Conformal harmonic coordinates

Abstract: We study conformal harmonic coordinates on Riemannian manifolds. These are coordinates constructed as quotients of solutions to the conformal Laplace equation. We show their existence under general conditions. We find that conformal harmonic coordinates are a close conformal analogue of harmonic coordinates. We prove up to boundary regularity results for conformal mappings. We show that Weyl, Cotton, Bach, and Fefferman-Graham obstruction tensors become elliptic operators in conformal harmonic coordinates if one normalizes the determinant of the metric. We give a corresponding elliptic regularity result, which includes an analytic case. We prove a unique continuation result for local conformal flatness for Bach and obstruction flat manifolds. We discuss and prove existence of conformal harmonic coordinates on Lorentzian manifolds. We prove unique continuation results for conformal mappings both on Riemannian and Lorentzian manifolds.