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. © 2011 J. Differential Geometry.