Uniqueness of complex geodesics and characterization of circular domains (Q1191450)

The authors study complex geodesics for a complex Finsler metric \(F\) on a complex \(n\)-dimensional manifold \(M\) and prove a uniqueness theorem. Under the additional hypotheses that \(M\) is taut and that at a point \(p\in M\) the Kobayashi and Carathéodory metrics agree, they construct an exponential map for the Kobayashi metric, with interesting properties, and describe the relationship between the indicatrix \(I_ p(M)=\{v\in T_ pM\mid F(v)<1\}\) and small geodesic balls. Finally, exploiting the connection between intrinsic metrics and the complex Monge-Ampère equation on \(M\), the authors give a characterization for circular domains in \(\mathbb{C}^ n\).