Lipschitz-Volume rigidity and Sobolev coarea inequality for metric surfaces
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2023
- DOI:
- 10.48550/arXiv.2305.07621
- arXiv:
- arXiv:2305.07621
- Bibcode:
- 2023arXiv230507621M
- Keywords:
-
- Mathematics - Metric Geometry;
- Mathematics - Complex Variables;
- Mathematics - Differential Geometry;
- Primary 53C23;
- 53C45;
- Secondary 30C65;
- 53A05
- E-Print:
- 28 pages