On complex geodesics of balanced convex domains (Q1089476)

Let E be a complex, locally convex, Hausdorff vector space and let D be a domain in E. A complex geodesic of D is a holomorphic map of the open unit disc \(\Delta\) of \({\mathbb{C}}\) into D which is an isometry with respect to the Carathéodory (or Kobayashi) pseudo-distances of \(\Delta\) and D. [See, e.g., \textit{E. Vesentini}, Complex Geodesics, Compos. Math. 44, 375- 394 (1981; Zbl 0488.30015)]. In this paper, questions of non-uniqueness for complex geodesics of a balanced convex domain D are investigated. The results obtained establish a precise relationship between the shape of the boundary of D at a point y and the structure of the family of complex geodesics ''near'' \(\xi\mapsto \xi y\). Moreover, a complete description of all the complex geodesics is given for the open unit ball of the space C(X) of all complex valued continuous functions on a compact Hausdorff space X.