Geodesic disks in the universal asymptotic Teichmüller space

Yao, Guowu

doi:10.1090/tran/7541

Geodesic disks in the universal asymptotic Teichmüller space
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Trans. Amer. Math. Soc. 371 (2019), 7707-7724 Request permission

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.

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