Geodesic disks in asymptotic Teichm\"uller space

TL;DR

This paper investigates the structure of geodesic disks in the universal asymptotic Teichmüller space, proving that infinitely many such disks pass through two points, thus addressing a key open problem.

Contribution

It establishes that in the universal asymptotic Teichmüller space, there are always infinitely many geodesic disks passing through any two points, resolving an open question.

Findings

Infinitely many geodesic disks pass through two points in the space.

Addresses an open problem in the structure of geodesic disks.

Focuses on the universal asymptotic Teichmüller space.

Abstract

Let $S$ be a hyperbolic Riemann surface. In a finite-dimensional Teichm\"uller space $T(S)$, it is still an open problem whether the geodesic disk passing through two points is unique. In an infinite-dimensional Teichm\"uller space it is also unclear how many geodesic disks pass through a Strebel point and the basepoint while we know that there are always geodesic disks passing through a non-Strebel point and the basepoint. In this paper, we answer the problem arising in the universal asymptotic Teichm\"uller space and prove that there are always infinitely many geodesic disks passing through two points.