\section{Introduction}

Integral currents and flat distance are classical notions from Geometric Measure Theory, being employed for instance  by Federer--Fleming to solve the Plateau problem  \cite{FF}. Ambrosio--Kirchheim \cite{AK} defined integral currents in metric measure spaces and applying this work  
Sormani--Wenger \cite{SW2} defined integral current spaces and the intrinsic flat distance.
The intrinsic flat distance between two compact oriented Riemannian manifolds of the same dimension $M_i$, endowed with their canonical currents $\curr{M_i}$, is defined as the infimum of the flat distances between the push-forwards of both currents, 
\[
\inf \Big \{d_{F}(\varphi_{1\sharp}\curr{M_1}, \varphi_{2\sharp}\curr{M_2})\Big \},
\]
where the infimum runs over all complete metric spaces $Z$ and distance preserving embeddings $\varphi_i: M_i \to Z$. 




The Sormani--Wenger intrinsic flat distance \cite{SW2} has shown to be an adequate notion to study several stability problems, such as the stability of the positive mass theorem or the stability of tori with almost non-negative scalar curvature. Sakovich--Sormani \cite{SS} obtained an intrinsic flat stability result for the positive mass theorem for complete rotationally symmetric asymptotically hyperbolic manifolds. Huang--Lee \cite{HL} showed stability of the positive mass theorem with respect to the Federer--Fleming flat distance for a class of asymptotically flat graphical manifolds.  Motivated by this work,  Huang--Lee--Sormani \cite{HLSa,HLSb} attempted to prove the stability of the positive mass theorem with respect to the intrinsic flat distance. The proof of their first result, \cite[{Theorem 1.3}]{HLSa}, consisted in reducing it to the following rigidity of mass-preserving $1$-Lipschitz maps into Euclidean space. 

\begin{problem*}
Let $R>0$ and $(\Omega_R,d_R,T_R)$ be an $n$-dimensional integral current space, and let $\psi:(\Omega_R,d_R)\to(\mathbb{R}^{n+1}, d_{Eucl})$ be a $1$-Lipschitz function with the following properties:
\begin{enumerate}
    \item $\psi_\# T_R=\curr{B_R \times \{0\}}$, where $B_R$ denotes a ball of radius $R$ in the $n$-dimensional Euclidean space 
    \item $\mathbf{M}(T_R)=\mathbf{M}(\psi_\# T_R)=\omega_n R^n$
    \item $\psi_{\mathrm{set}(\partial T_R)}:\mathrm{set}(\partial T_R) \to \mathbb{R}^{n+1}$ is bi-Lipschitz onto its image, $\partial B_R \times \{0\}$.
    \end{enumerate}
Then $\psi$ is an isometry between $(\Omega_R,d_R)$ and $(B_R,d_{Eucl})$, hence $(\Omega_R,d_R,T_R)= (B_R,d_{Eucl},\curr{B_R})$ as integral current spaces.
\end{problem*}

When dealing with Euclidean currents the result is a consequence of the area formula and classical rigidity for Lipschitz maps with gradient in $SO(n)$.
It was assumed to be true for Ambrosio--Kirchheim currents in the proof of \cite[{Theorem 1.3}]{HLSa}, but its validity is more subtle in this case, and the main contribution of this manuscript is to provide a proof of this claim.


\begin{theorem}[Rigidity of mass-preserving 1-Lipschitz maps]\label{thm-rigidity}
Let $(X,d,T)$ be an $n$-dimensional integral current space, and let $\psi:X\to\mathbb{R}^n$ be a $1$-Lipschitz function with the following properties:
\begin{enumerate}
    \item $\psi_\# T=\curr{B_1}$;
    \item $\mathbf{M}(T)=\mathbf{M}(\psi_\# T)$;
    \item $\psi$ is injective on $\mathrm{set}(\partial T)$, and $\psi(\mathrm{set}(\partial T))\subseteq \partial B_1$.
\end{enumerate}
Then $\psi$ is an isometry between $(X,d)$ and  $(B_1,d_{Eucl})$, hence $(X,d,T)= (B_1,d_{Eucl},\curr{B_1})$ as integral current spaces.
\end{theorem}


Theorem \ref{thm-rigidity} directly implies a positive answer to the Problem above, and therefore the original proof of \cite[{Theorem 1.3}]{HLSa} is valid. See Section \ref{sec-applications} for a sketch of this proof. We note that there is a different and rigorous  proof of  \cite[{Theorem 1.3}]{HLSa} provided
by Huang, Lee and the second named author 
\cite[{Theorem 3.2}]{HLP} that does not rely on the existence of such a $1$-Lipschitz function nor on solving the Problem above. The proof uses an intrinsic flat compactness result, \cite[{Theorem 3.4}]{HLP}, which is an easy corollary of \cite[{Theorem 4.2}]{AP} of Allen and the second named author. The latter extends the work of both of them and Sormani to manifolds with boundary. One can also find  a proof of \cite[{Theorem 1.3}]{HLSa} in \cite[{Section 7}]{AP} that applies \cite[{Theorem 4.2}]{AP}, under the added assumption that the manifolds are entire. To obtain a full proof of \cite[{Theorem 1.3}]{HLSa}, in \cite{HLP} the manifolds with inner boundary are carefully extended to manifolds homeomerphic to balls in Euclidean space and then one can apply \cite[{Theorem 3.4}]{HLP}.


