Spherical volume and spherical Plateau problem

I am grateful to Gérard Besson, Gilles Courtois, John Lott, Ian Agol, Jason Manning, Camillo De Lellis, Xin Zhou, Hyun Chul Jang, Luca Spolaor, Tamunonye Cheetham-West, Alexander Nolte, Luca Di Cerbo, Richard Bamler, Song Sun, Stéphane Sabourau, Alexander Nabutovsky, Ben Lowe, Shi Wang, Cosmin Manea and Zhenhua Liu for many insightful and stimulating discussions during the writing of this article.

A.S. was partially supported by NSF grant DMS-2104254. This research was conducted during the period A.S. served as a Clay Research Fellow.

Let $(E,d)$ be a complete metric space²²2The metric $d$ is allowed to take $\infty$ as value.. Let $n\geq 0$, and let $\mathcal{D}^{n}(E)$ be the set of $(n+1)$-tuples $(f,\pi_{1},...,\pi_{n})$ of Lipschitz functions on $E$ with $f$ bounded. As a suggestive reference to the finite dimensional theory of currents, $(f,\pi_{1},...,\pi_{n})$ is also usually denoted by $fd\pi_{1}\wedge...,\wedge d\pi_{n}$. Metric currents in the sense of Ambrosio-Kirchheim are a flexible generalization of oriented submanifolds:

The minimal Borel measure $\mu$ satisfying the above inequality is called the mass of $T$ and denoted by $\|T\|$. The total mass of $T$ is defined as

$$\mathbf{M}(T):=\|T\|(E),$$

and should be thought of as the “$n$-dimensional area” of $T$. Currents in the sense of Ambrosio-Kirchheim [AK00a] have finite mass by definition (and we will only consider such currents here), but there is a variant of this theory due to Lang [Lan11] which avoids the finite mass condition. The support $\operatorname{spt}(T)$ of $T$ is the support of the measure $\|T\|$ in the usual sense. The canonical set of $T$, called $\operatorname{set}(T)$, is the collection of points in $E$ with a positive lower density:

$$\operatorname{set}(T):=\{p\in E;\lim_{r\to 0^{+}}\|T\|(B(p,r))r^{-n}>0\}.$$

In general,

$$\operatorname{set}(T)\subset\operatorname{spt}(T)$$

and the inclusion is usually strict. The boundary $\partial T$ is defined by

$$\partial T(fd\pi_{1}\wedge...,\wedge d\pi_{n-1}):=T(1df\wedge d\pi_{1}\wedge...,\wedge d\pi_{n})$$

for all $fd\pi_{1}\wedge...,\wedge d\pi_{n-1}\in\mathcal{D}^{n-1}(E)$. The push-forward of $T$ by a Lipschitz map $\psi$ from $E$ to another complete metric space $E^{\prime}$ is given by

$$\psi_{\sharp}T(fd\pi_{1}\wedge...,\wedge d\pi_{n}):=T(f\circ\psi d(\pi_{1}\circ\psi)\wedge...\wedge d(\pi_{n}\circ\psi))$$

for all $fd\pi_{1}\wedge...,\wedge d\pi_{n}\in\mathcal{D}^{n}(E^{\prime})$. There is also a notion of restriction of $T$ to a Borel subset $A\subset E$:

$$(T\llcorner A)(fd\pi_{1}\wedge...,\wedge d\pi_{n}):=T(f\chi_{A}d\pi_{1}\wedge...,\wedge d\pi_{n})$$

where $\chi_{A}$ is the characteristic function of $A$ (the above is well-defined by an extension of the functional $T$).

