Small eigenvalues of hyperbolic surfaces with many cusps
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929533389635584 |
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| author | Hide, Will Thomas, Joe |
| author_facet | Hide, Will Thomas, Joe |
| contents | We study topological lower bounds on the number of small Laplacian eigenvalues on hyperbolic surfaces. We show there exist constants $a,b>0$ such that when $(g+1)<a\frac{n}{\log n}$, any hyperbolic surface of genus-$g$ with $n$ cusps has at least $b\frac{2g+n-2}{\log\left(2g+n-2\right)}$ Laplacian eigenvalues below $\frac{1}{4}$. We also show that, under certain additional constraints on the lengths of short geodesics, the lower bound can be improved to $b\left(2g+n-2\right)$ with the weaker condition $(g+1)<an$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_06093 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Small eigenvalues of hyperbolic surfaces with many cusps Hide, Will Thomas, Joe Spectral Theory Analysis of PDEs 58J50 We study topological lower bounds on the number of small Laplacian eigenvalues on hyperbolic surfaces. We show there exist constants $a,b>0$ such that when $(g+1)<a\frac{n}{\log n}$, any hyperbolic surface of genus-$g$ with $n$ cusps has at least $b\frac{2g+n-2}{\log\left(2g+n-2\right)}$ Laplacian eigenvalues below $\frac{1}{4}$. We also show that, under certain additional constraints on the lengths of short geodesics, the lower bound can be improved to $b\left(2g+n-2\right)$ with the weaker condition $(g+1)<an$. |
| title | Small eigenvalues of hyperbolic surfaces with many cusps |
| topic | Spectral Theory Analysis of PDEs 58J50 |
| url | https://arxiv.org/abs/2410.06093 |