Inverse spectral problems on a closed manifold

TL;DR

This paper investigates inverse spectral problems on closed Riemannian manifolds, demonstrating unique determination of the manifold from spectral data and boundary measurements, even with limited information.

Contribution

It introduces new inverse problems on manifolds, proving uniqueness results with minimal spectral data and conditions for reconstructing the manifold.

Findings

Unique determination of the manifold from eigenvalues and boundary data.

Reconstruction is possible with partial boundary data if the hypersurface has multiple components.

Conditions based on spectra of cut manifolds ensure identifiability.

Abstract

In this paper we consider two inverse problems on a closed connected Riemannian manifold $(M,g)$. The first one is a direct analog of the Gel'fand inverse boundary spectral problem. To formulate it, assume that $M$ is divided by a hypersurface $\Sigma$ into two components and we know the eigenvalues $\lambda_j$ of the Laplace operator on $(M,g)$ and also the Cauchy data, on $\Sigma$, of the corresponding eigenfunctions $\phi_j$, i.e. $\phi_j|_{\Sigma},\partial_\nu\phi_j|_{\Sigma}$, where $\nu$ is the normal to $\Sigma$. We prove that these data determine $(M,g)$ uniquely, i.e. up to an isometry. In the second problem we are given much less data, namely, $\lambda_j$ and $\phi_j|_{\Sigma}$ only. However, if $\Sigma$ consists of at least two components, $\Sigma_1, \Sigma_2$, we are still able to determine $(M,g)$ assuming some conditions on $M$ and $\Sigma$. These conditions are formulated in terms of the spectra of the manifolds with boundary obtained by cutting $M$ along $\Sigma_i$, $i=1,2$, and are of a generic nature. We consider also some other inverse problems on $M$ related to the above with data which is easier to obtain from measurements than the spectral data described.