Pull-back of differential forms by multi-valued Sobolev maps, and the quasiregularity of the multi-valued inverse of a quasiregular map

TL;DR

This paper develops a pull-back theory for differential forms under multi-valued Sobolev maps and demonstrates that the multi-valued inverse of a quasiregular map is itself quasiregular, revealing new regularity properties.

Contribution

It introduces a novel pull-back framework for differential forms on multi-valued maps and proves the quasiregularity of the multi-valued inverse of a quasiregular map.

Findings

Multi-valued inverse is a quasiregular ω-curve.

Higher Sobolev integrability of the inverse.

Quasiminimality of the multi-valued inverse.

Abstract

We use Almgren's framework of multi-valued maps to construct a multi-valued inverse $F:f(\Omega)\to \mathcal A_d(\mathbb R^n)$ of a quasiregular map $f:\Omega\to \mathbb R^n$ of finite degree $d$. We then develop a pull-back theory of differential forms on $\mathcal A_d(\mathbb R^n)$ by Sobolev maps, and use it to show that the multi-valued inverse is a quasiregular $\omega$-curve (in the sense of Pankka) with respect to a natural $n$-form $\omega$ (suitably interpreted). The pull-back theory is of independent interest, and allows us to conclude e.g. higher Sobolev integrability and quasiminimality of the multi-valued inverse.