A similar result to Theorem \ref{thm-rigidity} has been recently proven by Basso, Creutz and Soultanis \cite{BCS}, of which we became aware while in the final stages of the completion of this manuscript. There, condition (3) is replaced by
$\mathbf{M}( \partial T)=\mathbf{M}(\partial \curr{B_1})$. They include interesting examples of the necessity of the hypotheses. Furthermore, in their work $B_1$ can be taken to be any convex set in Euclidean space.  After this, we realized that our proof of Theorem \ref{thm-rigidity} also goes through for any convex set in Euclidean space. This implies that the stability with respect to intrinsic flat distance of fundamental domains of $3$-dimensional tori with scalar curvature converging to zero, as originally proven by Cabrera Pacheco, Ketterer and the second named author \cite[{Theorem 5.5}]{CKP-arxive}, holds. This was used to prove stability of $3$-dimensional graphical tori in \cite[{Theorem 1.4}]{CKP-arxive}. But we note that there is a rigorous and published version of the latter \cite{CPKP}(c.f. \cite[{Theorem 1.4}]{CKP-arxive2}), with a proof in the same spirit as the proof of \cite[{Theorem 3.2}]{HLP}. 

There are other rigidity results of volume preserving $1$-Lipschitz functions  \cite{CHS, Li, LiWang, BCG, CDNPSW} defined on spaces with no boundary that satisfy lower curvature bounds, such as scalar curvature bounds, sectional curvature bounds, Alexandrov spaces, Ricci limits and RCD spaces.  For spaces with boundary we also mention \cite{BI} and the references within.


\subsection*{Acknowledgements}

GDN received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme, grant agreement No 757254 (SINGULARITY).
RP acknowledges support from CONACyT Ciencia de Frontera 2019  CF217392 grant.



\section{Preliminaries}

We will denote by $B=B(0,1)$ the open Euclidean unit ball in $\mathbb{R}^n$, and by $B_r$ or $B(r)$ the open ball of radius $r$.

\subsection{Metric currents}

We refer to Ambrosio--Kirchheim \cite{AK} for the theory of currents in metric spaces. We will only deal with integral and rectifiable currents, and for this reason we omit the general definitions.

Given a Lipschitz curve $\gamma:[0,1]\to X$, with $(X,d)$ a complete metric space, we denote by $\curr{\gamma}$ the associated integral $1$-current with weight equal to 1. More in general, given an $\H^k$-rectifiable set $E \subset \mathbb{R}^n$, a simple unit $k$-vector field $\tau$ on $E$, and a multiplicity function $\theta:E\to \mathbb{R}$, we denote by $[E,\tau,\theta]$ the Euclidean $n$-current given by
\[
\langle[E,\tau,\theta],\omega\rangle=\int_E \langle\omega(x),\tau(x)\rangle \theta(x) d\H^k(x).
\]
We will denote for simplicity $[B,\tau_{std},1]$ (where $\tau_{std}=e_1\wedge\ldots\wedge e_n$ is the standard orientation of $\mathbb{R}^n$) by $\curr{B}$.
We will denote by $\mathbf{M}(T)$ the mass of the current $T$.

We have the following structure theorem for rectifiable currents in metric spaces, which we can also take as a definition.

\begin{theorem}[{\cite[Theorem 4.5]{AK}}]\label{thm:parametric_representation}
Every rectifiable $n$-current $T$ in a complete metric space $(X,d)$ can be represented as
\begin{equation}\label{eq:decomposition_rectifiable}
T= \sum_{i}(f_i)_{\sharp} \curr{\theta_i}\qquad\text{with}\quad \mathbf{M}(T)=\sum_i\mathbf{M}((f_i)_\#\curr{\theta_i})
\end{equation}
for a countable collection of biLipischitz maps $f_i:K_i\to X$, with $K_i$ compact in $\mathbb{R}^n$, $f_i(K_i)$ pairwise disjoint and $\theta_i\in L^1(K_i;\mathbb{R}\setminus\{0\})$. If $T$ is integral, then $\theta_i$ are integer-valued.
\end{theorem}

In the case of integral 1-currents we can say more, as we have the following structure result.

\begin{theorem}[Decomposition of integral $1$-currents {\cite[Theorem 5.3]{BDP}}]\label{thm:decomposition}
Every integral $1$-current $T$ in a complete metric space $(X,d)$ can be written as $T=\sum_i\curr{\gamma_i}$, where $\gamma_i$ are either injective Lipschitz curves or injective Lipschitz loops, and so that
\[
\mathbf{M}(T)=\sum_i \mathbf{M}(\curr{\gamma_i}),\qquad \mathbf{M}(\partial T)=\sum_i \mathbf{M}(\partial\curr{\gamma_i}).
\]
\end{theorem}

\subsection{Pushforward}
Given a Lipschitz map $f:\mathbb{R}^n\to\mathbb{R}^n$, and an $n$-current $T=[E,\tau_{std},\theta]$ (where $\tau_{std}$ denotes the standard orientation of $\mathbb{R}^n$), we recall that the pushforward $f_\#T$ equals $[f(E),\tau_{std},\tilde\theta]$, where
\begin{equation}\label{eq:pushforward_formula}
    \tilde\theta(y)=\sum_{x\in f^{-1}(y)} \mathrm{sign}(\det(df|_x)) \theta(x)\qquad\text{for $\H^n$-a.e. $y$}.
\end{equation}
It is a consequence of the area formula that the expression above is well-defined, since the cardinality of $f^{-1}(y)$ is finite for $\H^n$-a.e. $y$, and $f$ is differentiable at $\H^n$-a.e. point by Rademacher's theorem. 

Given two complete metric spaces $X,Y$, a Lipschitz function $f:X\to Y$, and an $n$-current $T$ in $X$ with finite mass, the pushforward $f_\#T$ is a well-defined $n$-current in $Y$ \cite[Definition 2.4]{AK}, and we have $\mathbf{M}(f_\# T)\leq \mathrm{Lip}(f)^n \mathbf{M}(T)$.

\subsection{Integral current spaces}

Here we briefly define integral current spaces, more details can be consulted in \cite{SW2} where they were originally defined together with the definition of intrisic flat convergence.

