Proper holomorphic images of strictly pseudoconvex domains

Journal of Geometric Analysis, 2003

Let f : D → D ′ be a proper holomorphic mapping between bounded domains D, D ′ in C 2 . Let M, M ′ be open pieces on ∂D, ∂D ′ respectively that are smooth, real analytic and of finite type. Suppose that the cluster set of M under f is contained in M ′ . It is shown that f extends holomorphically across M . This can be viewed as a local version of the Diederich-Pinchuk extension result for proper mappings in C 2 .

Complex Analysis and Operator Theory, 2016

In this paper, we generalize a recent work of Liu et al. from the open unit ball B n ⊂ C n to more general bounded strongly pseudoconvex domains with C 2 boundary. It turns out that part of the main result in this paper is in some certain sense just a part of results in a work of Bracci and Zaitsev. However, the proofs are significantly different: the argument in this paper involves a simple growth estimate for the Carathéodory metric near the boundary of C 2 domains and the well-known Graham's estimate on the boundary behavior of the Carathéodory metric on strongly pseudoconvex domains, while Bracci and Zaitsev use other arguments.

Indiana University Mathematics Journal, 1995

It is shown, that any proper holomorphic map f : D → D between bounded domains D, D C 2 with smooth real-analytic boundaries extends holomorphically to a neighborhood ofD.

Journal of the Korean Mathematical Society, 2012

Let D be an arbitrary domain in C n , n > 1, and M ⊂ ∂D be an open piece of the boundary. Suppose that M is connected and ∂D is smooth real-analytic of finite type (in the sense of D'Angelo) in a neighborhood ofM. Let f : D → C n be a holomorphic correspondence such that the cluster set cl f (M) is contained in a smooth closed real-algebraic hypersurface M ′ in C n of finite type. It is shown that if f extends continuously to some open peace of M , then f extends as a holomorphic correspondence across M. As an application, we prove that any proper holomorphic correspondence from a bounded domain D in C n with smooth real-analytic boundary onto a bounded domain D ′ in C n with smooth real-algebraic boundary extends as a holomorphic correspondence to a neighborhood ofD.

Proceedings of the American Mathematical Society, 1999

In the present paper, we generalize Wong-Rosay's theorem for proper holomorphic mappings with bounded multiplicity. As an application, we prove the non-existence of a proper holomorphic mapping from a bounded, homogenous domain in C n onto a domain in C n whose boundary contains strongly pseudoconvex points.

Proceedings of the American Mathematical Society, 2015

It is shown that if a proper holomorphic map f : C n → C N , 1 < n ≤ N , sends a pseudoconvex real analytic hypersurface of finite type into another such hypersurface, then any n − 1 dimensional component of the critical locus of f intersects both sides of M. We apply this result to the problem of boundary regularity of proper holomorphic mappings between bounded domains in C n .

Proceedings of the Steklov Institute of Mathematics, 2017

This expository paper is concerned with the properties of proper holomorphic mappings between domains in complex affine spaces. We discuss some of the main geometric methods of this theory, such as the Reflection Principle, the scaling method, and the Kobayashi-Royden metric. We sketch the proofs of certain principal results and discuss some recent achievements. Several open problems are also stated.

Chinese Annals of Mathematics Series B, 2007

The authors consider proper holomorphic mappings between smoothly bounded pseudoconvex regions in complex 2-space, where the domain is of finite type and admits a transverse circle action. The main result is that the closure of each irreducible component of the branch locus of such a map intersects the boundary of the domain in the union of finitely many orbits of the group action.

Illinois Journal of Mathematics, 2013

A piecewise smooth domain is said to have generic corners if the corners are generic CR manifolds. It is shown that a biholomorphic mapping from a piecewise smooth pseudoconvex domain with generic corners in complex Euclidean space that satisfies Condition R to another domain extends as a smooth diffeomorphism of the respective closures if and only if the target domain is also piecewise smooth with generic corners and satisfies Condition R. Further it is shown that a proper map from a domain with generic corners satisfying Condition R to a product domain of the same dimension extends continuously to the closure of the source domain in such a way that the extension is smooth on the smooth part of the boundary. In particular, the existence of such a proper mapping forces the smooth part of the boundary of the source to be Levi degenerate.

Canadian Journal of Mathematics, 2012