We are mainly interested in integral currents, which roughly speaking are currents $T$ such that both $T$ and $\partial T$ are the push-forward of a countable union of elementary currents by Lipschitz maps. An elementary current is obtained as follows: consider $\theta\in L^{1}(A;\mathbb{N})$ where $A\subset\mathbb{R}^{n}$, then define the following current in $\mathbb{R}^{n}$: for all $fd\pi_{1}\wedge...,\wedge d\pi_{n}\in\mathcal{D}^{n}(\mathbb{R}^{n})$,

$$\llbracket\theta\rrbracket(fd\pi_{1}\wedge...,\wedge d\pi_{n}):=\int_{A}\theta fd\pi_{1}\wedge...,\wedge d\pi_{n}.$$

Integer rectifiable currents and integral currents enjoy the following characterizations [AK00a, Theorem 4.5, Theorem 8.6], which we will take as definitions:

For instance, if $(M,g)$ is a complete oriented Riemannian manifold with compact boundary and finite volume, then it carries a natural $n$-dimensional integral current usually denoted by $\llbracket 1_{M}\rrbracket$, induced by “integration on $M$”.

Recall that a Borel set $S\subset E$ is countable $\mathcal{H}^{n}$-rectifiable if there is a sequence of Lipschitz functions $\varphi_{i}:A_{i}\to E$ where $A_{i}\subset\mathbb{R}^{n}$ is Borel, such that

(1)

$$\mathcal{H}^{n}(S\setminus\bigcup_{i}\varphi_{i}(A_{i}))=0$$

where $\mathcal{H}^{n}$ denotes the $n$-dimensional Hausdorff measure. It is proved in [AK00a, Theorem 4.6] that if $T$ is an $n$-dimensional integral current with a parametrization $(\{\varphi_{i}\},\{\theta_{i}\})$, then

$$\mathcal{H}^{n}\big{(}\operatorname{set}(T)\setminus\bigcup_{i=1}^{\infty}\varphi_{i}(A_{i})\cup\bigcup_{i=1}^{\infty}\varphi_{i}(A_{i})\setminus\operatorname{set}(T)\big{)}=0.$$

The canonical set $\operatorname{set}(T)$ is in particular a countably $\mathcal{H}^{n}$-rectifiable set in $E$ (contrarily to the support $\operatorname{spt}(T)$, in general). The functions $\theta_{i}$ determine a Borel function called the multiplicity function $\theta_{T}:E\to\mathbb{N}$, which is well-defined $\mathcal{H}^{n}$-almost everywhere. In the special case where $E$ is an infinite Riemannian dimensional manifold (i.e. locally modelled on a Hilbert space), it is shown in [AK00a, Theorem 9.5] that

$$\|T\|=\theta_{T}\mathcal{H}^{n}\llcorner\operatorname{set}(T).$$

That intuitive formula needs a correction factor in the case of general Banach spaces.

There are two fundamental notions of convergence for integral currents in a metric space: the weak and flat convergences. A sequence $\{T_{m}\}$ of $n$-dimensional integral currents in $E$ is said to converge weakly to some current $T$ if for all $fd\pi_{1}\wedge...,\wedge d\pi_{n}\in\mathcal{D}^{n}(E)$,

$$\lim_{m\to\infty}T_{m}(fd\pi_{1}\wedge...,\wedge d\pi_{n})=T(fd\pi_{1}\wedge...,\wedge d\pi_{n}).$$

In that case, an important result states that such a limit $T$ is also an integral current [AK00a, Theorem 8.5]. Besides, a fundamental property of the mass which makes it so useful in Plateau problems is that it is lower semicontinuous with respect to weak converge, see paragraph below [AK00a, Definition 3.6]: if $T_{m}$ converges weakly to $T$ then

$$\mathbf{M}(T)\leq\liminf_{m\to\infty}\mathbf{M}(T_{m}).$$

The sequence $\{T_{m}\}$ is said to converge to $T$ in the flat topology if there are sequences $\{U_{m}\},\{V_{m}\}$ of integral currents such that

$$T_{m}-T=U_{m}+\partial V_{m},\quad\lim_{m\to\infty}\mathbf{M}(U_{m})=\lim_{m\to\infty}\mathbf{M}(V_{m})=0.$$

Convergence in the flat topology implies convergence in the weak topology. A partial converse is proved in [Wen07].