First recall that for  $T\in\mathrm{I}_n(X)$, the canonical set of $T$ is defined as 
\[
\mathrm{set}(T):= \left\{x \in X \; : \; \liminf_{r \to 0} \frac{\|T\|(B(x,r))}{\omega_n r^n} > 0\right\}.
\]
This is the smallest set among Borel sets in $X$ in which $T$ is concentrated, up to $\H^{n-1}$-negligible sets \cite[{Theorem 4.6}]{AK}.

An integral current space, $(X,d,T)$, consists of a metric space $(X,d)$ with a current $T \in \mathrm{I}_n(\bar X)$, where $\bar X$ is the completion of $X$ and so that $\mathrm{set}(T)=X$. 

For an oriented compact Riemannian manifold $(M,g)$ we associate the integral current space $(M, d_g, \curr{M})$, where $d_g$ is the length distance given by $g$ and $\curr{M}$ is the integral current given by a bi-Lipschitz oriented and countable atlas of $M$, and choosing weight $\theta=1$ (c.f. \cite[{Remarks 2.8, 2.38}]{SW2}).  We often abuse notation and write  $\curr{M}$ or even $M$ instead of the whole triple. 

We mention that for integral current spaces $(X_i,d_i, T_i)$ of the same dimension, $i=1,2$, their intrinsic flat distance equals zero if and only if
there exists an isometry $\psi:X_1 \to X_2$ such that $\psi_\sharp T_1=T_2$. 


\subsection{Area factor} For a rectifiable $n$-current $[E,\tau,\theta]$ in a Euclidean space we have the following formula for its total mass:
\[
\mathbf{M}(T)=\int_E |\theta| d\H^n.
\]
However, for a rectifiable current in a metric space an extra factor appears, the so-called area factor $\lambda$. If $V$ is a Banach space of dimension $n$, the area factor of $V$ is defined to be 
\begin{align*}
\lambda_V= \tfrac{2^n} {\omega_n}   \sup\Big\{ \tfrac{  \H^n(B_1)} {\H^n (L(C)) } \, \vert \, & B_1 \subset L(C) \subset V, \, L:  \mathbb R^n \to V \, \text{ linear}\\
&
\text{and $C$ a Euclidean cube} \Big\}.
\end{align*}
Consider now a $w^*$-separable Banach space $Z$ and a countably $\H^n$-rectifiable
subset $E$ of $Z$, i.e. assume that 
\begin{equation*}
\H^n \left( E \backslash \bigcup_{i=1}^\infty f_i(A_i) \right) = 0
\end{equation*}
with $f_i: A_i \subset \mathbb{R}^n \to Z$ Lipschitz functions, $A_i$ Borel sets, $i \in \mathbb N$. The approximate tangent space to $E$ at a point $x$ is defined as 
\begin{equation*}
\mathrm{Tan}\,^n(E,x) = wd_y f_i(\mathbb{R}^n),
\end{equation*} 
for $y \in A_i$ such that  $f_i(y)=x$, $f_i$ metrically and $w^*$-differentiable at $y$, with $J_n(wd_y f_i) > 0$. 
By \cite{AK}, $\mathrm{Tan}\,^n(E,x)$ is well defined for $\H^n$-almost all $x \in E$.  Then one defines the area factor of $E$, $\lambda: E \to \mathbb{R}$, as the area factor of $\mathrm{Tan}\,^n(E, \cdot)$. 
This can be generalized to an arbitrary separable, countably $\H^n$-rectifiable subset $E$ of a metric space. Indeed, we may isometrically embed $E$ into a $w^*$-separable Banach space $Z$, $\iota:  E \to Z$, and for $\H^n$-almost every $x\in E$
the approximate tangent space of $E$ at $x$ is defined as
\begin{equation*}
\mathrm{Tan}\,^n(E,x) : = \mathrm{Tan}\,^n(\iota(E),\iota(x)).
\end{equation*}
Once more $\mathrm{Tan}\,^n(E,x)$ is uniquely determined $\H^n$-a.e. up to linear isometries and $\lambda: E \to \mathbb{R}$ can be defined. We thus have the following representation of the mass measure.

\begin{theorem}[{\cite[Theorem 9.5]{AK}}]
For a rectifiable $n$-current $T$ in a complete metric space $(X,d)$ the mass can be represented as
\begin{equation}\label{eq:mass_formula}
\|T\|=\lambda\theta\H^n\restrict E,\qquad \mathbf{M}(T)=\int_E \lambda(x)|\theta(x)|d\H^n(x),
\end{equation}
for some $\H^n$-rectifiable set $E$, some Borel $\H^n$-integrable real multiplicity $\theta:E\to \mathbb{R} \setminus \{0\}$, and where $\lambda$ is the area factor. If $T$ is an integral current then $\theta$ assumes integer values. Moreover, writing $T$ as in \eqref{eq:decomposition_rectifiable}, we have
\begin{equation}\label{eq:theta_formula}
\theta(x)=\theta_i(f_i^{-1}(x))\qquad\text{for $\|T\|$-a.e. $x\in f_i(K_i)$},
\end{equation}
and “$\,\|T\|$-a.e.” can be equivalently replaced with “$\,\H^n\restrict \mathrm{set}(T)$-a.e.”.
\end{theorem}


