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A Metric Version of Poincaré’s Theorem Concerning Biholomorphic Inequivalence of Domains

Lemmens, Bas (2022) A Metric Version of Poincaré’s Theorem Concerning Biholomorphic Inequivalence of Domains. Journal of Geometric Analysis, 32 . Article Number 160. ISSN 1050-6926. E-ISSN 1559-002X. (doi:10.1007/s12220-022-00893-4) (KAR id:81176)

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Abstract

We show that if Yj⊂Cnj is a bounded strongly convex domain with C3-boundary for j=1,…,q, and Xj⊂Cmj is a bounded convex domain for j=1,…,p, then the product domain ∏pj=1Xj⊂Cm cannot be isometrically embedded into ∏qj=1Yj⊂Cn under the Kobayashi distance, if p>q. This result generalises Poincaré’s theorem which says that there is no biholomorphic map from the polydisc onto the Euclidean ball in Cn for n≥2. The method of proof only relies on the metric geometry of the spaces and will be derived from a more general result for products of proper geodesic metric spaces with the sup-metric. In fact, the main goal of the paper is to establish a general criterion, in terms of certain asymptotic geometric properties of the individual metric spaces, that yields an obstruction for the existence of an isometric embedding between product metric spaces.

Item Type: Article
DOI/Identification number: 10.1007/s12220-022-00893-4
Uncontrolled keywords: Product metric spaces, Product domains, Kobayashi distance, Isometric embeddings, Metric compactification, Busemann points, Detour distance
Subjects: Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, Calculus
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Funders: Engineering and Physical Sciences Research Council (https://ror.org/0439y7842)
Depositing User: Bas Lemmens
Date Deposited: 11 May 2020 15:03 UTC
Last Modified: 17 Mar 2022 16:19 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/81176 (The current URI for this page, for reference purposes)
Lemmens, Bas: https://orcid.org/0000-0001-6713-7683
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