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arXiv · 2109.09019

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

Wlodzimierz Zwonek
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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.

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Wlodzimierz Zwonek. 2021-09-18. Complex geodesics in tube domains and their role in the study of harmonic mappings in the disc. https://arxiv.org/abs/2109.09019

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