Next we recall some versions of two particularly useful tools, the area and coarea formulas for countably $\mathcal{H}^{n}$-rectifiable sets. In this paragraph we assume for simplicity that $(E,d)$ is a separable complete infinite-dimensional Riemannian manifold, embedded isometrically inside an $\ell^{\infty}$ space $Y$ by a Kuratowski embedding, since only that case is needed here. Note that an $\ell^{\infty}$ space is a $w^{*}$-separable dual space in the sense of [AK00a]. Given a Lipschitz map $\psi:A\to E$ where $A\subset\mathbb{R}^{n}$ is Borel, for almost every $y\in A$, $\psi$ is $w^{*}$-differentiable at $y$ with a $w^{*}$-differential called $wd_{y}\psi$ in the sense of [AK00b][AK00a, Section 9]. The latter is a linear map from $T_{y}\mathbb{R}^{n}$ to $Y$. For almost every $y\in A$, $wd_{y}\psi$ is of full rank; in that case the image $Q:=wd_{y}\psi(T_{y}\mathbb{R}^{n})$ is called an approximate tangent $n$-plane at $p:=\psi(y)$. Such $Q$ is a linear $n$-plane inside $Y$, and in our case it is also a tangent $n$-plane of the manifold $E$. More generally, let $S\subset E$ be a countably $\mathcal{H}^{n}$-rectifiable set, with $\{\varphi_{i}:A_{i}\to E\}$ as in (1). At $\mathcal{H}^{n}$-almost every $p\in S$, there are $i$ and $y\in A_{i}$ such that $\varphi_{i}(y)=p$ and an approximate tangent $n$-plane $Q$ at $p$ exists in the sense above.

Let $S$ be a countably $\mathcal{H}^{n}$-rectifiable set in $E$. Given a Lispchitz map $g:S\to E^{\prime}$ where $E^{\prime}$ is another separable complete infinite-dimensional Riemannian manifold embedded isometrically in an $\ell^{\infty}$ space $Y^{\prime}$, there is a well-defined nonnegative number for $\mathcal{H}^{n}$-almost every $z\in S$, called the Jacobian of $g$ and denoted by $\mathbf{J}_{n}(d^{S}g_{z})$, such that the following area formula [AK00b, Theorem 8.2] holds for any Borel function $\theta:S\to[0,\infty]$:

(2)

$$\int_{S}\theta(p)\mathbf{J}_{n}(d^{S}g_{p})d\mathcal{H}^{n}(p)=\int_{E^{\prime}}\sum_{p\in S\cap g^{-1}(z)}\theta(p)d\mathcal{H}^{n}(z).$$

More concretely, suppose that for $\{\varphi_{i}:A_{i}\to E\}$ as (1), for $i$ and $y\in A_{i}$, $p:=\varphi_{i}(y)\in S$, and suppose that $\varphi_{i}$ is $w^{*}$-differentiable at $y$ with a $w^{*}$-differential of full rank, and the composition $g\circ\varphi_{i}:\mathbb{R}^{n}\to E^{\prime}$ is $w^{*}$-differentiable at $y$ (all of this holds for almost every $y\in A_{i}$). Let

$$Q:=wd_{y}\varphi_{i}(T_{y}\mathbb{R}^{n})$$

be the tangent $n$-plane at $p$. Then $g$ is tangentially differentiable at $p$ along $Q$ with tangential differential $d^{S}g_{p}:Q\to T_{g(p)}E^{\prime}$ satisfying:

$$d^{S}g_{p}=wd_{y}(g\circ\varphi_{i})\circ(wd_{y}\varphi_{i})^{-1}.$$

In our case where $E,E^{\prime}$ are infinite-dimensional Riemannian manifolds, $Q$ and $T_{g(p)}E^{\prime}$ are Hilbert linear spaces. The Jacobian $\mathbf{J}_{n}(dg_{p})$ is then equal to the absolute value of the usual Jacobian determinant of the linear map $d^{S}g_{p}$. In particular when $g$ is $\lambda$-Lipschitz then

$$\mathbf{J}_{n}(dg_{z})\leq\lambda^{n}$$

as expected. Given a tangent $n$-plane $Q$ of $S$ at $p$ as above, we often use the following more classical notations:

