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Inverse Spectral Problems for Collapsing Manifolds I: Uniqueness and Stability

Yaroslav Kurylev, Matti Lassas, Jinpeng Lu, Takao Yamaguchi

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.

Inverse Spectral Problems for Collapsing Manifolds I: Uniqueness and Stability

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 -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.

Paper Structure

This paper contains 19 sections, 47 theorems, 219 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Inverse problems for collapsing manifolds
  3. Main results
  4. Plan of the exposition
  5. Basic results in the theory of collapsing
  6. Properties of the limit spaces
  7. Measured Gromov-Hausdorff distance
  8. Smoothness of the density functions
  9. On properties of eigenfunctions
  10. Continuity of the direct map
  11. Spectral estimates in ${\overline{\frak F\frak M \frak M}}_p$ and ${\overline{\frak M \frak M}}_p$
  12. Spectral convergence on $\overline{{\frak F \frak M \frak M}}_p$ and $\overline{{ \frak M \frak M}}_p$
  13. Heat kernel convergence
  14. From the local spectral data to the metric-measure structure
  15. Blagovestchenskii identity on ${\overline{\frak M \frak M}}$
  16. ...and 4 more sections

Key Result

Theorem 1.4

Let $r,\Lambda,D >0, \,n \in {\mathbb Z}_+$. Denote by $\,\overline{\frak M \frak M}_p(n,\Lambda, D)$ the closure of the class of connected closed smooth pointed Riemannian manifolds defined by the conditions (basic cond 1) in the pointed measured Gromov-Hausdorff topology. Let $(X,p,\mu),(X',p',\mu where $H, \, H'$ are the heat kernels on $X,\, X'$. Then there exists an isometry $F:X\to X'$ satis

Figures (2)

  • Figure 1: Applications in manifold learning for almost collapsed manifolds. Left: Data points $X=\{x_j\in {\mathbb R}^3:\ j=1,2,\dots, 2048\}$ sampled from a 2-dimensional torus embedded in ${\mathbb R}^3$. The points are sampled along a helical curve. The distances between the points $x_j$ are defined using the flat metric of the torus $M_{R,r}=\mathbb S^1_R\times \mathbb S^1_r$, with the larger radius $R=10$ and the smaller radius $r=\frac{1}{2}$. When $r$ is small, the torus $M_{R,r}$ is 'almost collapsed' to the circle $\mathbb S^1_R$. Note that the points could also be sampled randomly, but we have used points sampled randomly from a helical curve on $M_{R,r}$ to make the visualization clearer. Right: The image of the 2-dimensional eigenfunction map $\Phi^{(2,2)}(X)$ associated to the eigenfunctions for an approximation of the heat kernel of $M_{R,r}$. The image $\Phi^{(2,2)}(X)\subset {\mathbb R}^2$ is close to a circle, that is, it is an approximation of the limiting space $\mathbb{S}^1_R$, see \ref{['eigenmap']}. We study the local version of Gel'fand's problem \ref{['Gelfand:1']}, where the heat kernel $H(x_j,x_{j'},t_\ell)$ are given at points $x_j$ that do not fill the whole manifold $M=M_{R,r}$ but only fill a possibly small metric ball $B\subset M$, for example, only the red part of the helical curve on the left picture. The data missing from the points $x\in M\setminus B$ are compensated by measuring the heat kernel at several times $t_\ell>0$, see Theorem \ref{['main uni thm']}. For details of the figure, see the part II of this paper LLY.
  • Figure 2: The set $N=N(x,\xi;\rho,s,\varepsilon)$ where $y=\gamma_{x,\xi}(\rho)$.

Theorems & Definitions (88)

  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Example 2.1
  • Lemma 2.2
  • Theorem 2.3: Fuk_JDG
  • Theorem 2.4: Fuk_JDG, Theorem 10.1
  • Corollary 2.5: Fuk:haus, Proposition 11.5
  • Lemma 2.6
  • ...and 78 more