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

Autor: Bas Lemmens
Rok vydání: 2022
Předmět:
Zdroj: The Journal of Geometric Analysis. 32
ISSN: 1559-002X
1050-6926
Popis: We show that if$$Y_j\subset \mathbb {C}^{n_j}$$Yj⊂Cnjis a bounded strongly convex domain with$$C^3$$C3-boundary for$$j=1,\dots ,q$$j=1,⋯,q, and$$X_j\subset \mathbb {C}^{m_j}$$Xj⊂Cmjis a bounded convex domain for$$j=1,\ldots ,p$$j=1,…,p, then the product domain$$\prod _{j=1}^p X_j\subset \mathbb {C}^m$$∏j=1pXj⊂Cmcannot be isometrically embedded into$$\prod _{j=1}^q Y_j\subset \mathbb {C}^n$$∏j=1qYj⊂Cnunder the Kobayashi distance, if$$p>q$$p>q. This result generalises Poincaré’s theorem which says that there is no biholomorphic map from the polydisc onto the Euclidean ball in$$\mathbb {C}^n$$Cnfor$$n\ge 2$$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.
Databáze: OpenAIRE
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