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.
Highlights
Numerous theorems in several complex variables are instances of results in metric geometry
Poincaré’s theorem which says that there is no biholomorphic map from the polydisc onto the Euclidean ball in Cn for n ≥ 2
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
Summary
Numerous theorems in several complex variables are instances of results in metric geometry. We shall see that a classic theorem due to Poincaré [22], which says that there is no biholomorphic map from the polydisc n onto the (open) Euclidean ball Bn in Cn if n ≥ 2, is a case in point. It is known [19, 29, 30] that there exists no surjective Kobayashi distance isometry of n onto Bn if n ≥ 2.
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