zbMATH — the first resource for mathematics

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.

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
Full Text: DOI EuDML