Optimal geodesics for boundary points of the Gardiner-Masur compactification

TL;DR

This paper proves the existence and uniqueness of optimal Teichmüller geodesics connecting boundary points in the Gardiner-Masur compactification, with implications for convergence of geodesic sequences.

Contribution

It establishes a unique optimal geodesic for boundary points in the Gardiner-Masur compactification and analyzes convergence properties of sequences of geodesics.

Findings

Unique optimal geodesic exists for boundary points that fill the surface.

Geodesics converge to boundary points in the compactification.

Sequences of geodesics passing through converging points also converge to a unique geodesic.

Abstract

The Gardiner-Masur compactification of Teichm\"uller space is homeomorphic to the horofunction compactification of the Teichm\"uller metric. Let $\xi$ and $\eta$ be a pair of boundary points in the Gardiner-Masur compactification that fill up the surface. We show that there is a unique Teichm\"uller geodesic which is optimal for the horofunctions corresponding to $\xi$ and $\eta$. In particular, when $\xi$ and $\eta$ are Busemann points that fill up the surface, the geodesic converges to $\xi$ in forward direction and to $\eta$ in backward direction. As an application, we show that if $\mathbf{G}_n$ is a sequence of Teichm\"uller geodesics passing through $X_n$ and $Y_n$ such that $X_n \to \xi$ and $Y_n \to \eta$, then $\mathbf{G}_n$ converges to a unique Teichm\"uller geodesic.