\subsection{Coarea formula}
A similar reasoning as in the previous discussion, that is, considering an embedding into a $w^*$-separable Banach space, allows to define the tangential differential $d^E\psi_x$ of a Lipschitz function $\psi:E\to\mathbb{R}^n$, with $E$ an $\H^n$-rectifiable set in a metric space $(X,d)$. From this, one also has the following version of the Coarea formula \cite[Theorem 9.4]{AK2}: for a Lipschitz map $\psi:X\to \mathbb{R}^n$, and an $\H^n$-rectifiable set $E\subset X$
\begin{equation}\label{eq:coarea_formula}
\int_E \theta(x) \mathbf{C_n}(d^E \psi_x)d\H^n(x)=\int_{\mathbb{R}^n}\bigg(\sum_{\psi{-1}(y)} \theta(x) \bigg)d\H^n(y).
\end{equation}
Here $\mathbf{C}_n(d^E\psi_x)$ is the coarea factor of the tangential differential of $f$ on $E$ \cite[Section 9]{AK2}. We will not give the precise definition of $\mathbf{C}_n$, since we will only use the following inequality: for a $1$-Lipschitz map $\psi:X\to\mathbb{R}^n$ and an $\H^n$-rectifiable set $E\subset X$ we have 
\begin{equation}\label{eq:lambda_versus_Jacobian}
\mathbf{C}_n (d^E\psi_x) \leq \lambda(x)\qquad\text{for $\H^n$-a.e. $x\in X$.}
\end{equation}
This inequality is proven in \cite[Lemma 9.2]{AK} for the Jacobian $\mathbf{J}_n(d^E\psi_x)$, but the version above follows if one observes that $\mathbf{J}_n(d^E\psi_x)=\mathbf{C}_n(d^E\psi_x)$ for an $n$-dimensional set $E$ and a function $\psi$ with values in $\mathbb{R}^n$ (see the discussion above \cite[Eq. (9.2)]{AK2}). 





\subsection{Slicing}
We summarise below some properties of the slicing operator, adapted to our situation (see Theorems 5.6, 5.7 and the proof of Lemma 5.9 in \cite{AK}).


\begin{proposition}\label{prop:slicing}
Let $T\in\mathrm{I}_n(X)$, and let $\pi\in \mathrm{Lip}(X,\mathbb{R}^{n-1})$. Then for $z \in  \mathbb{R}^{n-1}$
there exist currents $\langle T,\pi,z\rangle\in\mathrm{I}_{1}(X)$ (called slices), satisfying:
\begin{enumerate}
\item $\langle T,\pi,z\rangle$ is concentrated on $\mathrm{set}(T)\cap\pi^{-1}(z)$.

 
  
    \item The following identity between measures holds:
    \[
    \int_{\mathbb{R}^{n-1}} \|\langle T,\pi,z\rangle \|dz=\|T\restrict d\pi\|,
    \]
    and in particular the following inequality holds:
    \[
    \int_{\mathbb{R}^{n-1}} \mathbf{M}(\langle T,\pi,z\rangle)dz\leq \mathrm{Lip}(\pi)^{n-1}\mathbf{M}(T).
    \]
    \item   $\partial \langle T,\pi,z\rangle =(-1)^{n-1} \langle \partial T,\pi,z\rangle$ for $\H^{n-1}$-a.e. $z$.
  
   
   
   
    \item   If the map $\pi$ coincides with $p\circ\psi$, for a pair of Lipschitz maps  $\psi:X\to\mathbb{R}^n$ and $p:\mathbb{R}^n\to\mathbb{R}^{n-1}$, then
            \[
            \psi_\# \langle T,\pi,z\rangle=\langle \psi_\# T,p,z\rangle\qquad\text{for $\H^{n-1}$-a.e. $z\in\mathbb{R}^{n-1}$.}
            \]
\end{enumerate}

\end{proposition}





\section{Rigidity of mass-preserving 1-Lipschitz maps}\label{sec:rigidity}


We start with the following simple observation, that will be used several times throughout.

\begin{lemma}\label{lemma:preserving_subsets}
Let $(X,d)$ and $(Y,\rho)$ be complete metric spaces, and let $\psi:X\to Y$ be a $1$-Lipschitz map. Consider an $n$-current $T$ in $X$ with finite mass, and suppose that $\mathbf{M}(\psi_\# T)=\mathbf{M}(T)$. Then for every $\H^n$-measurable set $A\subset X$ we also have $\mathbf{M}(\psi_\#( T\restrict A))=\mathbf{M}(T\restrict A)$.
\end{lemma}

\begin{proof} Denote by $A^c:=X\setminus A$. Then
\begin{align*}
\mathbf{M}(\psi_\# T)&\leq \mathbf{M}(\psi_\#(T\restrict A))+\mathbf{M}(\psi_\#(T\restrict A^c))\\
&\leq \mathbf{M}(T\restrict A)+\mathbf{M}(T\restrict A^c)=\mathbf{M}(T),
\end{align*}
where we have used that the mass of every current is not increased by a $1$-Lipschitz map. It follows that all inequalities are equalities, and in particular $\mathbf{M}(\psi_\#(T\restrict A))=\mathbf{M}(T\restrict A)$.
\end{proof}


\begin{lemma}\label{lemma:curve_between_points}
Let $T$ be an integral $1$-current in a complete metric space $(X,d)$, such that $\partial T=\delta_b-\delta_a$, for some $a,b\in X$. Then:
\begin{enumerate}
    \item $\mathbf{M}(T)\geq d(a,b)$;
    \item If $\mathbf{M}(T)= d(a,b)$ then $T$ is a geodesic between $a$ and $b$.
\end{enumerate} 
\end{lemma}

\begin{proof}
(1) From the hypotheses and the Decomposition Theorem \ref{thm:decomposition}, we know that 
$T=\curr{\gamma} + \sum_i\curr{\gamma_i}$, where $\gamma:[0,1]\to X$  is an injective Lipschitz curve with $\gamma(0)=a$ and $\gamma(1)=b$, and $\gamma_i$ are at most countably many simple Lipschitz loops. Moreover,
\begin{equation}\label{eq:mass_bigger_than_distance}
\mathbf{M}(T)=\mathbf{M}(\curr{\gamma})+\sum_i \mathbf{M}(\curr{\gamma_i})\geq \mathbf{M}(\curr{\gamma})=\ell(\gamma)\geq d(a,b).
\end{equation}

