2025/03/26 by Assaf Bar-Natan, Ken’ichi Ohshika, Ken'Ichi Ohshika +4 · 1 citation
Engineering · Biochemistry, Genetics and Molecular Biology · Mathematics · #Advanced Materials and Mechanics #Cellular Mechanics and Interactions #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.2503.20404
We analyse the convex structure of the Finsler infinitesimal balls of the Thurston metric on Teichmüller 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 ≥ 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.