A metric version of Poincaré's theorem concerning biholomorphic inequivalence of domains

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 $j=1,\dots ,p$, 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 Poincaré's theorem which says that there is no biholomorphic map from the polydisc onto the Euclidean ball in ${\mathbb{C}}^n$ for $n\ge 2$. 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.

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References in corpus (2)

• The horofunction boundary of the Hilbert geometry
• Elements of a metric spectral theory

Cited by in corpus (1)

• Horofunction compactifications and duality