(2) If $\mathbf{M}(T)=d(a,b)$ then every inequality in \eqref{eq:mass_bigger_than_distance} is an equality. Therefore $T=\curr{\gamma}$, and $\gamma:[0,1]\to X$ is a Lipschitz injective curve with $\ell(\gamma)=d(\gamma(0),\gamma(1))$. This implies that $\gamma$ is a geodesic.
\end{proof}

\begin{lemma}[Essential injectivity]\label{lemma-essentialInj}
Suppose that $T$ is an integral $n$-current in a complete metric space $(X,d)$, and $\psi:X\to\mathbb{R}^n$ is a $1$-Lipschitz map such that $\mathbf{M}(\psi_\# T)=\mathbf{M}(T)$ and $\psi_\# T=\curr{B_1}$. Then:
\begin{enumerate}
   
    \item There exists an $\H^n$-measurable set $A\subset X$, with $\|T\|(X\setminus A)=0$, such that $\psi$ is injective on $A$;
    \item $T$ has multiplicity $1$ $\|T\|$-almost everywhere.
\end{enumerate}
\end{lemma}


\begin{proof}
By Theorem \ref{thm:parametric_representation} we can write $T$ as
\[
T=\sum_i (f_i)_\#\curr{\theta_i},
\]
therefore
\[
\psi_\#T=\sum_i (\psi\circ f_i)_\# \curr{\theta_i}.
\]
Since $\psi\circ f_i:\mathbb{R}^n\to\mathbb{R}^n$, we know from \eqref{eq:pushforward_formula} that 
\[
(\psi\circ f_i)_\# T=[\tilde E_i,\tau_{std},\tilde\theta_i]
\]
where $\tilde E_i=\psi(f_i(K_i))$, $\tau_{std}$ is the standard orientation of $\mathbb{R}^n$, and
\[
\tilde\theta_i(y)=\sum_{z\in (\psi\circ f_i)^{-1}(y)} \pm \theta_i(z)\qquad\text{for $\H^n$-a.e. $y\in \psi(f_i(K_i))$,}
\]
where the plus or minus sign depends on the sign of the jacobian of $\psi\circ f_i$ at the point $z$. It follows that
\[
\psi_\# T=[\psi(E), \tau_{std},\tilde \theta]
\]
where, thanks to \eqref{eq:theta_formula}, we can write
\[
\tilde \theta(y)=\sum_{i}\sum_{z\in (\psi\circ f_i)^{-1}(y)} \pm\theta_i(z)=\sum_{x\in\psi^{-1}(y)} \pm\theta(x)\qquad\text{for $\H^n$-a.e. $y\in \psi(E)$.}
\]
By the triangle inequality it follows that
\begin{equation}\label{eq:inequality_multiplicities}
|\tilde\theta(y)|\leq \sum_{x\in\psi^{-1}(y)} |\theta(x)|\qquad\text{for $\H^n$-a.e. $y\in \psi(E)$.}
\end{equation}
From this we derive the following estimate:
\begin{align*}
    \mathbf{M}(\psi_\#T)&=\int_{\psi( E)} |\tilde\theta (y)|d\H^n(y)\\
    &\leq \int_{\psi( E)}\sum_{x\in \psi^{-1}(y)} |\theta(x)|d\H^n(y)\\
    &=\int_E |\theta(x)| \mathbf{C}_n(d^E\psi_x) d\H^n(x)\\
    & \leq \int_E |\theta(x)| \lambda (x) d\H^n(x)\\
    &=\mathbf{M}(T)=\mathbf{M}(\psi_\#T).
\end{align*}
In the previous chain of inequalities we have used, in order, \eqref{eq:mass_formula}, \eqref{eq:inequality_multiplicities}, the coarea formula \eqref{eq:coarea_formula}, \eqref{eq:lambda_versus_Jacobian}, \eqref{eq:mass_formula} again, and the mass-preserving assumption. It follows that all inequalities are equalities, and in particular
\begin{equation}\label{eq:equality_multiplicities}
1=|\tilde\theta(y)|= \sum_{x\in\psi^{-1}(y)} |\theta(x)|\qquad\text{for $\H^n$-a.e. $y\in B$}
\end{equation}
because we know by assumption that $\psi_\#T=\curr{B}=[B,\tau,1]$. Now \eqref{eq:equality_multiplicities} implies both that the cardinality of $\psi^{-1}(y)$ is 1 and that $|\theta(\psi^{-1}(y))|=1$ for all points $y\in G$, for some measurable $G\subset B$ with full $\H^n$-measure. Setting $A:=\psi^{-1}(G)$ we obtain the desired properties. Indeed, $\psi$ is injective on $A$, $|\theta|=1$ on $A$, and moreover $\psi_\#(T\restrict A)=(\psi_\# T)\restrict G=\psi_\#T$, therefore by Lemma \ref{lemma:preserving_subsets} $\mathbf{M}(T\restrict A)=\mathbf{M}(\psi_\#(T\restrict A))=\mathbf{M}(\psi_\#T)=\mathbf{M}(T)$. It follows that $\|T\|(X\setminus A)=0$.
\end{proof}






        
        
