On second eigenvalues of closed hyperbolic surfaces for large genus
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866916779184357376 |
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| author | He, Yuxin Wu, Yunhui |
| author_facet | He, Yuxin Wu, Yunhui |
| contents | In this article, we study the second eigenvalues of closed hyperbolic surfaces for large genus. We show that for every closed hyperbolic surface $X_g$ of genus $g$ $(g\geq 3)$, up to uniform positive constants multiplications, the second eigenvalue $λ_2(X_g)$ of $X_g$ is greater than $\frac{\mathcal{L}_2(X_g)}{g^2}$ and less than $\mathcal{L}_2(X_g)$; moreover these two bounds are optimal as $g\to \infty$. Here $\mathcal{L}_2(X_g)$ is the shortest length of simple closed multi-geodesics separating $X_g$ into three components.
Furthermore, we also investigate the quantity $\frac{λ_2(X_g)}{\mathcal{L}_2(X_g)}$ for random hyperbolic surfaces of large genus. We show that as $g\to \infty$, a generic hyperbolic surface $X_g$ has $\frac{λ_2(X_g)}{\mathcal{L}_2(X_g)}$ uniformly comparable to $\frac{1}{\ln(g)}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_12919 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On second eigenvalues of closed hyperbolic surfaces for large genus He, Yuxin Wu, Yunhui Geometric Topology Differential Geometry Spectral Theory In this article, we study the second eigenvalues of closed hyperbolic surfaces for large genus. We show that for every closed hyperbolic surface $X_g$ of genus $g$ $(g\geq 3)$, up to uniform positive constants multiplications, the second eigenvalue $λ_2(X_g)$ of $X_g$ is greater than $\frac{\mathcal{L}_2(X_g)}{g^2}$ and less than $\mathcal{L}_2(X_g)$; moreover these two bounds are optimal as $g\to \infty$. Here $\mathcal{L}_2(X_g)$ is the shortest length of simple closed multi-geodesics separating $X_g$ into three components. Furthermore, we also investigate the quantity $\frac{λ_2(X_g)}{\mathcal{L}_2(X_g)}$ for random hyperbolic surfaces of large genus. We show that as $g\to \infty$, a generic hyperbolic surface $X_g$ has $\frac{λ_2(X_g)}{\mathcal{L}_2(X_g)}$ uniformly comparable to $\frac{1}{\ln(g)}$. |
| title | On second eigenvalues of closed hyperbolic surfaces for large genus |
| topic | Geometric Topology Differential Geometry Spectral Theory |
| url | https://arxiv.org/abs/2207.12919 |