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# Communications in Analysis and Geometry

## Volume 25 (2017)

### Number 2

### $p$-harmonic coordinates for Hölder metrics and applications

Pages: 395 – 430

DOI: http://dx.doi.org/10.4310/CAG.2017.v25.n2.a5

#### Authors

#### Abstract

We show that on any Riemannian manifold with Hölder continuous metric tensor, there exists a $p$-harmonic coordinate system near any point. When $p = n$ this leads to a useful gauge condition for regularity results in conformal geometry. As applications, we show that any conformal mapping between manifolds having $C^{\alpha}$ metric tensors is $C^{1+\alpha}$ regular, and that a manifold with $W^{1,n} \cap C^{\alpha}$ metric tensor and with vanishingWeyl tensor is locally conformally flat if $n \geq 4$. The results extend the works [LS14, LS16] from the case of $C^{1+\alpha}$ metrics to the Hölder continuous case. In an appendix, we also develop some regularity results for overdetermined elliptic systems in divergence form.

Received 4 August 2015

Published 4 August 2017