        
        




\begin{lemma}[Isometry on slices]\label{lemma:isometry_on_slices}
Let $(X,d,T)$ be an $n$-dimensional integral current space, and let $\psi:X\to \mathbb{R}^n$ be a $1$-Lipschitz map satisfying assumptions (1),(2),(3) of Theorem \ref{thm-rigidity}. Let $v\in\S^{n-1}$, and consider the orthogonal projection $p:\mathbb{R}^n\to v^\perp$. Denote by $T_z:=\langle T,\pi,z\rangle\in \mathrm{I}_1(X)$ the slices with respect to $\pi:=p\circ\psi$, for $z\in D:=B\cap v^\perp$. Then, for $\H^{n-1}$-a.e. $z\in D$, $T_z$ is the current associated with a geodesic, and $\psi$ is an isometry between $\mathrm{set}(T_z)$ and the segment $B\cap p^{-1}(\{z\})$.
\end{lemma}



\begin{proof}
For every $z\in D$, we define the 1-current $\gamma_z$ naturally associated with the segment $B\cap p^{-1}(\{z\})$ (say, with orientation parallel to $v$). Note that since  $\psi_\# T=\curr{B}$, we have $\langle \psi_\# T, p, z \rangle = \gamma_z$ (up to a sign). By item (4) of Proposition \ref{prop:slicing}, we have (up to a sign)
\[
\psi_\# T_z=\gamma_z\qquad\text{for $\H^{n-1}$-a.e. $z\in D$,}
\]
and since $\psi$ is $1$-Lipschitz we also obtain 
\[
\mathbf{M}(\gamma_z)\leq \mathbf{M}(T_z)\qquad\text{for $\H^{n-1}$-a.e. $z\in D$}.
\]
Then by item (2) of Proposition \ref{prop:slicing}
applied to $T_z$ and this inequality, we get
\begin{align*}
\int_{D}\mathbf{M}(T_z)d\H^{n-1}(z)& =  \mathbf{M}(T\restrict d\pi)    \\
& \leq  \mathbf{M}(T)=\mathbf{M}(\psi_\# T)\\
&=\int_D \mathbf{M}(\gamma_z)d\H^{n-1}(z)\\
&\leq \int_D \mathbf{M}(T_z)d\H^{n-1}(z).
\end{align*}
Therefore all inequalities are equalities, and specifically 
\[
\mathbf{M}(T_z)=\mathbf{M}(\gamma_z)\qquad\text{for $\H^{n-1}$-a.e. $z\in D$}.
\]
We now claim that
\begin{equation}\label{eq:mass_boundary_2}
\mathbf{M}(\partial T_z)=2\qquad\text{for $\H^{n-1}$-a.e. $z\in D$.}
\end{equation}
The inequality $\mathbf{M}(\partial T_z)\geq \mathbf{M}(\partial \gamma_z)\geq 2$ is immediate from the $1$-Lipschitz property of $\psi$.
To prove the other inequality, we first observe that, by slicing, for $\H^{n-1}$-a.e. $z\in D$ the $1$-current $T_z$ is integral. By \cite[Theorem 5.3]{BDP} (see Theorem \ref{thm:decomposition}) $T_z$ is the sum of at most countably many (currents associated with) injective Lipschitz curves. Moreover, by item (3) of Proposition \ref{prop:slicing} for $\H^{n-1}$-a.e. $z\in D$ we have
\[
\partial T_z=\partial \langle T,\pi,z\rangle=-\langle \partial T,\pi,z\rangle,
\]
so that by item (1) of Proposition \ref{prop:slicing} we infer that $\partial T_z$ is supported on $\mathrm{set}(\partial T)\cap \pi^{-1}(z)$. Since $\pi^{-1}(z)=\psi^{-1}(p^{-1}(z))$, and $p^{-1}(z)\cap \partial B$ has cardinality at most 2, from assumption (3) of Theorem \ref{thm-rigidity} we deduce that the support of $\partial T_z$ has at most 2 points (denote them by $a_z$ and $b_z$), and thus $\mathbf{M}(\partial T_z)=2$ (otherwise we would have $\mathbf{M}(\partial\gamma_z)>2$, which is impossible because the weight of $\gamma_z$ equals 1). This proves \eqref{eq:mass_boundary_2}.

We are now in the position to apply Lemma \ref{lemma:curve_between_points}. We know that 
\[
\mathbf{M}(T_z)=\mathbf{M}(\gamma_z)=|\psi(a_z)-\psi(b_z)|\leq d(a_z,b_z),
\]
therefore we conclude that
$d(a_z,b_z)=|\psi(b_z)-\psi(a_z)|$ and that $T_z$ is (represented by) a geodesic $\eta_z$ between $a_z$ and $b_z$. By the fact that $\eta_z$ is a geodesic, for every pair of points $x,x'$ in the support of $\eta_z$ we conclude that  $d(x,x')=|\psi(x)-\psi(x')|$.
\end{proof}


In the proof of the main theorem we will need the following elementary lemma.


\begin{lemma}\label{lemma:Lebesgue}
Let $A_1,A_2$ be subsets of $\mathbb{R}^n$ with positive Lebesgue measure. Then there exists $v\in\mathbb{S}^{n-1}$ such that, denoting by $p_v: \mathbb{R}^n \to \mathbb{R}^n$ the orthogonal projection on $v^\perp$, it holds
\[
\H^{n-1}(p_v(A_1)\cap p_v(A_2))>0.
\]
\end{lemma}

