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On Arnold's Transversality Conjecture for the Laplace-Beltrami Operator

Josef Greilhuber, Willi Kepplinger

Abstract

This paper is concerned with the structure of the set of Riemannian metrics on a connected manifold such that the corresponding Laplace-Beltrami operator has an eigenvalue of a given multiplicity. We introduce a simple geometric condition on metrics and their corresponding Laplace eigenfunctions, and show it is equivalent to the strong Arnold hypothesis. This hypothesis essentially posits that multiple Laplace eigenvalues split up under perturbation like those of symmetric matrices. We prove our condition is satisfied except on a set of infinite codimension, and use this to obtain non-crossing rules for Laplace eigenvalues. Furthermore, we show that our condition is satisfied on all metrics admitting eigenvalues of multiplicity at most six, and exhibit examples of metrics violating it.

On Arnold's Transversality Conjecture for the Laplace-Beltrami Operator

Abstract

This paper is concerned with the structure of the set of Riemannian metrics on a connected manifold such that the corresponding Laplace-Beltrami operator has an eigenvalue of a given multiplicity. We introduce a simple geometric condition on metrics and their corresponding Laplace eigenfunctions, and show it is equivalent to the strong Arnold hypothesis. This hypothesis essentially posits that multiple Laplace eigenvalues split up under perturbation like those of symmetric matrices. We prove our condition is satisfied except on a set of infinite codimension, and use this to obtain non-crossing rules for Laplace eigenvalues. Furthermore, we show that our condition is satisfied on all metrics admitting eigenvalues of multiplicity at most six, and exhibit examples of metrics violating it.
Paper Structure (21 sections, 19 theorems, 84 equations)

This paper contains 21 sections, 19 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Statement of the results.
  3. Preliminaries
  4. A local structure theorem for metrics with Laplace eigenvalues of higher multiplicity
  5. General setup for the construction of local defining functions and parameter transversality
  6. The case of the Laplace-Beltrami operator: Proof of Theorem \ref{['theorem:nondegenerate structure theorem']}
  7. Metrics with a (conformally) degenerate eigenvalue are rare
  8. Low multiplicities and examples
  9. Nonexistence of degenerate eigenvalues of low multiplicity
  10. Examples with many degenerate eigenvalues: Flat tori
  11. An example with no degenerate eigenvalues: The round $2$-sphere
  12. Further examples and observations
  13. Products of homogeneous spaces
  14. Disconnected Manifolds
  15. Comparison with the Hodge Laplacian
  16. ...and 6 more sections

Key Result

Theorem 1.2

Let $g$ be a Riemannian metric on a closed smooth manifold $M$. Suppose $\lambda$ is a nondegenerate eigenvalue of $\Delta_g$ with multiplicity $m$. Then there exist an open neighborhood $\mathcal{U}$ of $g$ in $\mathcal{G}(M)$, a submersion $\pi: \mathcal{U} \rightarrow \mathbb R^{\frac{m(m+1)}{2}} counted with multiplicity, for all $g' \in \mathcal{U}$. If $\lambda$ is conformally nondegenerate,

Theorems & Definitions (45)

  • Definition 1.0
  • Remark 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 2.1
  • Definition 2.2
  • Proposition 2.3
  • ...and 35 more