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Gromov hyperbolicity, the Kobayashi metric, and $ \mathbb{C}$-convex sets


Author: Andrew M. Zimmer
Journal: Trans. Amer. Math. Soc. 369 (2017), 8437-8456
MSC (2010): Primary 32F45, 53C23, 32F18
DOI: https://doi.org/10.1090/tran/6909
Published electronically: June 27, 2017
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Abstract: In this paper we study the global geometry of the Kobayashi metric on domains in complex Euclidean space. We are particularly interested in developing necessary and sufficient conditions for the Kobayashi metric to be Gromov hyperbolic. For general domains, it has been suggested that a non-trivial complex affine disk in the boundary is an obstruction to Gromov hyperbolicity. This is known to be the case when the set in question is convex. In this paper we first extend this result to $ \mathbb{C}$-convex sets with $ C^1$-smooth boundary. We will then show that some boundary regularity is necessary by producing in any dimension examples of open bounded $ \mathbb{C}$-convex sets where the Kobayashi metric is Gromov hyperbolic but whose boundary contains a complex affine ball of complex codimension one.


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Additional Information

Andrew M. Zimmer
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Email: aazimmer@uchicago.edu

DOI: https://doi.org/10.1090/tran/6909
Received by editor(s): September 28, 2014
Received by editor(s) in revised form: January 21, 2016
Published electronically: June 27, 2017
Additional Notes: This material is based upon work supported by the National Science Foundation under grant number NSF 1400919.
Article copyright: © Copyright 2017 American Mathematical Society

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