\begin{proof}
Given any two Lebesgue points $a_1\in A_1$ and $a_2\in A_2$, we prove that the conclusion holds with $v$ parallel to $a_2-a_1$. We can assume without loss of generality that $a_1$ is the origin and $a_2$ is the $n$-th basis vector, and thus $v=e_n$. In the Lebesgue density theorem we can equivalently replace balls by cylinders $C_r(a_i)=D_r(p_v(a_i))\times I_r$, where $D_r(\pi_v(a_i))\subset \mathbb{R}^{n-1}$ is a disk of radius $r$ centered at $\pi_v(a_i)$, and $I_r$ is a segment of length $2r$, so that we can assume
\begin{equation}\label{eq:Lebesgue}
\lim_{r\to 0}\frac{|A_i\cap C_r(a_i)|}{|C_r(a_i)|}=1,\qquad i=1,2.
\end{equation}
It follows that, for $i=1,2$
\[
\H^{n-1}(D_r\setminus p_v(A_i))\leq \frac{1}{2r}|C_r(a_i)\setminus A_i|=\H^{n-1}(D_r) \frac{|C_r(a_i)\setminus A_i|}{|C_r(a_i)|}.
\]
The ratio to the right goes to zero as $r\to 0$ by \eqref{eq:Lebesgue}, thus for every $\varepsilon>0$, for sufficiently small $r>0$
\[
\H^{n-1}(D_r\cap p_v(A_i))\geq (1-\varepsilon) \H^{n-1}(D_r).
\]
This ensures that for $r>0$ small enough $\H^{n-1}(D_r\cap p_v(A_1)\cap p_v(A_2))>0$, which implies the thesis.
\end{proof}



We finally come to the proof of the main theorem.
\begin{proof}[Proof of Theorem \ref{thm-rigidity}]
Take any two density points $x_1,x_2$ of $\mathrm{set}(T)\cap A$, where $A$ is the injectivity set defined in Lemma \ref{lemma-essentialInj}. We are going to show that $d(x_1,x_2)=|\psi(x_1)-\psi(x_2)|$.

Take any $\delta>0$. Since $x_1,x_2$ belong to $\mathrm{set}(T)$ then $T\restrict B(x_i,\delta)$ are non zero for $i=1,2$. From the mass-preserving property and Lemma \ref{lemma:preserving_subsets} we deduce that also $\psi_\#(T\restrict B(x_i,\delta))$ are non zero for $i=1,2$, and moreover from the $1$-Lipschitz property they are supported on $B(\psi(x_1),\delta)$ and $B(\psi(x_2),\delta)$ respectively. From the fact that the latter are non-trivial rectifiable $n$-dimensional currents in $\mathbb{R}^n$, their mass is absolutely continuous with respect to Lebesgue measure. This implies that $A_i:=\psi(\mathrm{set}(T)\cap A\cap B(x_i,\delta))$ is a set of positive measure for $i=1,2$. By applying Lemma \ref{lemma:Lebesgue}, we find $v\in \mathbb{S}^{n-1}$ such that $p_v(A_1)\cap p_v(A_2)$ has positive $\H^{n-1}$-measure. Combining this with Lemma \ref{lemma:isometry_on_slices} (with $p=p_v$), we find at least two points $y_1\in B(\psi(x_1),\delta)$ and $y_2\in B(\psi(x_2),\delta)$ such that $\psi$ is an isometry between a geodesic with extremes in $\psi^{-1}(y_1)$ and $\psi^{-1}(y_2)$, and the segment between $y_1$ and $y_2$. We deduce that
\[
d(\psi^{-1}(y_1),\psi^{-1}(y_2))=|y_1-y_2|
\]
and consequently, by triangle inequality
\[
\big|d(x_1,x_2)-|\psi(x_1)-\psi(x_2)|\big|\leq 4\delta.
\]
Here we used that $y_i\in B(\psi(x_i),\delta)$ and $\psi^{-1}(y_i)\in B(x_i,\delta)$ for $i=1,2$.
From the arbitrariness of $\delta>0$ we deduce that $d(x_1,x_2)=|\psi(x_1)-\psi(x_2)|$ for all $x_1,x_2$ density points of $\mathrm{set}(T)\cap A$. Since the latter is a set of full $\|T\|$-measure, it is dense in the closure of $\mathrm{set}(T)$, therefore the isometry property extends by density to $\mathrm{spt}(T)=\overline{\mathrm{set}(T)}$.
\end{proof}
































\section{Application: stability of the Positive Mass Theorem}\label{sec-applications}


The Positive Mass Theorem of Schoen-Yau and Witten~\cite{SY, W} states that any complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative ADM mass, and if the ADM mass is zero then the manifold must be the Euclidean space. 

Here we give some details of the intrinsic flat stability of the positive mass theorem proved by Huang--Lee--Sormani in \cite{HLSa}, indicating how to use Theorem \ref{thm-rigidity} in the proof. 
We first define the class of uniformly asymptotically flat graphical hypersurfaces of $\mathbb{R}^{n+1}$ with uniformly bounded depth and nonnegative scalar curvature for which stability is shown. 





\begin{definition}
For $n\ge3$, $r_0, \gamma, D>0$, and $\alpha<0$, define $\mathcal{G}_n(r_0,\gamma,D, \alpha)$ to be the space of all smooth complete Riemannian manifolds of nonnegative scalar curvature, $(M^n,g)$ that admit a smooth Riemannian isometric embedding $\psi:M\to \mathbb{R}^{n+1}$ such that for some open $U\subset B(r_0/2)\subset \mathbb{R}^n$, the image $\psi(M)$ is the graph of a function $f\in C^\infty(\mathbb{R}^{n}\smallsetminus \overline{U}) \cap C^0(\mathbb{R}^{n}\smallsetminus {U})$,
\begin{equation*}
    \psi(M)=\left\{(x,f(x)): \,\, x\in \mathbb{R}^n \smallsetminus U\right\},
\end{equation*}
with empty or minimal boundary, 
that is, either $U=\emptyset$ or $f$ is constant on each component of $\partial U$ and $\lim_{x \to \infty}|D f|(x) \to 0$. Assume that for almost every $h$, the level set $f^{-1}(h)\subset \mathbb{R}^n$ is strictly mean-convex and outward-minimizing,
where strictly mean-convex means that the mean curvature is strictly positive, and outward-minimizing means that any region of $\mathbb{R}^n$ that contains the region enclosed by $f^{-1}(h)$ must have perimeter at least as large as $\mathcal{H}^{n-1}( f^{-1}(h))$. Assume that $f$ satisfies the following uniform asymptotic flatness conditions:
\begin{equation*}
 |D f| \le \gamma
\textrm{ for }|x|\ge r_0/2  \textrm{ and }
\lim_{x\to\infty} |D f| =0. 
\end{equation*}
Assume that 
$f(x)$ approaches a constant as $x\to\infty$. 
If $n=3$ or $4$, additionally assume that the graph is asymptotically Schwarzschild,
\begin{equation*}
\exists \Lambda\in \mathbb{R} \textrm{ such that }
 \left|f(x) - (\Lambda+ S_m(|x|)) \right| \le \gamma |x|^\alpha 
 \textrm{ for } |x|\ge r_0.
\end{equation*}
Finally assume that the regions
\begin{equation*}
\Omega(r_0)=\psi^{-1}(\overline{B(r_0)}\times\mathbb{R})\,\,\,
\textrm{ and } \,\,\, \Sigma(r_0)=\partial\Omega(r_0)
\end{equation*}
have bounded depth
 \begin{equation*}
 \operatorname{Depth}(\Omega(r_0), \Sigma(r_0)) =\sup\left\{d_g(p,\Sigma(r_0)): p\in \Omega(r_0)\right\}
 \le D.
 \end{equation*}
\end{definition}