$$dg\big{|}_{Q}:=d^{S}g_{p},$$

$$|\operatorname{Jac}g\big{|}_{Q}|:=\mathbf{J}_{n}(d^{S}g_{p}).$$

Let $S$ be a countably $\mathcal{H}^{n}$-rectifiable set in $E$. Consider a Lipschitz function $\pi:S\to\mathbb{R}^{k}$ where $k\leq n$. At $\mathcal{H}^{n}$-almost every $p\in S$, there is a tangent $n$-plane $Q$ along which the tangential differential $d^{S}\pi_{p}$ exists. At such $p$, there is a nonnegative number called coarea factor and denoted by $\mathbf{C}_{k}(d^{S}\pi_{p})$ such that the following coarea formula [AK00b, Theorem 9.4] holds for any Borel function $\theta:S\to[0,\infty]$:

(3)

$$\int_{S}\theta(p)\mathbf{C}_{k}(d^{S}\pi_{p})d\mathcal{H}^{n}(p)=\int_{\mathbb{R}^{k}}\big{(}\int_{\pi^{-1}(z)}\theta(p)d\mathcal{H}^{n-k}(p)\big{)}d\mathcal{H}^{k}(z).$$

Besides, for $\mathcal{H}^{k}$-almost every $z\in\mathbb{R}^{k}$, the set $\pi^{-1}(z)\cap S$ is countably $\mathcal{H}^{n-k}$-rectifiable. The precise definition of the coarea factor $\mathbf{C}_{k}(d\pi_{z})$ is a bit more involved than the Jacobian [AK00b, Definition 9.1], nevertheless it is similar to what appears in the smooth finite dimensional case and when $\pi$ is $\lambda$-Lipschitz then

$$\mathbf{C}_{k}(d\pi_{z})\leq\lambda^{k}$$

as expected.

We move on to the slicing theorem for integral currents, which enables to construct lower dimensional integral currents out of a given integral current and a Lipschitz map. Again, we still assume for simplicity that $E$ is an infinite-dimensional Riemannian manifold (locally modelled on a Hilbert space). Let $S$ be a countably $\mathcal{H}^{n}$-rectifiable set in $E$. Given $fd\pi_{1}\wedge...\wedge d\pi_{n}\in\mathcal{D}^{n}(E)$, then at $\mathcal{H}^{n}$-almost every $p\in S$, an approximate tangent $n$-plane $Q$ exists, the tangential differentials $d^{S}(\pi_{j})_{p}$ along $Q$ exist and the $n$-covector $fd^{S}\pi_{1}\wedge...\wedge d^{S}\pi_{n}$ is well-defined. There is a notion of orientations on $S$ [AK00a, Section 9]. Roughly speaking, an orientation $\tau$ on $S$ endows each approximate tangent $n$-plane $Q$ of $S$ with an $n$-vector $\mathbf{e}_{1}\wedge...\wedge\mathbf{e}_{n}$ where $(\mathbf{e}_{1},...,\mathbf{e}_{n})$ is an orthonormal basis of $Q$ (recall that here $E$ is a manifold).

Let $T$ be an $n$-dimensional integral current in $E$, with multiplicity function $\theta_{T}$. There is an intrinsic description of $T$, based on the notion of orientation [AK00a, Theorem 9.1]. In fact there is an orientation $\tau_{T}$ on $\operatorname{set}(T)$ such that for all $fd\pi_{1}\wedge...\wedge d\pi_{n}\in\mathcal{D}^{n}(E)$,

$$T(fd\pi_{1}\wedge...\wedge d\pi_{n})=\int_{\operatorname{set}(T)}f(p)\theta_{T}(p)\langle d^{\operatorname{set}(T)}\pi_{1}\wedge...\wedge d^{\operatorname{set}(T)}\pi_{n},\tau_{T}\rangle d\mathcal{H}^{n}(p).$$

Conversely if $S$ is a countably $\mathcal{H}^{n}$-rectifiable set in $E$, if we have $\theta:E\to\mathbb{N}$ such that $\int_{S}\theta d\mathcal{H}^{n}<\infty$, and an orientation $\tau$ on $S$ then the current

$$\llbracket S,\theta,\tau\rrbracket$$

defined by the analogue of the formula above is an $n$-dimensional integral current.

