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Gel'fand's inverse problem under Ricci curvature bounds

Shouhei Honda, Jinpeng Lu

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.

Gel'fand's inverse problem under Ricci curvature bounds

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 spaces, namely, metric-measure spaces with synthetic Riemannian Ricci curvature bounded below by and dimension bounded above by . We establish the unique solvability of Gel'fand's inverse problem for the class of compact spaces whose regular set admits -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.
Paper Structure (10 sections, 28 theorems, 175 equations)

This paper contains 10 sections, 28 theorems, 175 equations.

Table of Contents

  1. introduction
  2. RCD spaces
  3. Definition
  4. Heat kernel
  5. Eigenfunctions and spectral datum
  6. Structure theory
  7. Wave equation on RCD spaces
  8. Unique continuation on RCD spaces
  9. Inverse problem for the heat kernel
  10. Quantitative Gel'fand's inverse problem under bounded Ricci curvature

Key Result

Theorem 1.1

For all $n \in \mathbb{N}$ and $\epsilon, r, v, D \in (0, \infty)$, there exists $\delta=\delta(n, \epsilon, r, v, D)>0$ such that the following holds. For two closed Riemannian manifolds $M_i \in \mathcal{M}(n, D, v)$ for $i=1,2$, if there exists a map $\psi:B \to M_2$ on an open ball $B\subset M_1 Furthermore, $\psi$ is obtained as the restriction of an $\epsilon$-Gromov-Hausdorff approximation

Theorems & Definitions (56)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1: RCD space
  • Definition 2.2: Non-collapsed RCD space DG
  • Theorem 2.3: Gaussian estimate
  • Proposition 2.4: Varadhan's asymptotics
  • Theorem 2.5
  • proof
  • Corollary 2.6
  • ...and 46 more