Generalizations of the theorems of Cartan and Greene-Krantz to complex manifolds. (English) Zbl 1070.32014

Do Duc Thai; Tran Hue Minh

Ill. J. Math. 48, No. 4, 1367-1384 (2004).

Cartan’s classical theorem on sequences of biholomorphic self-maps of a bounded domain in \(\mathbb{C}^{n}\) states that if the sequence converges locally uniformly, then the limit is either a biholomorphic self-map, or it maps the domain into the boundary. The latter happens precisely when the limit map has Jacobian determinant identical to zero on the domain [H. Cartan, Math. Z. 35, 760–773 (1932; JFM 58.0349.02)].
There is a generalization by Greene-Krantz where one assumes a sequence of biholomorphic maps from bounded domains \(\Omega_{j}\) to bounded domains \(A_{j}\). Under suitable convergence assumptions on the initial domains and on the target domains, a result analogous to Cartan’s theorem holds [R. E. Greene and S. G. Krantz, Complex Analysis II, Lect. Notes Math. 1276, 136–207 (1987; Zbl 0625.32024)].
In the paper under review, some generalizations of these results are obtained for domains (not necessarily bounded) in complex manifolds.

Reviewer: Emil J. Straube (College Station)

MSC:

32H02	Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
32D20	Removable singularities in several complex variables
32Q45	Hyperbolic and Kobayashi hyperbolic manifolds

Keywords:

biholomorphic mappings; Cartan’s theorem; unbounded domains; hyperbolic manifolds

Citations:

Zbl 0625.32024; JFM 58.0349.02