# A metric version of Poincare's theorem concerning biholomorphic inequivalence of domains

Lemmens, Bas (2020) A metric version of Poincare's theorem concerning biholomorphic inequivalence of domains. arXiv, . (Submitted) (KAR id:81176)

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