p-harmonic coordinates for Holder metrics and applications

摘要

We show that on any Riemannian manifold with Holder 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 C1+alpha regular, and that a manifold with W-1,W-n boolean AND C-alpha metric tensor and with vanishing Weyl tensor is locally conformally flat if n >= 4. The results extend the works [LS14, LS16] from the case of C1+alpha metrics to the Holder continuous case. In an appendix, we also develop some regularity results for overdetermined elliptic systems in divergence form.