)=\beta\kappa^2r^2z^{1+2\beta} f(\kappa r z^\beta) \quad(z>0).\] Their partial derivatives are \[\begin{align*} \partial_zF_g &=\kappa^2z^{1+2\beta} \bigl((2+2\beta)f(t)+\beta t f'(t)\bigr),\\ \partial_rF_g &=\kappa^3z^{2+3\beta}f'(t),\\ \partial_zF_\psi &=\beta\kappa^2r^2 \bigl((1+2\beta)z^{2\beta}f(t) +\beta\kappa r z^{3\beta}f'(t)\bigr),\\ \partial_rF_\psi &=\beta\kappa^2 \bigl(2r z^{1+2\beta}f(t) +\kappa r^2z^{1+3\beta}f'(t)\bigr). \end{align*}\] Their values and all these derivatives tend to zero as \(z\downarrow0\), uniformly for bounded \(r\). The smallest power of \(z\) in the derivative formulas is \(2\beta>0\). Consequently, extension by zero on \(z\le0\) makes both functions jointly \(C^1\), even if \(f(0)\) or \(f'(0)\) is nonzero. Their values and first derivatives again have polynomial growth on the range of \(r\).

These \(C^1\) extensions and polynomial bounds satisfy all the hypotheses of Lemma 22. That lemma gives membership in \(\mathcal H\) and the chain rules without a boundedness assumption on \(v\). On \(\{v>0\}\), differentiation gives (83). On \(\{v=0\}\), Proposition 18 gives \(Dv=0\) almost everywhere, while (82) gives \(\xi=0\). Thus these formulas hold \(\mu\)-almost everywhere on all of \(Y\). ◻

Proof. The extension of (26) to these tests is part of Proposition 18. Pointwise, with \(b=\sqrt{1-\lvert Dr\rvert^2}\), \[1-rb-\lvert Dr\rvert^2+\frac{r^2}{4} =\left(b-\frac r2\right)^2\ge0.\] Multiply by \(ng\ge0\) and combine this inequality with (26). All terms are integrable by boundedness of \(r\), \(\lvert Dr\rvert\le1\), finite \(\mu\), and \(g\in\mathcal H\). ◻

Proof. Fix the center and suppress its subscript on \(J\). For \(a>0\) let \(w(t)=(1+at^2)^{-(n-2)/2}\) as in Lemma 23. Since \(p(n-2)/2=n\), \[\int_Y w(s)^p\,d\nu=J(a).\] Testing (71) with \(vw(s)^2\) and using the two product formulas (83) gives \[ Q(vw(s)) =n\int_Y v^p w(s)^2\,d\mu +4\beta\int_Y w'(s)^2\lvert\xi\rvert^2\,d\mu. \tag{87}\] Indeed the terms \(w^2\lvert Dv\rvert^2+2ww'\langle Dv,\xi\rangle\) in \(\lvert D(vw)\rvert^2\) equal \(\langle Dv,D(vw^2)\rangle\). The quotient minimality of \(v\) gives, on the other hand, \[ Q(vw(s))\ge \frac{Q(v)}{\lVert v\rVert_p^2}\lVert vw(s)\rVert_p^2 =nW J(a)^{(n-2)/n}. \tag{88}\] The test \(vw(s)\) is nonzero because \(w\) is strictly positive and \(W>0\).

We next control the last integral in (87). Let \(f\) be smooth, bounded and nonnegative with bounded derivative, and use \(g,\psi\) from (84). Add (85) to (71) divided by \(4\beta\), with test \(\psi\). The identity \(v\psi=\beta r^2g\) cancels the two undifferentiated terms and yields \[ \int_Y\bigl(ng\lvert Dr\rvert^2+r\langle Dr,Dg\rangle +\langle Dv,D\psi\rangle\bigr)\,d\mu \le\frac n4\int_Y s^2f(s)v^p\,d\mu. \tag{89}\] Here we used \(v^{p-1}\psi=\beta s^2f(s)v^p\). For completeness, after removing the common factor \(\kappa^2\), the coefficients of \(\lvert Dr\rvert^2\), \(\langle Dr,Dv\rangle\), and \(\lvert Dv\rvert^2\) in the left integrand are respectively \[\begin{align*} &v^{2+2\beta}\bigl(nf+sf'\bigr),\\ &r v^{1+2\beta}\bigl((2+4\beta)f+2\beta sf'\bigr),\\ &\beta r^2v^{2\beta}\bigl((1+2\beta)f+\beta sf'\bigr). \end{align*}\] The scalar functions \(f,f'\) in this display are evaluated at \(s\). Since \[ 1+2\beta=n\beta,\qquad 2+4\beta=2n\beta=p, \tag{90}\] these are precisely the coefficients of \((nf(s)+sf'(s))\lvert\xi\rvert^2/\kappa^2\). Thus \[ \int_Y\bigl(nf(s)+sf'(s)\bigr)\lvert\xi\rvert^2\,d\mu \le\frac n4\int_Ys^2f(s)v^p\,d\mu. \tag{91}\]

