Communications in Analysis and Geometry

Volume 30 (2022)

Number 5

The Calderón problem for the conformal Laplacian

Pages: 1121 – 1184

DOI: https://dx.doi.org/10.4310/CAG.2022.v30.n5.a6

Authors

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

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

Mikko Salo (Department of Mathematics and Statistics, University of Jyväskylä, Finland)

Abstract

We consider a conformally invariant version of the Calderón problem, where the objective is to determine the conformal class of a Riemannian manifold with boundary from the Dirichlet-to-Neumann map for the conformal Laplacian. The main result states that a locally conformally real-analytic manifold in dimensions $\geq 3$ can be determined in this way, giving a positive answer to an earlier conjecture [LU02, Conjecture 6.3]. The proof proceeds as in the standard Calderón problem on a real-analytic Riemannian manifold, but new features appear due to the conformal structure. In particular, we introduce a new coordinate system that replaces harmonic coordinates when determining the conformal class in a neighborhood of the boundary.

Received 15 June 2018

Accepted 16 October 2019

Published 17 March 2023