Shape of filling-systole subspace in surface moduli space and critical points of systole function
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866910538266574848 |
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| author | Gao, Yue |
| author_facet | Gao, Yue |
| contents | This paper studies the space $X_g\subset \mathcal{M}_g$ consisting of surfaces with filling systoles and its subset, critical points of the systole function. In the first part, we obtain a surface with Teichmüller distance $\frac{1}{5}\log\log g$ to $X_g$ and in the second and third part, prove that most points in $\mathcal{M}_g$ have Teichmüller distance $\frac{1}{5}\log\log g$ to $X_g$ and Weil-Petersson distance $0.6521(\sqrt{\log g}-\sqrt{7\log\log g})$ respectively.Therefore we prove that the radius-$r$ neighborhood of $X_g$ is not able to cover the thick part of $\mathcal M_g$ for any fixed $r>0$. In last two parts, we get critical points with small and large (comparable to diameter of thick part of $\mathcal M_g$) distance respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_07732 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Shape of filling-systole subspace in surface moduli space and critical points of systole function Gao, Yue Geometric Topology This paper studies the space $X_g\subset \mathcal{M}_g$ consisting of surfaces with filling systoles and its subset, critical points of the systole function. In the first part, we obtain a surface with Teichmüller distance $\frac{1}{5}\log\log g$ to $X_g$ and in the second and third part, prove that most points in $\mathcal{M}_g$ have Teichmüller distance $\frac{1}{5}\log\log g$ to $X_g$ and Weil-Petersson distance $0.6521(\sqrt{\log g}-\sqrt{7\log\log g})$ respectively.Therefore we prove that the radius-$r$ neighborhood of $X_g$ is not able to cover the thick part of $\mathcal M_g$ for any fixed $r>0$. In last two parts, we get critical points with small and large (comparable to diameter of thick part of $\mathcal M_g$) distance respectively. |
| title | Shape of filling-systole subspace in surface moduli space and critical points of systole function |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2111.07732 |