A Calder\'on's problem for harmonic maps

TL;DR

This paper extends Calderón's inverse problem to harmonic maps between Riemannian manifolds, demonstrating how boundary measurements determine the metric and establishing energy rigidity results.

Contribution

It introduces higher linearization techniques to recover metrics from Dirichlet-to-Neumann maps for harmonic maps in various geometric settings.

Findings

Dirichlet-to-Neumann map determines the metric up to gauge in specific cases

Metrics on the target manifold have the same jets at a point if boundary data agree

Energy of harmonic maps uniquely determines the Dirichlet-to-Neumann map

Abstract

We study a version of Calder\'on's problem for harmonic maps between Riemannian manifolds. By using the higher linearization method, we first show that the Dirichlet-to-Neumann map determines the metric on the domain up to a natural gauge in three cases: on surfaces, on analytic manifolds, and in conformally transversally anisotropic manifolds on a fixed conformal class with injective ray transform on the transversal manifold. Next, using higher linearizations we obtain integral identities that allows us to show that the metrics on the target have the same jets at one point. In particular, if the target is analytic, the metrics are equal. We also prove an energy rigidity result, in the sense that the Dirichlet energies of harmonic maps determines the Dirichlet-to-Neumann map.