Regularity via minors and applications to conformal maps

TL;DR

This paper establishes that Sobolev maps with smooth minors are themselves smooth, and applies this to provide a simplified proof of Liouville's theorem for conformal maps under minimal regularity assumptions, especially in certain even dimensions.

Contribution

It proves a new regularity result for Sobolev maps based on minors and offers a simplified proof of Liouville's theorem for conformal maps in specific dimensions.

Findings

Sobolev maps with smooth minors are smooth when k and d are not both even

A simplified proof of Liouville's theorem for conformal maps in certain even dimensions

Regularity of W^{1,d/2} conformal maps between Riemannian manifolds under continuity

Abstract

We prove that if the minors of degree $k$ of a Sobolev map $\mathbb{R}^d \to \mathbb{R}^d$ are smooth then the map is smooth, when $k,d$ are not both even. We use this result to derive a simple, self-contained proof of the famous Liouville theorem for conformal maps, under the weakest possible regularity assumptions, in even dimensions which are not multiple of $4$. This is based on the approach taken by Iwaniec and Martin in [Acta Mathematica, 170(1):29--81, 1993]. We also prove the regularity of $W^{1,d/2}$ conformal maps between Riemannian manifolds, under the additional assumption of continuity.