Lipschitz volume rigidity for non-collapsed $\rm RCD$ spaces with boundary

Abstract

We study Lipschitz-volume rigidity for non-collapsed ${\rm RCD}(K,N)$ spaces, possibly with boundary. We prove that a measure-preserving $1$-Lipschitz map between such spaces restricts to an isometry from the interior of the domain onto its image, where the latter is equipped with the intrinsic length distance. We then characterize when this interior rigidity extends to an isometric bijection between the whole spaces, in terms of the behavior of the map on the boundaries. Along the way, we establish a rigidity result of independent interest in a more general class of rectifiable metric spaces.