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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
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