No abstract

Complex geodesics are fundamental constructs for complex analysis and as such constitute one of the most vital research objects within this discipline. In this paper, we formulate a rigorous description, expressed in terms of geometric properties of a domain, of all complex geodesics in a convex tube domain in C n containing no complex affine lines. Next, we illustrate the obtained result by establishing a set of formulas stipulating a necessary condition for extremal mappings with respect to the Lempert function and the KobayashiRoyden metric in a large class of bounded, pseudoconvex, complete Reinhardt domains: for all of them in C 2 and for those in C n whose logarithmic image is strictly convex in the geometric sense.

We describe all the complex geodesics in convex tube domains. In the case where the base of a convex tube domain does not contain any real line, the obtained description involves the notion of boundary measure of a holomorphic map and it is expressed in the language of real Borel measures on the unit circle. Applying our result, we calculate all complex geodesics in convex tube domains with unbounded base covering a special class of Reinhardt domains.