Pseudoconvex non-Stein domains in primary Hopf surfaces

Pseudoconvex domains spread over complex homogeneous manifolds

Using the concept of inner integral curves defined by Hirschowitz we generalize a recent result by Kim, Levenberg and Yamaguchi concerning the obstruction of a pseudoconvex domain spread over a complex homogeneous manifold to be Stein. This is then applied to study the holomorphic reduction of pseudoconvex complex homogeneous manifolds X=G/H. Under the assumption that G is solvable or reductive we prove that X is the total space of a G-equivariant holomorphic fiber bundle over a Stein manifold such that all holomorphic functions on the fiber are constant.

Strong Stein neighborhood bases

Let D be a smooth bounded pseudoconvex domain in C^n. We give several characterizations for the closure of D to have a strong Stein neighborhood basis in the sense that D has a defining function r such that {z\in C^n:r(z)<a} is pseudoconvex for sufficiently small a>0. We also show that this condition is invariant under proper holomorphic maps that extend smoothly to the boundary.

Uniformization of strictly pseudoconvex domains

It is shown that two strictly pseudoconvex Stein domains with real analytic boundaries have biholomorphic universal coverings provided that their boundaries are locally biholomorphically equivalent. This statement can be regarded as a higher dimensional analogue of the Riemann uniformization theorem.

每个格罗莫夫椭圆域都是亚纯伪凸的

Every Gromov elliptic domain is meromorphically pseudoconvex

该研究证明每个格罗莫夫椭圆域都是亚纯伪凸的，明确了Stein流形中域的格罗莫夫椭圆性与Oka、Stein等性质的等价关系，还得出两类Oka流形非椭圆及射影空间中特定域非椭圆的结论。

Obstructing pseudoconvex embeddings and contractible Stein fillings for Brieskorn spheres

A conjecture due to Gompf asserts that no nontrivial Brieskorn homology sphere admits a pseudoconvex embedding in ${\mathbb C}^2$, with either orientation. A related question asks whether every compact contractible 4-manifold admits the structure of a Stein domain. We verify Gompf's conjecture, with one orientation, for a family of Brieskorn spheres of which some are known to admit a smooth embedding in ${\mathbb C}^2$. With the other orientation our methods do not resolve the question, but do give rise to an example of a contractible, boundary-irreducible 4-manifold that admits no Stein struc…

Proper J-holomorphic discs in Stein domains of dimension 2

We prove the existence of global Bishop discs in a strictly pseudoconvex Stein domain in an almost complex manifold of complex dimension 2.

Strongly pseudoconvex handlebodies

We give an explicit construction of special strongly pseudoconvex domains in C^n of handlebody type, i.e., domains which are small tubes surrounding the union of a quadratic strongly pseudoconvex domain with an attached totally real handle. Among other results, we give another proof of Lemma 3.4.3. from the paper of Y. Eliashberg, "Topological characterization of Stein manifolds of dimension $>2$", Internat. J. Math., 1 (1990), 29--46.

Exposing points on the boundary of a strictly pseudoconvex or a locally convexifiable domain of finite 1-type

We show that for any bounded domain $Ω\subset\Cp ^n$ of 1-type $2k $ which is locally convexifiable at $p\in bΩ$, having a Stein neighborhood basis, there is a biholomorphic map $f:\barΩ\rightarrow \Cp ^n $ such that $f(p)$ is a global extreme point of type $2k$ for $f{(\barΩ)}$.

Plurisubharmonic exhaustion functions and almost complex Stein structures

We prove that a relatively compact pseudoconvex domain with smooth boundary in an almost complex manifold admits a bounded strictly plurisubharmonic exhaustion function. We use this result for the study of convexity and hyperbolicity properties of these domains and the contact geometry of their boundaries.

On the Steinness index

We introduce the concept of Steinness index related to the Stein neighborhood basis. We then show several results: (1) The existence of Steinness index is equivalent to that of strong Stein neighborhood basis. (2) On the Diederich-Fornæss worm domains in particular, we present an explicit formula relating the Steinness index to the well-known Diederich-Fornæss index. (3) The Steinness index is 1 if a smoothly bounded pseudoconvex domain admits finitely many boundary points of infinite type.

Manifolds of holomorphic mappings from strongly pseudoconvex domains

Let D be a bounded strongly pseudoconvex domain in a Stein manifold S and let Y be a complex manifold. We prove that the graph of any continuous map from the closure of D to Y which is holomorphic in D admits a basis of open Stein neighborhoods in S x Y. Using this we show that many classical spaces of maps from the closure of D to Y which are holomorphic in D are infinite dimensional complex manifolds which are modeled on locally convex topological vector spaces (Banach, Hilbert or Frechet).

