Maximal metric surfaces and the Sobolev-to-Lipschitz property

TL;DR

This paper characterizes maximal metric spheres, especially Ahlfors regular ones, via Sobolev-to-Lipschitz property and volume rigidity, and applies these ideas to solutions of the Plateau problem in metric spaces.

Contribution

It introduces a new maximality concept for metric spheres and links it to Sobolev-to-Lipschitz property and volume rigidity, with applications to Plateau problem solutions.

Findings

Maximal representatives of metric spheres are characterized by Sobolev-to-Lipschitz property.

Unique maximality for Ahlfors regular spheres is established.

Application to Plateau problem solutions yields a maximal intrinsic disc.

Abstract

We find maximal representatives within equivalence classes of metric spheres. For Ahlfors regular spheres these are uniquely characterized by satisfying the seemingly unrelated notions of Sobolev-to-Lipschitz property, or volume rigidity. We also apply our construction to solutions of the Plateau problem in metric spaces and obtain a variant of the associated intrinsic disc studied by Lytchak--Wenger, which satisfies a related maximality condition.