Rigidity of mass-preserving 1-Lipschitz maps from integral current spaces into R-n

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 fiat distance, which implies the stability of the positive mass theorem for graphical manifolds as originally formulated by Huang-Lee-Sormani. (c) 2023 Elsevier Inc. All rights reserved.

Automated summary

This article proves that for an n-dimensional integral current space and a 1-Lipschitz map, if the map maps this space onto the n-dimensional Euclidean ball while preserving the mass of the current and is injective on the boundary, then the map must be an isometry. As a consequence, we deduce a stability result with respect to the intrinsic fiat distance, which implies the stability of the positive mass theorem for graphical manifolds as originally formulated by Huang-Lee-Sormani.