A Psh Hopf Lemma for Domains with Cusp Conditions

We obtain a psh Hopf lemma for domains satisfying certain cusp conditions by using a sharp estimate for the Green function of a planar cusp along the axis. As an application, we obtain a negative psh exhaustion function with certain global growth estimate on a pseudoconvex domain with Hölder boundary.

On the Theory of Stein Manifolds

This paper examines the broad structure on Stein manifolds and how it generalizes the notion of a domain of holomorphy in $\mathbb C^n$. Along with this generalization, we see that Stein manifolds share key properties from domains of holomorphy, and we prove one of these major consequences. In particular, we investigate an equivalence, similar to domains of holomorphy and pseudoconvexity, on the class of manifolds. Then, we examine the canonical symplectic structure of Stein manifolds inherited from this equivalence, and how its symplectic topology develops.

Exposing boundary points of strongly pseudoconvex subvarieties in complex spaces

We prove that all locally exposable points in a Stein compact in a complex space can be exposed along a given curve to a given real hypersurface. Moreover, the exposing map for a boundary point can be sufficiently close to the identity map outside any fixed neighborhood of the point. We also prove a parametric version of this result for bounded strongly pseudoconvex domains in $\mathbb C^n$. For a bounded strongly pseudoconvex domain in $\mathbb C^n$ and a given boundary point of it, we prove that there is a global coordinate change on the closure of the domain which is arbitrarily close to th…

Approximation of holomorphic mappings on strongly pseudoconvex domains

Let D be a relatively compact strongly pseudoconvex domain in a Stein manifold, and let Y be a complex manifold. We prove that the set A(D,Y), consisting of all continuous maps from the closure of D to Y which are holomorphic in D, is a complex Banach manifold. When D is the unit disc in C (or any other topologically trivial strongly pseudoconvex domain in a Stein manifold), A(D,Y) is locally modeled on the Banach space A(D,C^n)=A(D)^n with n=dim Y. Analogous results hold for maps which are holomorphic in D and of class C^r up to the boundary for any positive integer r. We also establish the O…

Good Stein Neighborhood Bases and Regularity of the d-bar Neumann Problem

We show that if the closure of a smooth pseudoconvex domain admits a "sufficiently nice" Stein neighborhood basis, then the d-bar Neumann problem on the domain is globally regular.

Remarks on weakly pseudoconvex boundaries

In this paper, we consider the boundary M of a weakly pseudoconvex domain in a Stein manifold. We point out a striking difference between the local cohomology and the global cohomology of M, and illustrate this with an example. We also discuss the first and second Cousin problems, and the strong Poincare problem for CR meromorphic functions on the weakly pseudoconvex boundary M.

Smooth embeddings with Stein surface images

A simple characterization is given of open subsets of a complex surface that smoothly perturb to Stein open subsets. As applications, complex 2-space C^2 contains domains of holomorphy (Stein open subsets) that are exotic R^4's, and others homotopy equivalent to the 2-sphere but cut out by smooth, compact 3-manifolds. Pseudoconvex embeddings of Brieskorn spheres and other 3-manifolds into complex surfaces are constructed, as are pseudoconcave holomorphic fillings (with disagreeing contact and boundary orientations). Pseudoconcave complex structures on Milnor fibers are found. A byproduct of th…

The $L^2$-cohomology of a bounded smooth Stein Domain is not necessarily Hausdorff

We give an example of a pseudoconvex domain in a complex manifold whose $L^2$-Dolbeault cohomology is non-Hausdorff, yet the domain is Stein. The domain is a smoothly bounded Levi-flat domain in a two complex-dimensional compact complex manifold. The domain is biholomorphic to a product domain in $\mathbb{C}^2$, hence Stein. This implies that for $q>0$, the usual Dolbeault cohomology with respect to smooth forms vanishes in degree $(p,q)$. But the $L^2$-Cauchy-Riemann operator on the domain does not have closed range on $(2,1)$-forms and consequently its $L^2$-Dolbeault cohomology is not Hausd…

On defining functions for unbounded pseudoconvex domains

We show that every strictly pseudoconvex domain $Ω$ with smooth boundary in a complex manifold $\mathcal{M}$ admits a global defining function, i.e., a smooth plurisubharmonic function $φ\colon U \to \mathbb R$ defined on an open neighbourhood $U \subset \mathcal{M}$ of $\overlineΩ$ such that $Ω= \{φ< 0\}$, $dφ\neq 0$ on $bΩ$ and $φ$ is strictly plurisubharmonic near $bΩ$. We then introduce the notion of the core $\mathfrak{c}(Ω)$ of an arbitrary domain $Ω\subset \mathcal{M}$ as the set of all points where every smooth and bounded from above plurisubharmonic function on $Ω$ fails to be strictl…

Deformation of Einstein metrics and $L^2$ cohomology on strictly pseudoconvex domains

We construct new complete Einstein metrics on smoothly bounded strictly pseudoconvex domains in Stein manifolds. This is done by deforming the Kähler-Einstein metric of Cheng and Yau, the approach that generalizes the works of Roth and Biquard on the deformations of the complex hyperbolic metric on the unit ball. Recasting the problem into the question of vanishing of an $L^2$ cohomology and taking advantage of the asymptotic complex hyperbolicity of the Cheng-Yau metric, we establish the possibility of such a deformation when the dimension of the domain is larger than or equal to three.

