Fractional Calder\'{o}n problem on a closed Riemannian manifold

Abstract

Given a fixed $\alpha \in (0,1)$, we study the inverse problem of recovering the isometry class of a smooth closed and connected Riemannian manifold $(M,g)$, given the knowledge of a source-to-solution map for the fractional Laplace equation $(-\Delta_g)^\alpha u=f$ on the manifold subject to an arbitrarily small observation region $\mathcal O$ where sources can be placed and solutions can be measured. This can be viewed as a non-local analogue of the well known anisotropic Calder\'{o}n problem that is concerned with the limiting case $\alpha=1$. While the latter problem is widely open in dimensions three and higher, we solve the non-local problem in broad geometric generality, assuming only a local property on the a priori known observation region $\mathcal O$ while making no geometric assumptions on the inaccessible region of the manifold, namely $M\setminus \mathcal O$. Our proof is based on discovering a hidden connection to a variant of Carlson's theorem in complex analysis that allows us to reduce the non-local inverse problem to the Gel'fand inverse spectral problem.