Lines of minima and Teichmuller geodesics

TL;DR

This paper studies the geometry of hyperbolic surfaces minimizing certain length functions related to measured laminations, characterizes short curves, and estimates distances between special surfaces in Teichmüller space, revealing both bounded and unbounded behaviors.

Contribution

It provides a detailed characterization of short curves in minimal length surfaces and estimates the Teichmüller distance between these surfaces and geodesic surfaces, with bounds depending on surface topology.

Findings

Short curves in minimal length surfaces coincide with those in associated Teichmüller geodesics.

The Teichmüller distance between these surfaces can be arbitrarily large.

For certain punctured surfaces, this distance remains bounded regardless of parameters.

Abstract

For two measured laminations $\nu^+$ and $\nu^-$ that fill up a hyperbolizable surface $S$ and for $t \in (-\infty, \infty)$, let $L_t$ be the unique hyperbolic surface that minimizes the length function $e^t l(\nu^+) + e^{-t} l(\nu^-)$ on Teichmuller space. We characterize the curves that are short in $L_t$ and estimate their lengths. We find that the short curves coincide with the curves that are short in the surface $G_t$ on the Teichmuller geodesic whose horizontal and vertical foliations are respectively, $e^t \nu^+$ and $e^{-t} \nu^-$. By deriving additional information about the twists of $\nu^+$ and $\nu^-$ around the short curves, we estimate the Teichmuller distance between $L_t$ and $G_t$. We deduce that this distance can be arbitrarily large, but that if $S$ is a once-punctured torus or four-times-punctured sphere, the distance is bounded independently of $t$.