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

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.