Length of a Curve is Quasi-Convex Along a Teichmüller Geodesic

Abstract

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