Complex geodesics in convex tube domains II

TL;DR

This paper provides explicit formulas for complex geodesics in convex tube domains without affine lines and applies these results to characterize extremal mappings in certain bounded pseudoconvex Reinhardt domains.

Contribution

It offers a direct geometric description of all complex geodesics in convex tube domains and derives necessary conditions for extremal mappings in specific Reinhardt domains.

Findings

Explicit formulas for complex geodesics in convex tube domains.

Necessary conditions for extremal mappings in bounded pseudoconvex Reinhardt domains.

Applicability to domains in ^2 and certain ^n domains with convex logarithmic images.

Abstract

We give a description (direct formulas) of all complex geodesics in a convex tube domain in $\CC^n$ containing no complex affine lines, expressed in terms of geometric properties of the domain. We next apply that result to give formulas (a necessary condition) for extremal mappings with respect to the Lempert function and the Kobayashi-Royden metric in a big class of bounded, pseudoconvex, complete Reinhardt domains: for all of them in $\CC^2$ and for those of them in $\CC^n$ which logarithmic image is strictly convex in geometric sense.