p-harmonic coordinates for Hölder metrics and applications

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 vanishing Weyl tensor is locally conformally flat if $n\ge 4$. The results extend the works [LS14, LS15] 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.

26 pages