LipschitzVolume rigidity and Sobolev coarea inequality for metric surfaces
Abstract
We prove that every 1Lipschitz 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 nonsmooth 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 EsmayliIkonenRajala.
 Publication:

arXiv eprints
 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
 EPrint:
 28 pages