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

Vesa Julin (Department of Mathematics and Statistics, University of Jyväskylä, Finland)

Tony Liimatainen (Department of Mathematics and Statistics, University of Helsinki, Finland)

Mikko Salo (Department of Mathematics and Statistics, University of Jyväskylä, Finland)

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.

Full Text (PDF format)

Received 4 August 2015

Published 4 August 2017