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

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.


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