Geodesic disks in the universal asymptotic Teichmüller space
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Abstract:
Let $S$ be a hyperbolic Riemann surface. In a finite-dimensional Teichmüller space $T(S)$, it is still an open problem whether the geodesic disk passing through two points is unique. In an infinite-dimensional Teichmüller space it is also unclear how many geodesic disks pass through a Strebel point and the basepoint while we know that there are always infinitely many geodesic disks passing through a non-Strebel point and the basepoint. In this paper, we answer the problem arising in the universal asymptotic Teichmüller space and prove that there are always infinitely many geodesic disks passing through two points. Moreover, with the help of the Finsler structure on the asymptotic Teichmüller space, a variation formula for the asymptotic Teichmüller metric is obtained.References
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Additional Information
- Guowu Yao
- Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China
- MR Author ID: 711172
- Email: wallgreat@mail.tsinghua.edu.cn
- Received by editor(s): October 23, 2016
- Received by editor(s) in revised form: December 16, 2017, and February 7, 2018
- Published electronically: February 28, 2019
- Additional Notes: This work was supported by the National Natural Science Foundation of China (Grant No. 11771233).
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 7707-7724
- MSC (2010): Primary 30C75, 30C62
- DOI: https://doi.org/10.1090/tran/7541
- MathSciNet review: 3955533