The Calder\'on problem for the conformal Laplacian

TL;DR

This paper extends the Calderón problem to conformally invariant settings, showing that the conformal class of certain manifolds can be uniquely determined from boundary measurements, with new coordinate techniques introduced.

Contribution

It proves the unique determination of conformal classes of locally conformally real-analytic manifolds from Dirichlet-to-Neumann maps for the conformal Laplacian, confirming a longstanding conjecture.

Findings

Unique determination of conformal class in dimensions ≥ 3

Introduction of a new coordinate system for boundary determination

Extension of Calderón problem to conformally invariant operators

Abstract

We consider a conformally invariant version of the Calder\'on 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 by Lassas and Uhlmann (2001). The proof proceeds as in the standard Calder\'on 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.