N-harmonic coordinates and the regularity of conformal mappings
Liimatainen, T., & Salo, M. (2014). N-harmonic coordinates and the regularity of conformal mappings. Mathematical research letters, 21 (2), 341-361. doi:10.4310/MRL.2014.v21.n2.a11 Retrieved from http://arxiv.org/abs/1209.1285
Published inMathematical research letters
© International Press 2014.
This article studies the smoothness of conformal mappings between two Riemannian manifolds whose metric tensors have limited regularity. We show that any bi-Lipschitz conformal mapping or 1-quasiregular mapping between two manifolds with Cr metric tensors (r > 1) is a Cr+1 conformal (local) diffeomorphism. This result is due to Iwaniec , but we give a new proof of this fact. The proof is based on n-harmonic coordinates, a generalization of the standard harmonic coordinates that is particularly suited to studying conformal mappings. We establish the existence of a p-harmonic coordinate system for 1 < p < ∞ on any Riemannian manifold.