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Inverse Spectral Problems for Collapsing Manifolds I: Uniqueness and Stability

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arxiv 1209.5875 v3 pith:47TBRS6J submitted 2012-09-26 math.DG math.AP

Yaroslav Kurylev , Matti Lassas , Jinpeng Lu , Takao Yamaguchi This is my paper · ORCID
classification math.DGmath.AP
keywords manifoldsinverseproblemclassstabilityboundedcollapsingcurvature
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abstract

We consider the geometric inverse problem of determining a closed Riemannian manifold from measurements of the heat kernel in an open subset of the manifold. In this paper we analyze the stability of this problem in the class of $n$-dimensional Riemannian manifolds with bounded diameter and sectional curvature. It is well-known that a sequence in this class of manifolds can collapse to a lower dimensional stratified space when the injectivity radius of the sequence of manifolds goes to zero. We prove the uniqueness of the inverse problem on the limiting spaces of the collapsing manifolds. As a result, we obtain stability results for the inverse problem in the class of manifolds with bounded diameter and sectional curvature.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Inverse spectral problems with sparse data and applications to passive imaging on manifolds

    math.AP 2025-07 accept novelty 7.0 of 10

    Two theorems show that a potential on a closed Riemannian manifold is uniquely recovered from sparse eigenvalue data and eigenfunction restrictions to an open set, with applications to single-measurement passive imaging.

  2. Stability of Gel'fand's inverse interior spectral problem for Schr\"odinger operators

    math.AP 2025-07 conditional novelty 7.0 of 10

    Approximate eigen-data on an open subset determine a closed Riemannian manifold up to Lipschitz distance epsilon^(1/12) and a Lipschitz potential up to epsilon^(1/(80n)), giving double-logarithmic stability.

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