Abstract
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 manifoldM, it is possible to find purely differential geometric properties of the metric ensuring the existence of such a fibration in complex geodesies ofM. 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, withv ≠ 0, there is a (only a segment if the metric is not complete) complex geodesic passing throughp tangent tov 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abate, M.Iteration Theory of Holomorphic Maps on Taut Manifolds. Mediterranean Press, Cosenza, 1989.
Abate, M., and Patrizio, G. Uniqueness of complex geodesics and characterization of circular domains.Man. Math. 74, 277–297 (1992).
Bejancu, A.Finsler Geometry and Applications. Ellis Horwood Limited, Chichester, 1990.
Burbea, J. On the hessian of the Carathéodory metric.Rocky Mtn. Math. J. 8, 555–559 (1978).
Faran, J. J. Hermitian Finsler metrics and the Kobayashi metric.J. Diff. Geom. 31, 601–625 (1990).
Heins, M. On a class of conformai mappings.Nagoya Math. J. 21, 1–60 (1962).
Kobayashi, S.Hyperbolic Manifolds and Holomorphic Mappings. Dekker, New York, 1970.
Kobayashi, S. Negative vector bundles and complex Finsler structures.Nagoya Math. J. 57, 153–166 (1975).
Lempert, L. La métrique de Kobayashi et la représentation des domaines sur la boule.Bull. Soc. Math. France 109, 427–474(1981).
Pang, M. Y. Finsler metrics with the properties of the Kobayashi metric on convex domains.Publications Mathématiques 36, 131–155 (1992).
Royden, H. L. Complex Finsler metrics.Contemporary Mathematics. Proceedings of Summer Research Conference. American Mathematical Society, Providence, 1984, pp. 119–124.
Royden, H. L. The Ahlfors-Schwarz lemma: The case of equality.J. Anal. Math. 46, 261–270 (1986).
Royden, H. L., and Wong, P. M. Carathéodory and Kobayashi metrics on convex domains. Preprint (1983).
Rund, H.The Differential Geometry of Finsler Spaces. Springer-Verlag, Berlin, 1959.
Rund, H. Generalized metrics on complex manifolds.Math. Nach. 34, 55–77 (1967).
Suzuki, M. The intrinsic metrics on the domains in ℂn.Math. Rep. Toyama Univ. 6, 143–177 (1983).
Vesentini, E. Complex geodesics.Comp. Math. 44, 375–394 (1981).
Wong, B. On the holomorphic sectional curvature of some intrinsic metrics.Proc. Am. Math. Soc. 65, 57–61(1977).
Wu, H. A remark on holomorphic sectional curvature.Indiana Math. J. 22, 1103–1108 (1973).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Abate, M., Patrizio, G. Holomorphic curvature of Finsler metrics and complex geodesics. J Geom Anal 6, 341–363 (1996). https://doi.org/10.1007/BF02921655
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02921655