Holomorphic curvature of Finsler metrics and complex geodesics
  In his famous 1981 paper, Lempert proved that given a point in a strongly convex domain the complex geodesics (i.e., the extremal disks) for the Kobayashi metric passing through that point provide a very useful fibration of the domain. In this paper we address the question whether, given a smooth complex Finsler metric on a complex manifold, it is possible to give purely differential geometric properties of the metric ensuring the existence of such a fibration in complex geodesics of the manifold. We first discuss at some length the notion of holomorphic sectional curvature for a complex Finsler metric; then, using the differential equation of complex geodesics we obtained in a previous paper, we show that for every pair (point, tangent vector) there is a (only a segment if the metric is not complete) complex geodesic passing through the point tangent to the given vector iff the Finsler metric is K\"ahler, has constant holomorphic sectional curvature -4 and satisfies a simmetry condition on the curvature tensor. Finally, we show that a complex Finsler metric of constant holomorphic sectional curvature -4 satisfying the given simmetry condition on the curvature is necessarily the Kobayashi metric. 
Marco Abate and Giorgio Patrizio