A Metric Version of Poincar\xe9\u2019s Theorem Concerning Biholomorphic Inequivalence of Domains

Abstract

We show that if Y_jsubset mathbb {C}^{n_j} is a bounded strongly convex domain with C^3-boundary for j=1,dots ,q, and X_jsubset mathbb {C}^{m_j} is a bounded convex domain for j=1,ldots ,p, then the product domain prod _{j=1}^p X_jsubset mathbb {C}^m cannot be isometrically embedded into prod _{j=1}^q Y_jsubset mathbb {C}^n 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 mathbb {C}^n for nge 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.