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Gromov hyperbolicity and the Kobayashi metric on convex domains of finite type

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In this paper we prove necessary and sufficient conditions for the Kobayashi metric on a convex domain to be Gromov hyperbolic. In particular we show that for convex domains with \(C^\infty \) boundary being of finite type in the sense of D’Angelo is equivalent to the Gromov hyperbolicity of the Kobayashi metric. We also show that bounded domains which are locally convexifiable and have finite type in the sense of D’Angelo have Gromov hyperbolic Kobayashi metric. The proofs use ideas from the theory of the Hilbert metric.

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Acknowledgments

I would like to thank my advisor Ralf Spatzier for many helpful conversations and also Yves Benoist for explaining to me some of the motivation behind his work on the Hilbert metric. This material is based upon work supported by the National Science Foundation under Grant Number NSF 1045119.

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  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    Andrew M. Zimmer

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Zimmer, A.M. Gromov hyperbolicity and the Kobayashi metric on convex domains of finite type. Math. Ann. 365, 1425–1498 (2016). https://doi.org/10.1007/s00208-015-1278-9

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