Complex geodesics in tube domains and their role in the study of harmonic mappings in the disc

TL;DR

This paper investigates the structure of complex geodesics in tube domains over convex bases, providing explicit formulas, a Kobayashi-Royden metric expression, and applications to harmonic mappings and a generalized Radó-Kneser-Choquet theorem.

Contribution

It offers explicit forms of complex geodesics in certain tube domains and derives a formula for the Kobayashi-Royden metric, advancing the understanding of harmonic mappings in complex analysis.

Findings

Explicit form of complex geodesics in special cases

Effective formula for Kobayashi-Royden metric at the origin

Generalization of Radó-Kneser-Choquet theorem for harmonic mappings

Abstract

We continue the research on the structure of complex geodesics in tube domains over (bounded) convex bases. In some special cases a more explicit form of the geodesics than the existing ones are provided. As one of the consequences of our study an effective formula for the Kobayashi-Royden metric in the tube domain $T_{\mathbb B_n}$ at the origin is given. The results on the Kobayashi-Royden metric in a natural way provide versions of the Schwarz Lemma for harmonic mappings. We also present a result on harmonic mappings defined on the disc that may be seen as a generalisation of the Rad\'o-Kneser-Choquet Theorem for a class of harmonic bivalent mappings that lets understand better the geometry of complex geodesics in tube domains.