# A metric version of Poincare's theorem concerning biholomorphic inequivalence of domains

Lemmens, Bas (2020) A metric version of Poincare's theorem concerning biholomorphic inequivalence of domains. arXiv, . (Submitted) (KAR id:81176)

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## Abstract

We show that if $$Y_j\subset \mathbb{C}^{n_j}$$ is a bounded strongly convex domain with $$C^3$$-boundary for $$j=1,\dots,q$$, and $$X_j\subset \mathbb{C}^{m_j}$$ is a bounded convex domain for $$j=1,\ldots,p$$, then the product domain $$\prod_{j=1}^p X_j\subset \mathbb{C}^m$$ cannot be isometrically embedded into $$\prod_{j=1}^q Y_j\subset \mathbb{C}^n$$ under the Kobayashi distance, if $$p>q$$. This result generalises Poincar\'e's theorem which says that there is no biholomorphic map from the polydisc onto the Euclidean ball in $$\mathbb{C}^n$$ for $$n\geq 2$$.

The method of proof only relies on the metric geometry of the spaces and will be derived from a 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 Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, Calculus Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science Bas Lemmens 11 May 2020 15:03 UTC 16 Feb 2021 14:12 UTC https://kar.kent.ac.uk/id/eprint/81176 (The current URI for this page, for reference purposes) https://orcid.org/0000-0001-6713-7683
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