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A metric analogue of Hartogs’ theorem

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Abstract

In this paper we prove a metric version of Hartogs’ theorem where the holomorphic function is replaced by a locally symmetric Hermitian metric. As an application, we prove that if the Kobayashi metric on a strongly pseudoconvex domain with \({\mathcal {C}}^2\) smooth boundary is a Kähler metric, then the universal cover of the domain is the unit ball.

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Acknowledgements

We are indebted to Stefan Nemirovskiĭ for comments on an earlier version of this paper. In particular, he suggested that Kerner’s theorem [Ker61] could be used to improve our results and this was indeed the case. We also thank the referee for his helpful remarks which improved this paper.

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Authors and Affiliations

  1. Université Grenoble Alpes, CNRS, IF, 38000, Grenoble, France

    Hervé Gaussier

  2. Department of Mathematics, University of Wisconsin-Madison, Madison, WI, 53706, USA

    Andrew Zimmer

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  1. Hervé Gaussier
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Correspondence to Hervé Gaussier.

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H. Gaussier: Partially supported by ERC ALKAGE.

A. Zimmer: Partially supported by Grants DMS-2105580 and DMS-2104381 from the National Science Foundation.

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Gaussier, H., Zimmer, A. A metric analogue of Hartogs’ theorem. Geom. Funct. Anal. 32, 1041–1062 (2022). https://doi.org/10.1007/s00039-022-00615-6

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