To bound the energy term in (87), we want \(nf(t)+tf'(t)=w'(t)^2\) in (91). Integrating this first-order equation leads to the choice \[ f(t)=\int_0^1\tau^{n-1}w'(\tau t)^2\,d\tau \qquad(t\in\mathbb R). \tag{92}\] This is smooth and nonnegative. Both \(w'\) and \(w''\) are bounded on \(\mathbb R\), so \(f\) and \(f'(t)=2\int_0^1\tau^nw'(\tau t)w''(\tau t)\,d\tau\) are bounded. Integrating the derivative of \(\tau^nw'(\tau t)^2\) gives \[ nf(t)+tf'(t)=w'(t)^2. \tag{93}\] The boundary term at \(\tau=0\) is zero. Combining (87), (88), (91), and (93), and dividing by \(nW\), we obtain \[ J(a)^{(n-2)/n} \le\int_Y\bigl(w(s)^2+\beta s^2f(s)\bigr)\,d\nu. \tag{94}\]

The current and variational estimates have now reduced the comparison to (94). We evaluate its scalar right side and solve the resulting differential inequality for the transform. Put \(b=as^2\ge0\) and \(q=(n-2)/2>0\). Since \[w'(t)^2=(n-2)^2a^2t^2(1+at^2)^{-n},\] the substitution \(u=\tau^2\) in (92) gives \[w(s)^2+\beta s^2f(s) =(1+b)^{-(n-2)} +q b^2\int_0^1\frac{u^{n/2}}{(1+bu)^n}\,du.\] The exact derivative identity \[\frac{\,d}{\,du}\left(\frac{u^q}{(1+bu)^{n-2}}\right) =q\frac{u^{q-1}(1-b^2u^2)}{(1+bu)^n}\] has an integrable right side on \((0,1)\); this includes \(n=3\), when \(q-1=-1/2\). Integrating it shows that \[ w(s)^2+\beta s^2f(s) =q\int_0^1\frac{u^{q-1}}{(1+as^2u)^n}\,du. \tag{95}\] In particular this expression lies in \((0,1]\). Tonelli’s theorem therefore turns (94) into \[ J(a)^{(n-2)/n}\le G(a),\qquad G(a)=q\int_0^1J(au)u^{q-1}\,du. \tag{96}\]

We have \(G(0)=1\) and \(G(a)>0\). For \(a>0\), changing variable \(t=au\) gives \[G(a)=q a^{-q}\int_0^a J(t)t^{q-1}\,dt, \qquad aG'(a)=q\bigl(J(a)-G(a)\bigr).\] Continuity of \(J\) suffices for this differentiation. The finite second moment in (80) also allows differentiation at zero from the right, because \[\left|\frac{\partial}{\partial a}(1+as^2)^{-n}\right| \le ns^2\qquad(a\ge0).\] Consequently \[ J'(0+)=-nL_y, \qquad G'(0+)=\frac{q}{q+1}J'(0+)=-(n-2)L_y. \tag{97}\] Set \(h=G^{-1/q}=G^{-2/(n-2)}\). For \(a>0\), \[ah'=h-JG^{-n/(n-2)}\ge h-1,\] where the last step follows from (96). Thus \[\left(\frac{h(a)-1}{a}\right)' =\frac{ah'(a)-h(a)+1}{a^2}\ge0.\] By (97), the quotient on the left has limit \(2L_y\) at zero. It follows that \(h(a)\ge1+2aL_y\). Finally (96) gives \[J(a)\le G(a)^{n/(n-2)}=h(a)^{-n/2} \le(1+2aL_y)^{-n/2}.\] At \(a=0\) both sides equal one. If \(L_y=0\), then \(s=0\) \(\nu\)-almost everywhere, so \(J=G=h=1\) and the same conclusion holds. No higher chord moment is needed in the scalar comparison. ◻

Proof. Use the disjoint chart representation of Proposition 8. For one chart \(\phi_i:E_i\to Z_i\subset Y\), write \[u_i=v\circ\phi_i, \qquad \rho_i=\lvert\theta_i\rvert J_i, \qquad J_i=\sqrt{\det G_i},\] and extend \(u_i\) and \(\rho_i\) by zero to \(\mathbb R^n\). The bi-Lipschitz bounds give a positive lower bound for \(J_i\) on this chart almost everywhere. Since \(\lvert\theta_i\rvert\ge1\), \(v\in L^p(\mu)\) and \(\mu(Y)<\infty\), \(u_i\) is Lebesgue integrable. The function \(\rho_i\) is integrable as well, by the mass representation. Thus ordinary Lebesgue differentiation applies to both functions and to \(1_{E_i}\).