Now consider a Lipschitz map $\pi:\operatorname{spt}(T)\to\mathbb{R}^{k}$ where $k\leq n$. Then for each $x\in\mathbb{R}^{k}$, there is an integral current $\langle T,\pi,x\rangle$ called sliced current [AK00a, Theorems 5.6 and 5.7], characterized by the fact that for all $fd\pi_{1}\wedge...\wedge d\pi_{n-k}\in\mathcal{D}^{n-k}(E)$ and $\psi\in C_{c}(\mathbb{R}^{k})$,

(4)

$$\big{[}\int_{\mathbb{R}^{k}}\langle T,\pi,x\rangle\psi(x)dx\big{]}(fd\pi_{1}\wedge...\wedge d\pi_{n-k})=T(f.(\psi\circ\pi)d\pi\wedge d\pi_{1}...\wedge d\pi_{n-k})$$

where if $\pi=(u_{1},...,u_{k})$ and $u_{j}$ are the coordinate functions, then $d\pi:=du_{1}\wedge...\wedge du_{k}$. Besides, with the notation $T=\llbracket\operatorname{set}(T),\theta_{T},\tau_{T}\rrbracket$ defined above, for almost every $x\in\mathbb{R}^{k}$ we have an orientation $\tau_{x}$ of $\operatorname{set}(T)\cap\pi^{-1}(x)$ such that

(5)

$$\langle T,\pi,x\rangle=\llbracket\operatorname{set}(T)\cap\pi^{-1}(x),\theta_{T},\tau_{x}\rrbracket,$$

see [AK00a, Theorem 9.7]. The fact that

$$\operatorname{spt}(\partial\langle T,\pi,x\rangle)\subset\operatorname{spt}(\partial T),$$

follows from [AK00a, Theorem 5.6 (i)] applied to the boundary $\partial T$, or alternatively by iterating [AK00a, Theorem 5.6 (iii), Lemma 5.3].

We end this section with the definitions of integral current spaces, the intrinsic flat topology of Sormani-Wenger, and Wenger’s compactness theorem [SW11].

A simple example of integral current space is given by a complete, oriented Riemannian $n$-manifold $(M,g)$ with compact boundary and finite volume: the metric space is $M$ endowed with the geodesic distance $\operatorname{dist}_{g}$ induced by $g$, and the integral current structure $\llbracket 1_{M}\rrbracket$ is the natural integral current induced by integration on $M$. Recall that we allow the metric to assume $\infty$ as value, and so we allow $M$ to be non-connected. In general, an integral current in a metric space $E$ determines uniquely an integral current space. The boundary $\partial C$ is the integral current space induced by $\partial T$.

If $C:=(X,d,T)$, $C^{\prime}:=(X^{\prime},d^{\prime},T^{\prime})$ are two integral current spaces such that there is an isometry $\varphi:\overline{X}\to\overline{X^{\prime}}$ with $\varphi_{\sharp}T=T^{\prime}$ (such a $\varphi$ is called a current preserving isometry [SW11, Definition 3.26]), then we say that $C$ and $C^{\prime}$ are isomorphic.

In the same way that Gromov-Hausdorff convergence generalizes Hausdorff convergence by allowing arbitrary isometric embeddings in a complete metric space, the notion of intrinsic flat convergence generalizes flat convergence as follows [SW11, Theorem 4.2].

The complete metric space $\mathbf{Z}$ can be taken to be a Banach space if the metrics $d_{m}$, $d_{\infty}$ only assume finite values, by the standard Kuratowski embedding. Actually the above notion of convergence is induced by a metric called the intrinsic flat distance $\mathbf{d}_{\mathcal{F}}$ and defined in [SW11, Definition 1.1]: given two integral current spaces of dimension $n$, $C:=(X,d,S)$ and $C^{\prime}:=(X^{\prime},d^{\prime},S^{\prime})$, set

$$\mathbf{d}_{\mathcal{F}}(C,C^{\prime}):=\inf\{\mathbf{M}(U)+\mathbf{M}(V)\}$$

where the infimum is taken over all complete metric spaces $(Z,d)$ and all integral currents $U$, $V$ in $Z$ such that there are isometric embeddings $\varphi:(\overline{X},d)\to Z$, $\varphi^{\prime}:(\overline{X^{\prime}},d^{\prime})\to Z$ with

$$\varphi_{\sharp}(S)-(\varphi^{\prime})_{\sharp}(S^{\prime})=U+\partial V.$$

By [SW11, Theorem 3.27], $\mathbf{d}_{\mathcal{F}}(C,C^{\prime})=0$ if and only if $C$ and $C^{\prime}$ are isomorphic.

