The Lipschitz-volume rigidity problem for metric manifolds

TL;DR

This paper establishes a rigidity result for 1-Lipschitz maps of non-zero degree between metric and Riemannian manifolds, expanding understanding of geometric constraints via degree theory and Lipschitz-volume methods.

Contribution

It introduces a Lipschitz-volume rigidity theorem for metric manifolds using degree theory and advances in integral currents, providing new insights into metric geometry.

Findings

Proves a rigidity theorem for 1-Lipschitz maps with non-zero degree.

Utilizes degree theory and Lipschitz-volume rigidity for integral currents.

Extends rigidity results to metric manifolds homeomorphic to closed oriented manifolds.

Abstract

We prove a Lipschitz-volume rigidity result for $1$-Lipschitz maps of non-zero degree between metric manifolds (metric spaces homeomorphic to a closed oriented manifold) and Riemannian manifolds. The proof is based on degree theory and recent developments of Lipschitz-volume rigidity for integral currents.