具有极小光滑性的强拟凸域上Bergman投影的交换子

The Commutator of the Bergman Projection on Strongly Pseudoconvex Domains with Minimal Smoothness

本文研究了$C^2$类极小光滑性强拟凸域上Bergman投影交换子的有界性与紧性，利用平均振荡条件和二进Berezin变换等方法给出了等价刻画。

强伪凸域上Hölder-Zygmund类全纯纤维丛的Oka原理

The Oka principle for holomorphic fibre bundles of Holder-Zygmund classes on strongly pseudoconvex domains

本文证明了在Stein流形中具有光滑边界的紧强伪凸域上，若纤维是Oka流形，则每个连续截面同伦于一个在内部全纯的Hölder-Zygmund类截面，并建立了参数化h-原理，进而得到向量丛和主丛分类的Oka原理。

Bergman-Einstein metrics, hyperbolic metrics and Stein spaces with spherical boundaries

In this new version, we give an affirmative solution to a conjecture of Cheng proposed in 1979 which asserts that the Bergman metric of a smoothly bounded strongly pseudoconvex domain in $\mathbb{C}^n, n\geq 2,$ is Kähler-Einstein if and only if the domain is biholomorphic to the ball. We establish versions of various classical theorems that are used in the solution for Stein spaces. Among other things, we construct a hyperbolic metric over a Stein space with spherical boundary. We also prove the Q. K. Lu type uniformization theorem for Stein spaces with isolated normal singularities.

Strongly pseudoconvex domains as subvarieties of complex manifolds

In this paper (a sequel to B. Drinovec Drnovsek and F. Forstneric, Holomorphic curves in complex spaces, Duke Math. J. 139 (2007), 203-253) we obtain existence and approximation results for closed complex subvarieties that are normalized by strongly pseudoconvex Stein domains. Our sufficient condition for the existence of such subvarieties in a complex manifold is expressed in terms of the Morse indices and the number of positive Levi eigenvalues of an exhaustion function on the manifold. Examples show that our condition cannot be weakened in general. Optimal results are obtained for subvariet…

更高维度的蠕虫区域

Higher dimensional worm domains

本文提出构造一类边界包含给定 Stein 流形的光滑有界伪凸区域的方法，可指定余维数和 D'Angelo 类，并在此类区域下探讨 $\overline\partial$-Neumann 问题正则性理论的开放问题。

通过拓扑同位素构造 Stein 曲面

Creating Stein surfaces by topological isotopy

本文结合 Freedman 拓扑与 Eliashberg 全纯理论，证明复曲面中驯服拓扑嵌入的 2-复形经小拓扑同位素后可作为不可数多个 Stein 正则邻域的交集，并借此研究拓扑伪凸性与伪凹性。

Sobolev 与 Hölder 估计：$\mathbb C^2$ 中有限型伪凸域上的 $\overline \partial$ 方程

Sobolev and Hölder estimates for the $\overline \partial$ equation on pseudoconvex domains of finite type in $\mathbb C^2$

本文通过构造全纯支撑函数，证明了 $\mathbb C^2$ 中有限型伪凸域上 $\overline \partial$ 方程在 Sobolev 和 Hölder-Zygmund 空间中的几乎尖锐估计，扩展了先前结果。

由其内射映射确定的雷因哈特域

Reinhardt domains determined by their endomorphisms

本文研究二维伪凸雷因哈特域，证明其内射映射同构时在双全息或反双全息意义上等价，并展示所有能收缩到适当嵌入空心复线的斯特恩流形由其内射映射半群确定。

有界伪凸域与管型域上的Bergman型投影的最新进展

Recent advances in Bergman type projections in bounded pseudoconvex and tubular domains

本文为综述，汇总多复变中有界伪凸域、对称锥上管型域等场景下不同维数函数空间间Bergman型投影的最新成果，推广经典结论，提出新研究问题并展望其应用场景。

Weakly complete domains in Grauert type surfaces

The aim of this short note is to investigate the geometry of weakly complete subdomains of Grauert type surfaces, i.e. open connected sets D, sitting inside a Grauert type surface X, which admit a smooth plurisubharmonic exhaustion function. We prove that they are either modifications of Stein spaces or Grauert type surfaces themselves and we apply these results to the special case of Hopf surfaces.