Inverse Spectral Problems for Collapsing Manifolds I: Uniqueness and Stability

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.