On a minimal factorization conjecture

arXiv (Cornell University), 2022

We solve the problem of simultaneously embedding properly holomorphically into C 2 a whole family of n-connected domains Ωr ⊂ P 1 such that none of the components of P 1 \ Ωr reduces to a point, by constructing a continuous mapping Ξ : r {r} × Ωr → C 2 such that Ξ(r, •) : Ωr ֒→ C 2 is a proper holomorphic embedding for every r. To this aim, a parametric version of both the Andersén-Lempert procedure and Carleman's Theorem is formulated and proved.

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.

Proceedings - Mathematical Sciences, 2018

Let D, D be arbitrary domains in C n and C N respectively, 1 < n ≤ N , both possibly unbounded and M ⊆ ∂ D, M ⊆ ∂ D be open pieces of the boundaries. Suppose that ∂ D is smooth real-analytic and minimal in an open neighborhood ofM and ∂ D is smooth real-algebraic and minimal in an open neighborhood ofM. Let f : D → D be a holomorphic mapping such that the cluster set cl f (M) does not intersect D. It is proved that if the cluster set cl f (p) of some point p ∈ M contains some point q ∈ M and the graph of f extends as an analytic set to a neighborhood of (p, q) ∈ C n × C N , then f extends as a holomorphic map to a dense subset of some neighborhood of p. If in addition, M = ∂ D, M = ∂ D and M is compact, then f extends holomorphically across an open dense subset of ∂ D.

The Journal of Geometric Analysis

We solve the problem of simultaneously embedding properly holomorphically into $${\mathbb {C}}^2$$ C 2 a whole family of n-connected domains $$\Omega _r\subset \mathbb P^1$$ Ω r ⊂ P 1 such that none of the components of $$\mathbb P^1\setminus \Omega _r$$ P 1 \ Ω r reduces to a point, by constructing a continuous mapping $$\Xi :\bigcup _r\{r\}\times \Omega _r\rightarrow {\mathbb {C}}^2$$ Ξ : ⋃ r { r } × Ω r → C 2 such that $$\Xi (r,\cdot ):\Omega _r\hookrightarrow {\mathbb {C}}^2$$ Ξ ( r , · ) : Ω r ↪ C 2 is a proper holomorphic embedding for every r. To this aim, a parametric version of both the Andersén–Lempert procedure and Carleman’s Theorem is formulated and proved.

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.

The Annals of Mathematics, 2001

We prove (a weak version of) Arnold's Chord Conjecture in [2] using Gromov's "classical" idea in [9] to produce holomorphic disks with boundary on a Lagrangian submanifold. Arnold's Chord Conjecture. In this paper we prove the following theorem which was conjectured by Arnold [2]:

Advances in Mathematics, 2017

Let X, Y be smooth algebraic varieties of the same dimension. Let f, g : X −→ Y be finite regular mappings. We say that f, g are equivalent if there exists a regular automorphism Φ ∈ Aut(X) such that f = g • Φ. Of course if f, g are equivalent, then they have the same discriminants (i.e., the same set of critical values) and the same geometric degree. We show that conversely, for every hypersurface V ⊂ Y and every k ∈ N, there are only a finite number of non-equivalent finite regular mappings f : X → Y such that the discriminant D(f) equals V and µ(f) = k. As one of applications we show that if f : X → Y is a finite mapping of topological degree two, then there exists a regular automorphism Φ : X → X which acts transitively on the fibers of f and Y = X/G, where G = {id, Φ} and f is equivalent to the canonical projection X → X/G. We prove the same statement in the local (and sometimes global) holomorphic situation. In particular we show that if f : (C n , 0) → (C n , 0) is a proper and holomorphic mapping of topological degree two, then there exist biholomorphisms Ψ, Φ : (C n , 0) → (C n , 0) such that Ψ • f • Φ(x1, x2,. .. , xn) = (x 2 1 , x2,. .. , xn). Moreover, for every proper holomorphic mapping f : (C n , 0) → (C n , 0) with smooth discriminant there exist biholomorphisms Ψ, Φ : (C n , 0) → (C n , 0) such that Ψ • f • Φ(x1, x2,. .. , xn) = (x k 1 , x2,. .. , xn), where k = µ(f).

arxiv, 2022

Note to a paper of M. Tanabe concerning the classical theorem of M. De Franchis and F. Severi.

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.

Mathematische Zeitschrift, 1978