The structure of complex Finsler manifolds is studied when the Finsler metric has the property of the Kobayashi metric on convex domains: (real) geodesics locally extend to complex curves (extremal disks). It is shown that this property of the Finsler metric induces a complex foliation of the cotangent space closely related to geodesics. Each geodesic of the metric is then shown to have a unique extension to a maximal totally geodesic complex curve Σ which has properties of extremal disks. Under the additional conditions that the metric is complete and the holomorphic sectional curvature is -4, Σ coincides with an extremal disk and a theorem of Faran is recovered: the Finsler metric coincides with the Kobayashi metric.
@article{urn:eudml:doc:41163,
title = {Finsler metrics with propierties of the Kobayashi metric on convex domains.},
journal = {Publicacions Matem\`atiques},
volume = {36},
year = {1992},
pages = {131-155},
zbl = {0754.53054},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41163}
}
Pang, Myung-Yull. Finsler metrics with propierties of the Kobayashi metric on convex domains.. Publicacions Matemàtiques, Tome 36 (1992) pp. 131-155. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41163/