The asymptoticity of extremal length in Teichmüller space
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911441868554240 |
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| author | Lyu, Zhiyang Qi, Yi |
| author_facet | Lyu, Zhiyang Qi, Yi |
| contents | We study the asymptotic behavior of extremal length along Teichmüller rays. Specifically, we determine the limit of extremal length along a Teichmüller ray and obtain an explicit expression for this limit, which complements a related formula established by Cormac Walsh. Building on this result and Kerckhoff's formula, we establish a formula for the limiting Teichmüller distance between two points moving along arbitrary pairs of Teichmüller rays. Furthermore, we derive a necessary and sufficient condition for two Teichmüller rays to be asymptotic. Finally, by shifting the initial points of the Teichmüller rays along their associated Teichmüller geodesics, we show that the minimum of the limiting Teichmüller distance coincides with the detour metric between the endpoints of the rays on the horofunction boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_06314 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The asymptoticity of extremal length in Teichmüller space Lyu, Zhiyang Qi, Yi Complex Variables Geometric Topology 30F60, 32G15, 57M50 We study the asymptotic behavior of extremal length along Teichmüller rays. Specifically, we determine the limit of extremal length along a Teichmüller ray and obtain an explicit expression for this limit, which complements a related formula established by Cormac Walsh. Building on this result and Kerckhoff's formula, we establish a formula for the limiting Teichmüller distance between two points moving along arbitrary pairs of Teichmüller rays. Furthermore, we derive a necessary and sufficient condition for two Teichmüller rays to be asymptotic. Finally, by shifting the initial points of the Teichmüller rays along their associated Teichmüller geodesics, we show that the minimum of the limiting Teichmüller distance coincides with the detour metric between the endpoints of the rays on the horofunction boundary. |
| title | The asymptoticity of extremal length in Teichmüller space |
| topic | Complex Variables Geometric Topology 30F60, 32G15, 57M50 |
| url | https://arxiv.org/abs/2511.06314 |