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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Unbounded visibility domains, the end compactification, and applications
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by Gautam Bharali and Andrew Zimmer
Trans. Amer. Math. Soc. 376 (2023), 5949-5988
DOI: https://doi.org/10.1090/tran/8944
Published electronically: May 31, 2023

Abstract:

In this paper we study when the Kobayashi distance on a Kobayashi hyperbolic domain has certain visibility properties, with a focus on unbounded domains. “Visibility” in this context is reminiscent of visibility, seen in negatively curved Riemannian manifolds, in the sense of Eberlein–O’Neill. However, we do not assume that the domains studied are Cauchy-complete with respect to the Kobayashi distance, as this is hard to establish for domains in $\mathbb {C}^d$, $d \geq 2$. We study the various ways in which this property controls the boundary behavior of holomorphic maps. Among these results is a Carathéodory-type extension theorem for biholomorphisms between planar domains—notably: between infinitely-connected domains. We also explore connections between our visibility property and Gromov hyperbolicity of the Kobayashi distance.
References
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Bibliographic Information
  • Gautam Bharali
  • Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
  • MR Author ID: 684958
  • Email: bharali@iisc.ac.in
  • Andrew Zimmer
  • Affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706
  • MR Author ID: 831053
  • Email: amzimmer2@wisc.edu
  • Received by editor(s): July 12, 2022
  • Received by editor(s) in revised form: March 1, 2023
  • Published electronically: May 31, 2023
  • Additional Notes: The first author was supported by a UGC CAS-II grant (Grant No. F.510/25/CAS-II/2018(SAP-I)). The second author was partially supported by grants DMS-2105580 and DMS-2104381 from the National Science Foundation.
  • © Copyright 2023 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 376 (2023), 5949-5988
  • MSC (2020): Primary 32F45, 53C23; Secondary 32Q45, 53C22, 32A40, 32H50
  • DOI: https://doi.org/10.1090/tran/8944
  • MathSciNet review: 4630764