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A Schwarz lemma for correspondences and applications. (English) Zbl 1062.32006
Author’s abstract: A version of the Schwarz lemma for correspondences is studied. Two applications are obtained namely, the ‘non-increasing’ property of the Kobayashi metric under correspondences and a weak version of the Wong-Rosay theorem for convex, finite type domains admitting a ‘non-compact’ family of proper correspondences.
Let us recall the definition of correspondences (from the Introduction): Let $${\mathcal D}$$ and $${\mathcal D}'$$ be bounded domains in $$\mathbb{C}^p$$ and $${\mathbb{C}}^n$$, respectively. A complex analytic set $$A\subset{\mathcal D}\times{\mathcal D}'$$ of pure dimension $$p$$ that satisfies $$\overline A\cap({\mathcal D}\times\partial{\mathcal D}')=\emptyset$$ is called a correspondence.

##### MSC:
 32B15 Analytic subsets of affine space 32A19 Normal families of holomorphic functions, mappings of several complex variables, and related topics (taut manifolds etc.) 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
##### Keywords:
correspondences; normal families; Kobayashi metric
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