Above, for any $m>0$, the function $S_m :  \mathbb{R}^n \smallsetminus B((2m)^{1/(n-2)})   \to \mathbb{R}$ is such that its graph corresponds to the Riemannian isometric embedding of one end of the spatial $n$-dimensional Schwarzschild manifold of ADM mass $m>0$ into $\mathbb{R}^{n+1}$ such that its minimal boundary lies in the plane $\mathbb{R}^n\times \left\{0\right\}$.

   

	
	




Huang--Lee--Sormani showed that the preimages of the intersections
of the graph $\psi_j(M_j)$ with the cylinder $\overline{B(r)} \times \mathbb{R}$ converge. 



\begin{theorem}   
Let $M_j \in\mathcal{G}_n(r_0,\gamma,D, \alpha)$ be a sequence of asymptotically flat manifolds 
so that $m_{ADM}(M_j) \to 0$. 
  Then for any $r  \geq r_0$, the sequence $  \Omega_j(r)$ subconverges in intrinsic flat sense to 
  $ \overline{B(r)}$.
 
 \end{theorem}
 
Now we review the proof given by \cite{HLSa}.


\begin{proof}



In the proof of Theorem 3.1 in \cite{HLSa} uniform upper bounds for 
$\operatorname{diam}(\Omega_j(r))$, $\operatorname{vol}(\partial \Omega_j(r))$ and $\operatorname{vol}(\Omega_j(r))$,
that only depend on the parameters $n,r_0, \gamma, D, \alpha$ and $r$, are obtained. Hence, it is concluded by applying Wenger's compactness theorem and Sormani's Arzela-Ascoli's theorem, that one gets subconvergence with respect to the intrinsic flat distance of $\Omega_j(r)$ to an integral current space $(\Omega_r, d_r, T_r)$
and a $1$-Lipschitz function $\psi_r : \Omega_r \to \overline{B(r)} \times \mathbb{R}$, which is the limit of the corresponding subsequence of $1$-Lipschitz functions $\psi_j|_{\Omega_j(r)}$.





In Corollary 4.4  in \cite{HLSa}  it is shown that 
\begin{equation}
\label{limsupVolume}
\limsup_{j \to \infty}  \operatorname{vol}(\Omega_j(r)) = \operatorname{vol}(B(r)).
\end{equation}
In Lemma 4.5 in \cite{HLSa} it is shown that 
$\psi_r ( \Omega_r ) = \overline{B(r)} \times \{0\}$, and in particular, 
$\psi_r ( \mathrm{set}(\partial T_r)) = \partial \overline{B(r)} \times \{0\}$. Then in 
Lemma 5.1 in \cite{HLSa}, it is shown that 
$\psi_r|_{\mathrm{set}(\partial T_r) }: \mathrm{set}(\partial T_r) \to \partial \overline{B(r)} \times \{0\}$ is bi-Lipschitz and in particular
$\partial \curr{ B(r)\times \left\{0\right\}}=\psi_{r\#}(\partial T_r)$. Then the following series of inequalities are shown in the proof of Theorem 1.3 in \cite{HLSa}:
\begin{align*}
\operatorname{vol}(B(r))&\le \mathbf{M}(\psi_{r\#}  T_r) \\
& \le \mathbf{M}(T_r) \label{t35}\\
 &\le \liminf_{j\to\infty} \mathbf{M}(\curr{\Omega_j(r)}   )\\
& =\liminf_{j\to\infty}\operatorname{vol}(\Omega_j(r)) \\
&= \operatorname{vol}(B(r)),
\end{align*}
where the minimizing property of the disk among Euclidean integral currents with the same boundary is used, the fact that $\mathrm{Lip}(\psi_r)\le 1$, lower semicontinuity of mass, and \eqref{limsupVolume}. It is then concluded in the proof of Theorem 1.3 in \cite{HLSa} that equality in the first step implies that $\psi_{r\#}T_r=\curr{ B(r)\times \left\{0\right\}}$ (since the disk is the \textit{unique} mass minimizer), and then we also have equality in the second inequality for the $1$-Lipschitz function $\psi_r:\Omega_r\to  B(r)\times\left\{0\right\}$. Applying now Theorem \ref{thm-rigidity} (with an obvious rescaling sending $B(r)$ to $B(1)$) we obtain that $\psi_r$ must be an isometry, and thus the limit integral current space $(\Omega_r,d_r,T_r)$ is isometric to $(B(r),d_{Eucl},\curr{B(r)})$. This ends the proof of the the stability property.
\end{proof}







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