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Localization of the Kobayashi metric and applications

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Abstract

In this paper we introduce a new class of domains— log-type convex domains, which have no boundary regularity assumptions. Then we will localize the Kobayashi metric in log-type convex subdomains. As an application, we prove a local version of continuous extension of rough isometric maps between two bounded domains with log-type convex Dini-smooth boundary points. Moreover we prove that the Teichmüller space \({\mathcal {T}}_{g,n}\) is not biholomorphic to any bounded pseudoconvex domain in \(\mathbb C^{3g-3+n}\) which is locally log-type convex near some boundary point.

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Acknowledgements

The authors would like to thank Yunhui Wu and Liyou Zhang for their precious advices and for many stimulating discussions. We would also like to thank the referee for a careful reading and valuable comments.

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Authors and Affiliations

  1. HLM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Jinsong Liu & Hongyu Wang

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Jinsong Liu & Hongyu Wang

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  1. Jinsong Liu
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  2. Hongyu Wang
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Correspondence to Jinsong Liu.

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Jinsong Liu is supported by NSF of China No.11925107, No.11671057 and No.11688101.

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Liu, J., Wang, H. Localization of the Kobayashi metric and applications. Math. Z. 297, 867–883 (2021). https://doi.org/10.1007/s00209-020-02538-0

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