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Short geodesics and multiplicities of eigenvalues of hyperbolic surfaces

Xiang He, Yunhui Wu, Haohao Zhang

TL;DR

The article investigates how the multiplicities of Laplacian eigenvalues on closed hyperbolic surfaces grow with genus by tying them to the number of short geodesics and by exploiting thick-thin decompositions. It introduces a mass-distribution framework that concentrates eigenfunction mass on the thick part and uses heat-kernel operator methods to bound multiplicities, producing sublinear bounds for the first eigenvalue and for small eigenvalues under mild thin-part controls. The main results include a sublinear bound m(λ1) ≲ g / log log g under favorable short-geodesic counts, and a general bound m(λ) ≤ C√λ g + 24 I_ε that leads to o(g) behavior for λ1 under certain conditions. These findings advance understanding of Colin de Verdière's conjecture in the high-genus hyperbolic setting and connect spectral multiplicities to geometric data of short geodesics.

Abstract

In this paper, we obtain upper bounds on the multiplicity of Laplacian eigenvalues for closed hyperbolic surfaces in terms of the number of short closed geodesics and the genus $g$. For example, we show that if the number of short closed geodesics is sublinear in $g$, then the multiplicity of the first eigenvalue is also sublinear in $g$. This makes new progress on a conjecture by Colin de Verdière in the mid 1980s.

Short geodesics and multiplicities of eigenvalues of hyperbolic surfaces

TL;DR

The article investigates how the multiplicities of Laplacian eigenvalues on closed hyperbolic surfaces grow with genus by tying them to the number of short geodesics and by exploiting thick-thin decompositions. It introduces a mass-distribution framework that concentrates eigenfunction mass on the thick part and uses heat-kernel operator methods to bound multiplicities, producing sublinear bounds for the first eigenvalue and for small eigenvalues under mild thin-part controls. The main results include a sublinear bound m(λ1) ≲ g / log log g under favorable short-geodesic counts, and a general bound m(λ) ≤ C√λ g + 24 I_ε that leads to o(g) behavior for λ1 under certain conditions. These findings advance understanding of Colin de Verdière's conjecture in the high-genus hyperbolic setting and connect spectral multiplicities to geometric data of short geodesics.

Abstract

In this paper, we obtain upper bounds on the multiplicity of Laplacian eigenvalues for closed hyperbolic surfaces in terms of the number of short closed geodesics and the genus . For example, we show that if the number of short closed geodesics is sublinear in , then the multiplicity of the first eigenvalue is also sublinear in . This makes new progress on a conjecture by Colin de Verdière in the mid 1980s.

Paper Structure

This paper contains 10 sections, 24 theorems, 180 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Thick-thin decomposition
  3. Mass distribution of eigenfunctions
  4. Proof of Theorem \ref{['thm:mt-1']}
  5. Rescaling the metric
  6. Reduce the problem to the thick part
  7. Estimate of the integral on the thick part
  8. Estimate on $Y_1$
  9. Estimate on $Y_2$
  10. Proof of Theorem \ref{['thm:mt-2']}

Key Result

Theorem 1.1

Let $X_g$ be a closed hyperbolic surface of genus $g$. Then there exists a uniform constant $K\geq 1$ such that for any constant $\epsilon\in (0,1)$, the multiplicity of $\lambda_1(X_g)$ satisfies

Figures (4)

  • Figure 1: A closed hyperbolic surface $\mathcal{X}_g$ of genus $g$, where $\gamma_i\in\mathcal{N}_{\epsilon_0}(\mathcal{X}_g)$ and $N_{\epsilon_0}(\mathcal{X}_g)=o(g)$ for some fixed $\epsilon_0>0$. So $m(\lambda_1(\mathcal{X}_g))=o(g)$.
  • Figure 2: A closed hyperbolic surface $\mathcal{Y}_g$ of genus $g$ separated by $g+1$ geodesics into two pieces, where $\gamma_i\in\mathcal{N}_{\epsilon}(\mathcal{Y}_g)$ and for each $i$, $\ell(\gamma_i)\to0$ as $g\to\infty$. So $m(\lambda_1(\mathcal{Y}_g))=o(g)$.
  • Figure 3: An illustration for $Y_1$ and $Y_2$
  • Figure 4: An illustration for the position of $\alpha$.

Theorems & Definitions (51)

  • Theorem 1.1
  • Remark
  • Theorem 1.2
  • Remark
  • Lemma 2.1: Collar Lemma, Buser1992
  • Definition 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 41 more