Short geodesics and multiplicities of eigenvalues of hyperbolic surfaces
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908685729529856 |
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| author | He, Xiang Wu, Yunhui Zhang, Haohao |
| author_facet | He, Xiang Wu, Yunhui Zhang, Haohao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_10988 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Short geodesics and multiplicities of eigenvalues of hyperbolic surfaces He, Xiang Wu, Yunhui Zhang, Haohao Differential Geometry Spectral Theory 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. |
| title | Short geodesics and multiplicities of eigenvalues of hyperbolic surfaces |
| topic | Differential Geometry Spectral Theory |
| url | https://arxiv.org/abs/2507.10988 |