One of the fundamental properties of integral current spaces is the following compactness theorem [Wen11][SW11, Theorem 4.19]:

The above compactness result will be necessary when defining spherical Plateau solutions in Subsection 3.1.

We end this section with a useful approximation theorem for integral currents in spherical manifolds by polyhedral chains. This result can be simply deduced from the analogous result in finite dimensions, which in turn is completely standard.

Let $(N,g_{N})$ be an infinite-dimensional Riemannian manifold which is locally isometric to the unit sphere of an infinite-dimensional Hilbert space. In our context, a $k$-dimensional integral current $P$ in $(N,g_{N})$ is called a polyhedral chain of dimension $k$ if there are smoothly embedded totally geodesic $k$-simplices $S_{1},...,S_{m}\subset N$ endowed with an orientation, and integers $a_{j}$ so that

$$P=\sum_{j=1}^{m}a_{j}\llbracket 1_{S_{j}}\rrbracket.$$

Here is an outline of proof. Consider an open ball $B$ centered at the origin in the Hilbert space $\ell^{2}(\mathbb{N})$ where $\mathbb{N}$ is the set of natural numbers, and consider an integral current $C$ with compact support in $B$. For any integer $L\geq 1$, we can use the orthogonal projection onto $\ell^{2}(\{1,...,L\})$ to map $C$ to an integral current $C_{1}$ compactly supported inside the finite dimensional space $B\cap\ell^{2}(\{1,...,L\})$. If $L$ is chosen large enough, $C_{1}$ will be as close to $C$ as we wish in the sense of Lemma 1.6. Now $C_{1}$ is also an integral current in the sense of Federer-Fleming (see the appendix of [AK00a]), so the usual approximation results ([Fed69, Sections 4.1 and 4.2], [DP14]) in finite dimension can be applied and give the desired polyhedral chain approximation $P_{1}$ in $B\cap\ell^{2}(\{1,...,L\})$. This construction clearly generalizes to small balls in $(N,g_{N})$ if $N$ is separable, which can always be assumed to be true since $\operatorname{spt}(C)$ is compact. In the general case where the support of $C$ is not contained in a small ball of $N$, one can argue using a partition of unity and construct via an interpolation map a current $C_{2}$ arbitrarily close to $C$ in the sense of Lemma 1.6, which is locally finite dimensional in the following sense: for any $x\in\operatorname{spt}(C_{2})$, there is a ball $B_{x}$ containing $x$ such that $\operatorname{spt}(C_{2})\cap B_{x}$ is contained in a totally geodesic finite dimensional plane of $B_{x}$. The rest is standard as before.

Let $(M,g_{0})$ be a closed oriented hyperbolic manifold. Let $(\tilde{M},g_{0})$ be its universal cover, namely the hyperbolic $n$-space. Let $\Gamma:=\pi_{1}(M)$. The latter acts properly cocompactly, freely and properly on $(\tilde{M},g_{0})$. Let ${S^{\infty}}$ be the unit sphere in the Hilbert space $\ell^{2}(\Gamma)$, on which $\Gamma$ acts freely and properly by isometries via the regular representation $\lambda_{\Gamma}:\Gamma\to\operatorname{End}(\ell^{2}(\Gamma))$, and let ${S^{\infty}}/\lambda_{\Gamma}(\Gamma)$ be the quotient manifold endowed with the standard round metric (see Subsection 3.1 for more details).

