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

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.

arXiv comment16 pages and 1 figure, accepted by the Journal of Mathematical Analysis and Applications (JMAA)