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Homeomorphic extension of quasi-isometries for convex domains in \({\mathbb {C}}^d\) and iteration theory

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We study the homeomorphic extension of biholomorphisms between convex domains in \({\mathbb {C}}^d\) without boundary regularity and boundedness assumptions. Our approach relies on methods from coarse geometry, namely the correspondence between the Gromov boundary and the topological boundaries of the domains and the dynamical properties of commuting 1-Lipschitz maps in Gromov hyperbolic spaces. This approach not only allows us to prove extensions for biholomorphisms, but for more general quasi-isometries between the domains endowed with their Kobayashi distances.

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Acknowledgements

The authors thank the referees for their very useful comments which improved the original version.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133, Rome, Italy

    Filippo Bracci

  2. Univ. Grenoble Alpes, CNRS, IF, 38000, Grenoble, France

    Hervé Gaussier

  3. Department of Mathematics, Louisiana State University, Baton Rouge, LA, USA

    Andrew Zimmer

Authors
  1. Filippo Bracci
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  2. Hervé Gaussier
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  3. Andrew Zimmer
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Correspondence to Hervé Gaussier.

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Communicated by Ngaiming Mok.

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F. Bracci: Partially supported by the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

H. Gaussier: Partially supported by ERC ALKAGE.

A. Zimmer: Partially supported by the National Science Foundation under grants DMS-1760233 and DMS-1904099.

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Bracci, F., Gaussier, H. & Zimmer, A. Homeomorphic extension of quasi-isometries for convex domains in \({\mathbb {C}}^d\) and iteration theory. Math. Ann. 379, 691–718 (2021). https://doi.org/10.1007/s00208-020-01954-1

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  • DOI: https://doi.org/10.1007/s00208-020-01954-1

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