Short geodesics and multiplicities of eigenvalues of hyperbolic surfaces

2025/07/15 by Xiang He, He, Xiang, Yunhui Wu +3
Mathematics · #Geometric Analysis and Curvature Flows #Analytic and geometric function theory #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2507.10988

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.

Citations

• A sharp lower bound on the small eigenvalues of surfaces
• Two counterexamples to a conjecture of Colin de Verdière on multiplicity
• Maximal multiplicity of Laplacian eigenvalues in negatively curved surfaces
• Linear programming bounds for hyperbolic surfaces
• On second eigenvalues of closed hyperbolic surfaces for large genus

Related

• Small eigenvalues of hyperbolic surfaces with many cusps
• On second eigenvalues of closed hyperbolic surfaces for large genus
• The Klein quartic maximizes the multiplicity of the first positive eigenvalue of the Laplacian
• Maximal multiplicity of Laplacian eigenvalues in negatively curved surfaces
• Two counterexamples to a conjecture of Colin de Verdière on multiplicity