It is well-known that the distance functions on $(\tilde{M},g_{0})$ satisfy the following Hessian lower bound: for any fixed $w\in\tilde{M}$, at any point different from $w$ we have

$$Dd\operatorname{dist}_{g_{0}}(w,.)\geq\operatorname{Id}-d\operatorname{dist}_{g_{0}}(w,.)\otimes d\operatorname{dist}_{g_{0}}(w,.).$$

Distance functions are not smooth so in the setting of hyperbolic manifolds, it is more convenient from a technical point of view to work with modified distance functions called $\rho_{w}$. Fix a smooth strictly convex increasing function

$$\varkappa:[0,\infty)\to[0,\infty)$$

such that $\lim_{t\to\infty}t^{-1}\varkappa(t)=1$, $\varkappa^{\prime}(t)<1$ for all $t$, and for any $w\in\tilde{M}$, the composition

$$\rho_{w}(.):=\varkappa(\operatorname{dist}_{g_{0}}(w,.))$$

is smooth everywhere and satisfies

(6)

$$Dd\rho_{w}\geq\operatorname{Id}-d\rho_{w}\otimes d\rho_{w}.$$

Such a function exists, for example if we set

$$\varkappa(t)=\frac{1}{c}\log(\cosh(ct))$$

where $c$ is a positive constant, it is an exercise to check that (6) holds whenever $c$ is large enough.

The barycenter map is well-defined: the modified distance functions $\rho_{\gamma.o}$ are strictly convex, moreover $\mathcal{B}_{f}$ tends to infinity uniformly as $x\to\infty$, so that the point where $\mathcal{B}_{f}$ attains its minimum exists and is unique. The subset $\mathbb{S}^{+}\subset{S^{\infty}}$ is invariant by $\Gamma$, and $\mathrm{Bar}$ is $\Gamma$-equivariant. The quotient map

$$\mathbb{S}^{+}/\lambda_{\Gamma}(\Gamma)\to M$$

is also denoted by $\mathrm{Bar}$.

The regularity of the barycenter map

$$\mathrm{Bar}:\mathbb{S}^{+}/\lambda_{\Gamma}(\Gamma)\to M$$

is not completely immediate. To avoid discussing such issues, we will only consider the barycenter map restricted to the support of polyhedral chains in $\mathbb{S}^{+}/\lambda_{\Gamma}(\Gamma)$. Recall that by definition (see Subsection 1.7), a $k$-dimensional polyhedral chain $P$ in $\mathbb{S}^{+}/\lambda_{\Gamma}(\Gamma)$ is a $k$-dimensional integral current that can be written as

$$P=\sum_{j=1}^{m}a_{j}\llbracket 1_{S_{j}}\rrbracket$$

where $a_{j}$ are integers and

$$S_{1},...,S_{m}\subset\mathbb{S}^{+}/\lambda_{\Gamma}(\Gamma)\subset{S^{\infty}}/\lambda_{\Gamma}(\Gamma)$$

are finitely many smoothly embedded totally geodesic $k$-simplices endowed with an orientation. In particular the support of a polyhedral chain is by definition a finite union of totally geodesic simplices. Given a polyhedral chain $P$ as above, each simplex $S_{j}$ lifts to a simplex $\tilde{S}_{j}$ in $\mathbb{S}^{+}\subset\ell^{2}(\Gamma)$. We claim that there is a finite set $\mathbf{S}_{j}\subset\Gamma$ depending only on $\tilde{S}_{j}$ such that any element of $\tilde{S}_{j}$ is a function with support in $\mathbf{S}_{j}$. Indeed, the $k$-simplex $\tilde{S}_{j}$ is the convex hull of its extremal points $f_{0},...,f_{k}\in\mathbb{S}^{+}$. If $\mathbf{S}_{j}$ denotes the union of the finite supports of $f_{0},...,f_{k}$, then any linear combination of those functions has support contained in $\mathbf{S}_{j}$, and that proves the claim. Now given a polyhedral chain $P$ in $\mathbb{S}^{+}/\lambda_{\Gamma}(\Gamma)$, one can check without difficulty that the restriction

$$\mathrm{Bar}:\operatorname{spt}(P)\to M$$

is continuous (and smooth on each simplex by the discussion below).