Non-convexity of extremal length

Abstract

With respect to every Riemannian metric, the Teichmüller metric, and the Thurston metric on Teichmüller space, we show that there exist measured foliations on surfaces whose extremal length functions are not convex. The construction uses harmonic maps to $\mathbb{R}$-trees and minimal surfaces in ${\mathbb{R}}^n.$