$n$-harmonic coordinates and the regularity of conformal mappings

TL;DR

This paper demonstrates that conformal mappings between Riemannian manifolds with limited regularity are smoother than initially assumed, using a new proof based on $n$-harmonic coordinates.

Contribution

It provides a new proof that bi-Lipschitz and $1$-quasiregular conformal mappings are $C^{r+1}$ smooth, employing $n$-harmonic coordinates for manifolds with $C^r$ metrics.

Findings

Bi-Lipschitz conformal mappings are $C^{r+1}$ smooth.

Existence of $p$-harmonic coordinate systems on Riemannian manifolds.

New proof technique based on $n$-harmonic coordinates.

Abstract

This article studies the smoothness of conformal mappings between two Riemannian manifolds whose metric tensors have limited regularity. We show that any bi-Lipschitz conformal mapping or $1$-quasiregular mapping between two manifolds with $C^r$ metric tensors ($r > 1$) is a $C^{r+1}$ conformal (local) diffeomorphism. This result was proved in [12, 27, 33], but we give a new proof of this fact. The proof is based on $n$-harmonic coordinates, a generalization of the standard harmonic coordinates that is particularly suited to studying conformal mappings. We establish the existence of a $p$-harmonic coordinate system for $1 < p < \infty$ on any Riemannian manifold.