Convex structures of the unit tangent spheres in Teichm{\"u}ller space

Abstract

We analyse the convex structure of the Finsler infinitesimal balls of the Thurston metric on Teichm{\"u}ller space. We analyse the convex structure of the Finsler unit ball in the tangent space at each point of Teichm\''uller space of a closed surface of genus $\geq 2$ equipped with Thurston's metric. We obtain a characterisation of faces, exposed faces and extreme points of such a unit sphere. In particular, we prove that every face has a unique naturally associated chain-recurrent geodesic lamination such that this face consists of the unit tangent vectors which are linear combinations of stretch vectors along maximal chain-recurrent geodesic laminations containing the given one. We show that a face is exposed if and only if its associated chain-recurrent geodesic lamination is the support of a measured lamination. Furthermore, we show that a point on a tangent unit sphere is an extreme point if and only if it is a stretch vector along some maximal chain-recurrent geodesic lamination. The last result gives an affirmative answer to a conjecture whose answer was known positively in the case where the surface is either the once-punctured torus or the 4-punctured sphere. Our main results also provide an alternative approach to the topological part of the infinitesimal rigidity result concerning Thurston's metric and the equivariance property of stretch vectors.