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# Mathematical Research Letters

## Volume 21 (2014)

### Number 2

### $n$-Harmonic coordinates and the regularity of conformal mappings

Pages: 341 – 361

DOI: http://dx.doi.org/10.4310/MRL.2014.v21.n2.a11

#### Authors

#### Abstract

This paper 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 $(r \gt 1)$ is a $C^{r+1}$ conformal (local) diffeomorphism. This result is due to Iwaniec [12], 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 $1 \lt p \lt \infty$ on any Riemannian manifold.