Lipschitz-Volume rigidity and Sobolev coarea inequality for metric surfaces

TL;DR

We address Lipschitz–Volume rigidity in two dimensions: any area-preserving $1$-Lipschitz map from a closed metric surface $X$ to a closed Riemannian surface $Y$ with $\mathcal{H}^2(X)=\mathcal{H}^2(Y)$ is an isometry. The proof combines a sharp coarea inequality for continuous Sobolev functions on metric surfaces with a uniformization-based quasiconformal parametrization by a Riemannian surface, enabling length-control and injectivity arguments. We establish that such maps preserve length along almost every curve and, under reciprocal and regularity hypotheses on $Y$, are injective and quasiconformal; when $Y$ is upper Ahlfors $2$-regular the map becomes a bounded-length-distortion (BLD) homeomorphism, and in the smooth target case, an isometry. The paper thereby extends rigidity phenomena to non-smooth target surfaces and provides a versatile framework linking coarea, metric differentiability, and 2D uniformization for area-preserving maps.

Abstract

We prove that every 1-Lipschitz map from a closed metric surface onto a closed Riemannian surface that has the same area is an isometry. If we replace the target space with a non-smooth surface, then the statement is not true and we study the regularity properties of such a map under different geometric assumptions. Our proof relies on a coarea inequality for continuous Sobolev functions on metric surfaces that we establish, and which generalizes a recent result of Esmayli--Ikonen--Rajala.

Theorems & Definitions (59)

• Theorem 1.1
• Corollary 1.2
• Example 1.3
• Theorem 1.4
• Theorem 1.6
• Theorem 2.1: Coarea inequality and formula
• Lemma 2.2
• proof
• Lemma 2.3
• proof