Non-convexity of extremal length

With respect to every Riemannian metric, the Teichm\"uller metric, and the Thurston metric on Teichm\"uller 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 $\mathbb{R}^n.$