On the anisotropic Calder\'on's problem

TL;DR

This paper proves that the Riemannian metric of a compact manifold with boundary can be uniquely identified by boundary measurements, advancing the understanding of inverse problems in differential geometry.

Contribution

It establishes the uniqueness of the Riemannian metric from the Dirichlet-to-Neumann map in dimensions three and higher, for smooth boundary manifolds.

Findings

Unique determination of the metric up to boundary-fixing isometry

Extension of Calderón's problem to anisotropic Riemannian metrics

Advancement in inverse boundary value problems for geometric analysis

Abstract

We prove that the Riemannian metric on a compact manifold of dimension $n\geq 3$ with smooth boundary can be uniquely determined, up to an isometry fixing the boundary, by the Dirichlet-to-Neumann map associated to the Laplace-Beltrami operator.