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

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.

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Published 1 August 2014