Uniqueness and Stability in Inverse Spectral Problems for Collapsing Manifolds

We consider a geometric inverse problems associated with interior measurements: Assume that on a closed Riemannian manifold $(M, h)$ we can make measurements of the point values of the heat kernel on some open subset $U \subset M$. Can these measurements be used to determine the whole manifold $M$ and metric $h$ on it? In this paper we analyze the stability of this reconstruction in a class of $n$-dimensional manifolds which may collapse to lower dimensions. In the Euclidean space, stability results for inverse problems for partial differential operators need considerations of operators with non-smooth coefficients. Indeed, operators with smooth coefficients can approximate those with non-smooth ones. For geometric inverse problems, we can encounter a similar phenomenon: to understand stability of the solution of inverse problems for smooth manifolds, we should study the question of uniqueness for the limiting non-smooth case. Moreover, it is well-known, that a sequence of smooth $n$-dimensional manifolds can collapse to a non-smooth space of lower dimension. To analyze the stability of inverse problem in a class of smooth manifolds with bounded sectional curvature and diameter, we study properties of the spaces which occur as limits of these collapsed manifolds and study uniqueness of inverse problems on collapsed manifolds. Combining these, we obtain stability results for inverse problems in the class of smooth manifolds with bounded sectional curvature and diameter