Short geodesics and multiplicities of eigenvalues of hyperbolic surfaces

Xiang He; Yunhui Wu; Haohao Zhang

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arXiv · 2507.10988

Short geodesics and multiplicities of eigenvalues of hyperbolic surfaces

Xiang He·Yunhui Wu·Haohao Zhang

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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 Verdi\`ere in the mid 1980s.

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Xiang He, Yunhui Wu, Haohao Zhang. 2025-07-15. Short geodesics and multiplicities of eigenvalues of hyperbolic surfaces. https://arxiv.org/abs/2507.10988

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