Abstract
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 S which has properties of extremal disks. Under the additional conditions that the metric is complete and the holomorphic sectional curvature is -4, S coincides with an extremal disk and a theorem of Faran is recovered: the Finsler metric coincides with the Kobayashi metric.
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