p-harmonic coordinates for Hölder metrics and applications
Julin, V., Liimatainen, T., & Salo, M. (2017). p-harmonic coordinates for Hölder metrics and applications. Communications in Analysis and Geometry, 25 (2), 395-430. doi:10.4310/CAG.2017.v25.n2.a5
Published inCommunications in Analysis and Geometry
© the Authors, 2017. This is a final draft version of an article whose final and definitive form has been published by International Press. Published in this repository with the kind permission of the publisher.
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 α metric tensors is C 1+α regular, and that a manifold with W1,n ∩ C α metric tensor and with vanishing Weyl tensor is locally conformally flat if n ≥ 4. The results extend the works [LS14, LS15] from the case of C 1+α metrics to the H¨older continuous case. In an appendix, we also develop some regularity results for overdetermined elliptic systems in divergence form.