For \(\nu\)-almost every \(y=\phi_i(z)\), we may therefore require all of the following: \(z\) is a density point of \(E_i\); the centered metric differential is the Euclidean norm \(\lvert\cdot\rvert_{G_0}\), with \(G_0=G_i(z)>0\); and \(u_i,\rho_i\) have Lebesgue values \[ v_0=v(y)>0, \qquad \rho_0=\lvert\theta_i(z)\rvert\sqrt{\det G_0}>0. \tag{100}\] Indeed the exceptional coordinate sets are Lebesgue-null and therefore have zero chart mass, while \(\{v=0\}\) has zero \(\nu\)-mass. There are only countably many charts. The representative of \(v\) agrees with its Lebesgue value at almost every such point. Fix a center with these properties.

Let \(\ell=\liminf_{a\to\infty}a^{n/2}J_y(a)\). If \(\ell=\infty\), the desired lower bound is automatic. Otherwise, choose \(a_j\to\infty\) realizing this lower limit and put \(\varepsilon_j=a_j^{-1/2}\). On every bounded set of \(h\in\mathbb R^n\), the functions \[u_i(z+\varepsilon_jh),\qquad \rho_i(z+\varepsilon_jh),\qquad 1_{E_i}(z+\varepsilon_jh)\] converge in measure to \(v_0,\rho_0,1\), respectively. For example, this follows by changing variables in the defining mean convergence at the Lebesgue point \(z\). A diagonal subsequence over bounded balls makes all three convergences hold almost everywhere in \(\mathbb R^n\). The subsequence still realizes \(\ell\).

At almost every \(h\), the points \(z+\varepsilon_jh\) consequently belong to \(E_i\) for all sufficiently large \(j\). The centered metric differential then gives \[ R_j(h):=\varepsilon_j^{-1} d\bigl(y,\phi_i(z+\varepsilon_jh)\bigr) \mathrel{\begingroup \let\hookrightarrowP PaperOriginalhookrightarrow\endcsname \let\mapstoP PaperOriginalmapsto\endcsname \let\longmapstoP PaperOriginallongmapsto\endcsname \let\longrightarrowP PaperOriginallongrightarrow\endcsname \let\LongrightarrowP PaperOriginalLongrightarrow\endcsname \let\LongleftrightarrowP PaperOriginalLongleftrightarrow\endcsname \pdfliteral direct{/Span << /ActualText <FEFF27F6> >> BDC}% PPaperOriginallongrightarrow\endcsname\pdfliteral direct{EMC}\endgroup}\lvert h\rvert_{G_0}. \tag{101}\] When the chart point is absent, define the integrand below to be zero. Restricting the nonnegative integral defining \(J_y\) to this chart and changing variables gives, for every fixed bounded ball \(D_R\subset\mathbb R^n\) centered at zero, \[W a_j^{n/2}J_y(a_j) \ge\int_{D_R} \frac{1_{E_i}(z+\varepsilon_jh) u_i(z+\varepsilon_jh)^p\rho_i(z+\varepsilon_jh)} {\bigl(1+\kappa^2R_j(h)^2 u_i(z+\varepsilon_jh)^{2\beta}\bigr)^n}\,dh.\] The factor \(a_j^{n/2}\) has canceled the Jacobian \(\varepsilon_j^n\). Fatou’s lemma, (100), and (101) now imply \[W\ell\ge v_0^p\rho_0 \int_{D_R}\bigl(1+v_0^{4\beta}\lvert h\rvert_{G_0}^2\bigr)^{-n} \,dh.\] This step uses no uniform integrability of the rescaled densities. Letting \(R\to\infty\) and applying monotone convergence gives the same bound with \(D_R\) replaced by \(\mathbb R^n\).

The linear substitution \(H=v_0^{2\beta}G_0^{1/2}h\) has Jacobian \(v_0^{2n\beta}\sqrt{\det G_0}\). Since \(2n\beta=p\), we obtain \[\begin{align*} W\ell &\ge \lvert\theta_i(z)\rvert \int_{\mathbb R^n}(1+\lvert H\rvert^2)^{-n}\,dH\\ &=\lvert\theta_i(z)\rvert\,2^{-n}s_n \ge2^{-n}s_n. \end{align*}\] The middle equality is the stereographic integral (61). The final inequality uses the nonzero integer multiplicity \(\lvert\theta_i(z)\rvert\ge1\). This proves (98) without a bound on the multiplicity.

If \(L_y>0\), Proposition 25 and (98) give \[\frac{s_n}{2^nW}\le(2L_y)^{-n/2}.\] Rearranging yields (99). If \(L_y=0\), that upper bound is immediate. Its strict inequality follows from \(W<s_n\). ◻