Holomorphic Curvature of Finsler Metrics and Complex Geodesics

If D is a bounded convex domain in C , then the work of Lempert [L] and Royden-Wong [RW] (see also [A]) show that given any point p ∈ D and any non-zero tangent vector v ∈ C at p, there exists a holomorphic map φ:U → D from the unit disk U ⊂ C into D passing through p and tangent to v in p which is an isometry with respect to the hyperbolic distance of U and the Kobayashi distance of D. Furthermore if D is smooth and strongly convex then given p and v this holomorphic disk is uniquely determined. For a general complex manifold it is hard to determine whether or not such complex curves, called complex geodesics in [V], exist. Therefore it is natural to investigate the special properties enjoyed by the Kobayashi metric of a strongly convex domain. In this case it is known that the Kobayashi metric is a strongly pseudoconvex smooth complex Finsler metric. Furthermore, for a suitable notion of holomorphic curvature (see below), this metric in convex domain has negative constant holomorphic curvature (cf. [W], [S], [R]). In [AP] it was started a systematic differential geometrical study of complex geodesics in the framework of complex Finsler metrics. As in ordinary Riemannian geometry it is natural to study geodesics as solutions of an extremal problem and not as globally lengthminimizing curves, so in our case the natural notion turns out to be the one of geodesic complex curves, i.e., of holomorphic maps from the unit disk into the manifold sending geodesics for the hyperbolic metric into geodesics for the given Finsler metric. For instance, the annulus in C has no complex geodesics in the sense of [V], whereas the usual universal covering map is a geodesic complex curve in the previous sense. It was shown in [AP] that geodesic complex curves for complex Finsler metrics satisfy a system of partial differential equations and, under suitable hypotheses, it was given an uniqueness theorem. Here we shall be concerned with the question of existence. The further ingredient needed to attack this problem is the notion of holomorphic curvature of complex Finsler metrics. Given a complex manifoldM and a complex Finsler metric F :T M → IR, i.e., a nonnegative upper semicontinuous function such that