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arXiv · 2511.00863

The asymptoticity of pairs of Teichmüller rays

Guangming Hu·Zhiyang Lyu·Hideki Miyachi·Yi Qi
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Abstract

In this paper, we study the limit of Teichmüller distance between two points along a pair of Teichmüller rays. We obtain an explicit formula for the limiting Teichmüller distance when the vertical measured foliations of the quadratic differentials are finite sums of weighted simple closed curves and uniquely ergodic measures. The limit is expressed in terms of ratios of the corresponding moduli and the Teichmüller distance between the limit surfaces when the vertical measured foliations are absolutely continuous. Consequently, two Teichmüller rays are asymptotic if and only if their vertical measured foliations are modularly equivalent and their limit surfaces coincide, which implies a main result of Masur on the asymptoticity of Teichmüller rays determined by uniquely ergodic quadratic differentials. Furthermore, we prove that the infimum of the limiting Teichmüller distances can be represented in terms of the distance between the limit surfaces of the Teichmüller rays and the detour metric of their endpoints on the Gardiner-Masur boundary, when the initial points of the rays vary along the Teichmüller geodesics.

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Guangming Hu, Zhiyang Lyu, Hideki Miyachi, Yi Qi. 2025-11-02. The asymptoticity of pairs of Teichmüller rays. https://arxiv.org/abs/2511.00863

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