Gel'fand's inverse problem under Ricci curvature bounds

TL;DR

This paper extends Gel'fand's inverse problem to non-smooth metric measure spaces with Ricci curvature bounds, proving unique solvability and stability results in a broad geometric setting.

Contribution

It establishes the unique solvability of Gel'fand's inverse problem for ${\rm RCD}(K,N)$ spaces with regular sets, and demonstrates stability in Riemannian manifolds with curvature bounds.

Findings

Unique solvability of Gel'fand's inverse problem in ${\rm RCD}(K,N)$ spaces.

Stability results for inverse problems in bounded Ricci curvature Riemannian manifolds.

Results apply to Einstein orbifolds and non-smooth boundary manifolds.

Abstract

The classical Gel'fand's inverse problem asks whether a Riemannian manifold is uniquely determined by the knowledge of the heat kernel on any open subset of the manifold. We study this inverse problem in the non-smooth setting in the framework of ${\rm RCD}(K,N)$ spaces, namely, metric-measure spaces with synthetic Riemannian Ricci curvature bounded below by $K$ and dimension bounded above by $N$. We establish the unique solvability of Gel'fand's inverse problem for the class of compact ${\rm RCD}(K,N)$ spaces whose regular set admits $C^1$-Riemannian structure. As an application, we obtain the stability of Gel'fand's inverse problem in the class of closed Riemannian manifolds with bounded Ricci curvature, diameter and volume bounded from below. We note that the results are new even for Einstein orbifolds and (weighted) Riemannian manifolds with non-smooth boundary.