A converse to the Oka principle over certain complex Banach manifolds

First Kajiwara then Leiterer gave geometric or cohomological criteria in the spirit of the Grauert-Oka principle for an open subset D of a Stein manifold M to be itself Stein. We give here criteria analogous to Leiterer's, e.g., for a relatively open subset D of a closed complex Hilbert submanifold M of separable Hilbert space to be itself biholomorphic to a closed complex Hilbert submanifold of separable Hilbert space.     

On the disk theorem

We construct an example of a 2-dimensional Stein normal space X with one singular point x ₀ such that X\{x ₀} is simply connected and it satisfies the disk condition. This answers a question raised by Forn?ss and Narasimhan. We also prove that any increasing union of Stein open sets contained in a Stein space of dimension 2 satisfies the disk condition. Starting from the above example we exhibit, without using deformation theory, a new type of 2-dimensional holes which cannot be filled.     

On the meromorphic convexity of normality domains in a Stein manifold

Let X be a Stein manifold. Then we prove that for any family ℱ⊂?(X) the normality domain Dℱ) is a meromorphically ?(X)-convex open set of X. Received: 4 November 1999     

Positive open book decompositions of Stein fillable 3-manifolds along prescribed links

It is known by Loi and Piergallini that a closed, oriented, smooth 3-manifold is Stein fillable if and only if it has a positive open book decomposition. In the present paper we will show that for every link L in a Stein fillable 3-manifold there exists an additional knot L^′ to L such that the link L∪L^′ is the binding of a positive open book decomposition of the Stein fillable 3-manifold. To prove the assertion, we will use the divide, which is a generalization of real morsification theory of complex plane curve singularities, and 2-handle attachings along Legendrian curves.     

The Levi problem for Riemann domains over Stein spaces with isolated singularities

We consider the following question: Let \({p:Y \rightarrow X}\) be an unbranched Riemann domain and assume that X is a Stein space and p is a Stein morphism. Does it follow that Y is Stein ? We show that the answer is affirmative if X has isolated singularities. This generalizes a result of Andreotti and Narasimhan.     

Globalizations, fiber bundles, and envelopes of holomorphy

We describe a relationship between globalizations of local holomorphic actions on Stein manifolds induced by global actions of certain non-compact Lie groups, and holomorphic fiber bundles with smooth Stein base and fiber and connected structure group. To this end we prove a univalence result for particular Stein Riemann domains with a free and properly discontinuous action of a discrete group of biholomorphisms. We then derive some consequences on the existence of Stein globalizations. Received August 11, 1998; in final form March 8, 1999     

Realizing connected Lie groups as automorphism groups of complex manifolds

We show that every connected real Lie group can be realized as the full automorphism group of a Stein hyperbolic complex manifold.     

Wu's theorem for Kähler–Finsler spaces

In this paper, we study strongly convex Kähler–Finsler manifolds. We prove two theorems: A strongly convex Kähler–Berwald manifold with a pole is a Stein manifold if it has nonpositive horizontal radial flag curvature; A strongly convex Kähler–Finsler manifold with a complex pole is a Stein manifold if it has nonpositive horizontal radial flag curvature.     

Stein domains in complex surfaces

Let S be a closed connected real surface and π: S→X a smooth embedding or immersion of S into a complex surface X. We denote by I(π) (resp. by I_±(π) if S is oriented) the number of complex points of π (S)∪X counted with algebraic multiplicities. Assuming that I(π)≤0 (resp. I_±(π)≤0 if S is oriented) we prove that π can be C⁰ approximated by an isotopic immersion π₁: S→X whose image has a basis of open Stein neighborhood in X which are homotopy equivalents to π₁ (S). We obtain precise results for surfaces in and find an immersed symplectic sphere in with a Stein neighborhood.     

The -equation and the Hartogs phenomenon on weakly q-pseudoconcave C R manifolds

We show that the Hartogs phenomenon holds in minimal, weakly 2-pseudoconcave generic C R submanifolds of a Stein manifold with trivial normal bundle. We also prove some results concerning the local and/or global solvability of the tangential Cauchy-Riemann equations for smooth forms and currents on weakly q-pseudoconcave C R manifolds.     

