The theory of quasiregular mappings in metric spaces: progress and challenges

TL;DR

This paper surveys recent advances in quasiregular mappings within metric spaces, focusing on geometric properties, definitions, and conditions for their equivalence, highlighting ongoing progress and challenges in the field.

Contribution

It introduces various natural definitions of quasiregular mappings in metric spaces and analyzes conditions under which these definitions are quantitatively equivalent.

Findings

Geometric porosity of the branch set in metric measure spaces

Equivalence conditions for different definitions of quasiregular mappings

Progress in understanding quasiregular mappings in general metric spaces

Abstract

We survey the recent developments in the theory of quasireg- ular mappings in metric spaces. In particular, we study the geometric porosity of the branch set of quasiregular mappings in general metric measure spaces, and then, introduce the various natural definitions of quasiregular mappings in general metric measure spaces, and give con- ditions under which they are quantitatively equivalent.