Finsler Geometry and applications – Finsler

The project concerns Finsler manifolds that are not Riemannian. Some new questions on this subject arose recently and, and they are being considered by several researchers in France. The aim of this project is to put together the techniques and the efforts that are made separately by these researchers. Our approach to Finsler geometry is at the same time metrical and dynamical. There are four themes that arouse naturally in the preparation of this program: Hilbert geometry, Finsler dynamics, systolic geometry and the metric structure of Teichmüller spaces. In these domains, Finsler geometry has several facts: differentiable, metrical, projective, etc., and it makes profound links between several branches in mathematics: convex geometry, dynamical systems, geometric group theory, discrete group representations, etc. The importance of this subject started in France about 20 years ago and it is increasing at a high speed. Several PhD theses on this subject were delivered in Strasbourg, Paris, Chambéry, Grenoble, Montpellier and elsewhere. The project includes mathematicians working in several universities, and the following questions are highlighted as being interesting at the same time for several people in the group. The questions are interrelated and they show at the same time the coherence of the project. (1) The study of low-regularity Finsler dynamics (to which the metric -- as compared to differentiable -- methods apply. (2) The study of the systolic volume function on moduli spaces of convex projective structures. (3) The study of the Funk and the Hilbert metrics on Teichmüller spaces, when these spaces are embedded in finite- or infinite-dimensional Banach spaces. (4) Finlser geodesic flows on moduli spaces (Teichmüller, moduli if projective structures, etc). (5) Comparison of the Liouville volume on the unit tangent space of a Finsler manifold with the length of the shortest closed geodesic, and the characterization of Finsler structures for which this comparison is optimal. (6) Dynamical characterization of hyperbolic geometry among compact manifolds carrying a strictly convex projective structure, using the geodesic flow, and the characterization of Hilbert geometries whose volume entropy i maximal and those for which it is zero. (7) The study of relations between various invariants associated to closed geodesic functions and volume in Finsler manifolds: systole, diameter, stable norm, volume entropy, etc. and the study of geodesic asomptotics for Finsler manifolds. (8) Distribution and comparison of systolic constants. (9) The study of the Homes-Thompson volume of the unit norm balls that appear as stable norms of surfaces of a given genus. (10) Study the minimal entropy of compact Finsler manifolds, in parallel with that of symmetric spaces of rank at least two, and the relation with growth of groups: The Lehmer conjecture. (11) The conjecture of Donaldson-Tian-Yau on the existence of Kähler-Einstein manifolds in a Finsler setting. The common denominator of these questions is Finsler geometry, and the project creates a unifying frame in order to work on them.