Envelopes of holomorphy and holomorphic discs

The envelope of holomorphy of an arbitrary domain in a two-dimensional Stein manifold is identified with a connected component of the set of equivalence classes of analytic discs immersed into the Stein manifold with boundary in the domain. This implies, in particular, that for each of its points the envelope of holomorphy contains an embedded (non-singular) Riemann surface (and also an immersed analytic disc) passing through this point with boundary contained in the natural embedding of the original domain into its envelope of holomorphy. Moreover, it says, that analytic continuation to a neighbourhood of an arbitrary point of the envelope of holomorphy can be performed by applying the Continuity Principle once. Another corollary concerns representation of certain elements of the fundamental group of the domain by boundaries of analytic discs. A particular case is the following. Given a contact three-manifold with Stein filling, any element of the fundamental group of the contact manifold whose representatives are contractible in the filling can be represented by the boundary of an immersed analytic disc.     

On a problem of Bremermann concerning Runge domains

In this paper we give an example of a bounded Stein domain in , with smooth boundary, which is not Runge and whose intersection with every complex line is simply connected.This work was supported by Marie Curie International Reintegration Grant no. 013023 and by the Romanian Ministry of Education and Research grant 2-CEx06-11-10/25.07.06.     

Open book decompositions and stable Hamiltonian structures

We show that every open book decomposition of a contact 3-manifold can be represented (up to isotopy) by a smooth R$\mathbb{R}$-invariant family of pseudoholomorphic curves on its symplectization with respect to a suitable stable Hamiltonian structure. In the planar case, this family survives small perturbations, and thus gives a concrete construction of a stable finite energy foliation that has been used in various applications to planar contact manifolds, including the Weinstein conjecture (Abbas et al., 2005) [2] and the equivalence of strong and Stein fillability (Wendl, to appear) [20].     

Parametric H-principle for holomorphic immersions with approximation

We prove the parametric homotopy principle for holomorphic immersions of Stein manifolds into Euclidean space and the homotopy principle with approximation on holomorphically convex sets. We write an integration by parts like formula for the solution f to the problem LfΣ_|=g, where L is a holomorphic vector field, semi-transversal to analytic variety Σ.     

Representation of holomorphic functions on coverings of pseudoconvex domains in Stein manifolds via integral formulas on these domains

The classical integral representation formulas for holomorphic functions defined on pseudoconvex domains in Stein manifolds play an important role in the constructive theory of functions of several complex variables. In this paper, we will show how to construct similar formulas for certain classes of holomorphic functions defined on coverings of such domains.     

Smoothly parameterized Čech cohomology of complex manifolds

A Stein covering of a complex manifold may be used to realize its analytic cohomology in accordance with the Čech theory. If however, the Stein covering is parameterized by a smooth manifold rather than just a discrete set, then we construct a cohomology theory in which an exterior derivative replaces the usual combinatorial Cech differential. Our construction is motivated by integral geometry and the representation theory of Lie groups.     

Plurisubharmonic Defining Functions,Good Vector Fields,and Exactness of a Certain One Form

 We show that the approaches to global regularity of the -Neumann problem via the methods listed in the title are equivalent when the conditions involved are suitably modified. These modified conditions are also equivalent to one that is relevant in the context of Stein neighborhood bases and Mergelyan type approximation. Received 14 May 2001; in revised form 8 November 2001     

Large discrete sets in Stein manifolds

We construct infinite discrete subsets of Stein manifolds with remarkable properties. These generalize results of Rosay and Rudin on discrete subsets of . Received August 21, 1998; in final form October 11, 1999 / Accepted Published online February 5, 2001     

Properties of compact complex manifolds carrying closed positive currents

We show that a compact complex manifold is Moishezon if and only if it carries a strictly positive, integral (1, 1)-current. We then study holomorphic line bundles carrying singular hermitian metrics with semi-positive curvature currents, and we give some cases in which these line bundles are big. We use these cases to provide sufficient conditions for a compact complex manifold to be Moishezon in terms of the existence of certain semi-positive, integral (1,1)-currents. We also show that the intersection number of two closed semi-positive currents of complementary degrees on a compact complex manifold is positive when the intersection of their singular supports is contained in a Stein domain. The first author was partially supported by National Science Foundation Grant Nos. DMS-8922760 and DMS-9204273. The second author was partially supported by National Science Foundation Grant Nos. DMS-9001365 and DMS-9204037.