The Lipschitz-volume rigidity problem for metric manifolds

TL;DR

This work proves a Lipschitz-volume rigidity theorem for 1-Lipschitz maps of non-zero degree from a metric space X, homeomorphic to a closed oriented manifold, to a closed oriented Riemannian manifold M, under the conditions \mathcal{H}^n(X)=\mathcal{H}^n(M) and porous sets in X having measure zero. The authors combine degree theory with recent Lipschitz-volume rigidity results for integral currents, leveraging rectifiability results (via Bate) to obtain an integral current on X and invoking Züst’s rigidity for integral currents to deduce that the map is an isometric homeomorphism. The approach generalizes classical rigidity beyond smooth domains and connects metric-geometry techniques with geometric measure theory, yielding an isometry under mass-equality and porosity assumptions and highlighting the role of the metric fundamental class. This provides a robust analytic framework for Lipschitz-volume rigidity in metric-manifold contexts with potential applications to analysis on metric spaces and geometric measure theory.

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.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Theorem 2.2
• proof
• Theorem 2.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof