Mappings of finite distortion on metric surfaces

TL;DR

This paper studies mappings of finite distortion on metric surfaces, extending classical theorems to more general metric spaces and introducing new tools like lower gradients to analyze non-homeomorphic maps.

Contribution

It introduces lower gradients for metric space analysis and extends key theorems on distortion and Sobolev properties to metric surfaces.

Findings

Mappings with integrable distortion are continuous, open, and discrete.

Injective mappings have Sobolev inverses under certain conditions.

Extends classical Euclidean results to general metric surfaces.

Abstract

We investigate basic properties of mappings of finite distortion $f:X \to \mathbb{R}^2$, where $X$ is any metric surface, i.e., metric space homeomorphic to a planar domain with locally finite $2$-dimensional Hausdorff measure. We introduce lower gradients, which complement the upper gradients of Heinonen and Koskela, to study the distortion of non-homeomorphic maps on metric spaces. We extend the Iwaniec-\v{S}ver\'ak theorem to metric surfaces: a non-constant $f:X \to \mathbb{R}^2$ with locally square integrable upper gradient and locally integrable distortion is continuous, open and discrete. We also extend the Hencl-Koskela theorem by showing that if $f$ is moreover injective then $f^{-1}$ is a Sobolev map.