Length of a curve is quasi-convex along a Teichmuller geodesic

TL;DR

This paper proves that extremal and hyperbolic lengths of simple closed curves are quasi-convex functions along Teichmuller geodesics, implying quasi-convexity of metric balls in Teichmuller space.

Contribution

It establishes the quasi-convexity of extremal and hyperbolic lengths along Teichmuller geodesics, a new geometric property in Teichmuller theory.

Findings

Extremal length is quasi-convex along Teichmuller geodesics.

Hyperbolic length is quasi-convex along Teichmuller geodesics.

Metric balls in Teichmuller space are quasi-convex.

Abstract

We show that for every simple closed curve \alpha, the extremal length and the hyperbolic length of \alpha are quasi-convex functions along any Teichmuller geodesic. As a corollary, we conclude that, in Teichmuller space equipped with the Teichmuller metric, balls are quasi- convex.