Communications in Analysis and Geometry

Volume 31 (2023)

Number 8

Conformal harmonic coordinates

Pages: 2101 – 2155

DOI: https://dx.doi.org/10.4310/CAG.2023.v31.n8.a8

Authors

Matti Lassas (Department of Mathematics and Statistics, University of Helsinki, Finland)

Tony Liimatainen (Department of Mathematics and Statistics, University of Jyväskylä, Finland; and Department of Mathematics and Statistics, University of Helsinki, Finland)

Abstract

We study conformal harmonic coordinates on Riemannian and Lorentzian manifolds, which are coordinates constructed as quotients of solutions to the conformal Laplace equation. We show existence of conformal harmonic coordinates under general conditions and find that the coordinates are a 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 are elliptic operators in conformal harmonic coordinates if one also normalizes the determinant of the metric. We give a corresponding elliptic regularity results, including the analytic case. We prove a unique continuation result for Bach and obstruction flat manifolds, which are conformally flat near a point. We prove unique continuation results for conformal mappings both on Riemannian and Lorentzian manifolds.

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The authors were supported by the Finnish Centre of Excellence in Inverse Modelling and Imaging (Academy of Finland grant 284715).

Received 16 June 2020

Accepted 15 March 2021

Published 10 August 2024