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 M, it is possible to find purely differential geometric properties of the metric ensuring the existence of such a fibration in complex geodesies of M. We first discuss at some length the notion of holomorphic sectional curvature for a complex Finsler metric; then, using the differential equation of complex geodesies we obtained in [AP], we show that for every pair (p;v) ∈T M, with v ≠ 0, there is a (only a segment if the metric is not complete) complex geodesic passing through p tangent to v iff the Finsler metric is Kähler, has constant holomorphic sectional curvature -4, and its curvature tensor satisfies a specific simmetry condition-which are the differential geometric conditions we were after. 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.