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

2021/09/18 by Zwonek, Wlodzimierz
#31A05 #32F45 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2109.09019

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 Bn} 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 Radó-Kneser-Choquet Theorem for a class of harmonic bivalent mappings that lets understand better the geometry of complex geodesics in tube domains.

Related

• Complex geodesics in convex tube domains II
• Complex geodesics in convex tube domains
• Regularity of complex geodesics and (non)-Gromov hyperbolicity of convex tube domains
• The Kobayashi metric, extremal discs, and biholomorphic mappings
• The Gehring-Hayman type theorems on complex domains