Vertical Limits of Graph Domains

TL;DR

This paper investigates the boundary behavior of Teichmüller geodesics in the universal Teichmüller space, revealing that their limits depend on both vertical and horizontal foliations of associated quadratic differentials.

Contribution

It demonstrates that the limits of Teichmüller geodesics in the Thurston boundary are influenced by both foliations, providing new insights into the boundary structure of the universal Teichmüller space.

Findings

Limits depend on both vertical and horizontal foliations.

Teichmüller geodesics have boundary limits influenced by quadratic differentials.

Results enhance understanding of the boundary behavior in Teichmüller theory.

Abstract

We consider the limiting behavior of Teichm\"uller geodesics in the universal Teichm\"uller space $T(\mathbb{H})$. Our main result states that the limits of the Teichm\"uller geodesics in the Thurston's boundary of $T(\mathbb{H})$ may depend on both vertical and horizontal foliation of the corresponding holomorphic quadratic differential.