Rigidity of mass-preserving $1$-Lipschitz maps from integral current spaces into $\mathbb{R}^n$

TL;DR

This paper proves that certain mass-preserving 1-Lipschitz maps from integral current spaces to Euclidean balls are isometries, establishing a rigidity result and a stability theorem related to the positive mass theorem.

Contribution

It introduces a rigidity theorem for mass-preserving 1-Lipschitz maps from integral current spaces, extending stability results in geometric analysis.

Findings

Mass-preserving 1-Lipschitz maps are isometries under specified conditions.

A stability result with respect to the intrinsic flat distance is established.

Implications for the positive mass theorem for graphical manifolds are demonstrated.

Abstract

We prove that given an $n$-dimensional integral current space and a $1$-Lipschitz map, from this space onto the $n$-dimensional Euclidean ball, that preserves the mass of the current and is injective on the boundary, then the map has to be an isometry. We deduce as a consequence a stability result with respect to the intrinsic flat distance, which implies the stability of the positive mass theorem for graphical manifolds as originally formulated by Huang--Lee--Sormani.