A Comparison of Period Coordinates and Teichmüller Distance
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arXiv
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| Format: | Preprint |
| Published: |
2017
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| _version_ | 1866910578825494528 |
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| author | Frankel, Ian |
| author_facet | Frankel, Ian |
| contents | Let $QD^1(\mathcal{M}_{g,n})$ be the unit cotangent bundle of the moduli space of Riemann surfaces $\mathcal{M}_{g,n}$. There is a metric $d_E$ on $QD^1(\mathcal{M}_{g,n})$ that is locally bi-Lipschitz to the Euclidean metrics defined by systems of period coordinates coming from of short and moderate-length saddle connections. We show the following: if $\mathcal{M}_{g,n}$ is equipped with the Teichmüller metric $d_T$, then the projection $(QD^1(\mathcal{M}_{g,n}),d_E) \to (\mathcal{M}_{g,n},d_T)$ is locally a Hölder map. We give a lower bound on the exponent in terms of $g$ and $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1712_00140 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | A Comparison of Period Coordinates and Teichmüller Distance Frankel, Ian Geometric Topology Dynamical Systems Let $QD^1(\mathcal{M}_{g,n})$ be the unit cotangent bundle of the moduli space of Riemann surfaces $\mathcal{M}_{g,n}$. There is a metric $d_E$ on $QD^1(\mathcal{M}_{g,n})$ that is locally bi-Lipschitz to the Euclidean metrics defined by systems of period coordinates coming from of short and moderate-length saddle connections. We show the following: if $\mathcal{M}_{g,n}$ is equipped with the Teichmüller metric $d_T$, then the projection $(QD^1(\mathcal{M}_{g,n}),d_E) \to (\mathcal{M}_{g,n},d_T)$ is locally a Hölder map. We give a lower bound on the exponent in terms of $g$ and $n$. |
| title | A Comparison of Period Coordinates and Teichmüller Distance |
| topic | Geometric Topology Dynamical Systems |
| url | https://arxiv.org/abs/1712.00140 |