p-harmonic coordinates for H\"older metrics and applications

Abstract

We show that on any Riemannian manifold with H\"older 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 $n \geq 4$. The results extend the works [LS14, LS15] from the case of $C^{1+\alpha}$ metrics to the H\"older continuous case. In an appendix, we also develop some regularity results for overdetermined elliptic systems in divergence form.