"Non-convex balls in the Teichmüller metric"  

Abstract: Let S be a closed oriented surface of genus g and let P be a finite subset of S. The Teichmüller space T(S,P) is the set of all complex structures on S up to conformal diffeomorphisms homotopic to the identity rel P. The Teichmüller distance between two points in T(S,P) is defined as the logarithm of the least amount that diffeomorphisms of S homotopic to the identity rel P can distort angles with respect to these two complex structures. The Teichmüller distance is complete, uniquely geodesic, and through any tangent direction in T(S,P) passes an isometric copy of the hyperbolic plane. Despite these nice properties, we show that there exist non-convex balls in T(S,P) whenever 3g-3+|P| is at least 4. The first step of the proof is to translate the problem into a question on extremal length, which is easier to tackle. This is joint work with Kasra Rafi.