The anisotropic Calder{\'o}n problem on 3-dimensional conformally St{\"a}ckel manifolds

TL;DR

This paper solves the anisotropic Calderón problem for 3D conformally Stäckel manifolds, demonstrating that the metric can be uniquely recovered from boundary measurements, advancing inverse problems in geometric analysis.

Contribution

It provides the first uniqueness result for the Calderón problem on conformally Stäckel manifolds in three dimensions, linking boundary data to the metric.

Findings

Unique determination of the metric from boundary data.

Explicit characterization of conformally Stäckel manifolds.

Advancement in inverse boundary value problems.

Abstract

Conformally St{\"a}ckel manifolds can be characterized as the class of n-dimensional pseudo-Riemannian manifolds (M, G) on which the Hamilton-Jacobi equation G($\nabla$u, $\nabla$u) = 0 for null geodesics and the Laplace equation --$\Delta$ G $\psi$ = 0 are solvable by R-separation of variables. In the particular case in which the metric has Riemannian signature, they provide explicit examples of metrics admitting a set of n--1 commuting conformal symmetry operators for the Laplace-Beltrami operator $\Delta$ G. In this paper, we solve the anisotropic Calder{\'o}n problem on compact 3-dimensional Riemannian manifolds with boundary which are conformally St{\"a}ckel, that is we show that the metric of such manifolds is uniquely determined by the Dirichlet-to-Neumann map measured on the boundary of the manifold, up to dieomorphims that preserve the boundary.