Klas Diederich, Alexandre Sukhov
We prove that a pseudoholomorphic diffeomorphism between two almost complex manifolds with boundaries satisfying some pseudoconvexity type condition cannot map a pseudoholomorphic disc in the boundary to a single point. This can be viewed as an almost complex analogue of a well-known theorem of J. E. Fornaess.
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A unified summary is given of the existence theory of Stein manifolds in all dimensions, based on published and pending literature. Eliashberg's characterization of manifolds admitting Stein structures requires an extra delicate hypothesis in complex dimension 2, which can be eliminated by passing to the topological setting and invoking Freedman theory. The situation is quite similar if one asks which open subsets of a fixed complex manifold can be made Stein by an isotopy. As an application of these theorems, one can construct uncountably many diffeomorphism types of exotic R^4's realized as …
The density property for a Stein manifold X implies that the group of holomorphic diffeomorphisms of X is infinite-dimensional and, in a certain well-defined sense, as large as possible. We prove that if G is a complex semisimple Lie group of adjoint type and K is a reductive subgroup, then G/K has the density property. This theorem is a non-trivial extension of an earlier result of ours, which handles the case of complex semi-simple Lie groups. We also establish the density property for some other complex homogeneous spaces by ad hoc methods. Finally, we introduce a lifting method that extend…
Holomorphic approximation for Hamiltonian diffeomorphism groups via the Symplectic Density Property
该研究证明全纯灵活Stein流形的余切丛具有辛密度性质，进而推导得出实余切丛的哈密顿微分同胚可由复化余切丛的复辛自同构实现Carleman意义下的全纯逼近。
We show that the Diederich-Fornaess index of a domain in a Stein manifold is invariant under CR-diffeomorphisms. For this purpose we also improve CR-extension theorem.
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。
We study exotic smoothings of open 4-manifolds using the minimal genus function and its analog for end homology. While traditional techniques in open 4-manifold smoothing theory give no control of minimal genera, we make progress by using the adjunction inequality for Stein surfaces. Smoothings can be constructed with much more control of these genus functions than the compact setting seems to allow. As an application, we expand the range of 4-manifolds known to have exotic smoothings (up to diffeomorphism). For example, every 2-handlebody interior (possibly infinite or nonorientable) has an e…
We give a characterization of the complex Euclidean space C^n, proving a global Newlander-Nirenberg's theorem on the representability of complex structures close to the standard complex structure on C^n.
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.
The complex structure of the Teichmüller space of circle diffeomorphisms in the Zygmund smooth class
该研究针对导数属于Zygmund类的圆周微分同胚的Teichmüller空间，通过证明Schwarzian导数映射是全纯分裂浸没，为其提供了复Banach流形结构。
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.
The real analytic structure of the Teichmüller space of circle diffeomorphisms with Zygmund continuous derivatives
本文研究了具有 Zygmund 连续导数的圆 diffeomorphisms 的 Teichmüller 空间的真实解析结构，通过统一化和 Besov 空间上的 composition 操作方法，得出了该空间与 Zygmund 连续函数空间的等价性结论。
This survey on the topology of Stein manifolds is an extract from our recent joint book. It is compiled from two short lecture series given by the first author in 2012 at the Institute for Advanced Study, Princeton, and the Alfred Renyi Institute of Mathematics, Budapest.
A solvmanifold with a left-invariant complex structure whose universal cover is not Stein
构造单连通可解李群\(G\)及其左不变复结构\(J\)，使其不是斯坦因流形，为长谷川关于单连通幺模可解李群斯坦因性质的猜想提供反例。
Based on the quasiconformal theory of the universal Teichmüller space, we introduce the Teichmüller space of diffeomorphisms of the unit circle with $α$-Hölder continuous derivatives as a subspace of the universal Teichmüller space. We characterize such a diffeomorphism quantitatively in terms of the complex dilatation of its quasiconformal extension and the Schwarzian derivative given by the Bers embedding. Then, we provide a complex Banach manifold structure for it and prove that its topology coincides with the one induced by local $C^{1+α}$-topology at the base point.
The complex structure of the Teichmüller space of circle diffeomorphisms in the Zygmund smooth class II
本研究第二部分在前期建立复Banach流形结构基础上，证明预Schwarzian导数映射像为实解析圆盘丛，为小Zygmund类子空间建立对应结果并构造商空间，证明其嵌入良定义且单射以赋予复结构。
In this paper we introduce the notion of a formal complex contact structure on an odd dimensional complex manifold. Our main result is that every formal complex contact structure on a Stein manifold $X$ is homotopic to a holomorphic contact structure on a Stein domain $Ω\subset X$ which is diffeotopic to $X$. We also prove a parametric h-principle in this setting, analogous to Gromov's h-principle for contact structures on smooth open manifolds. On Stein threefolds we obtain a complete homotopy classification of formal complex contact structures. Our methods also furnish a parametric h-princip…
We study topology of leaves of 1-dimensional singular holomorphic foliations of Stein manifolds. We prove that for a generic foliation all leaves, except for at most countably many, are contractible, the rest are topological cylinders. We show that a generic foliation is complex Kupka-Smale.
Let X be a Stein manifold, A a closed complex subvariety of X, and f a continuous map from X to a complex manifold Y whose restriction to A is holomorphic. After a homotopic deformation of the Stein structure outside a neighborhood of A in X (and of its smooth structure when X is a Stein surface)we find a holomorphic map from X to Y which agrees with f on A and which is homotopic to f relative to A. The analogous results in the case when the variety A is empty have been obtained in the preprint math.CV/0507212.
We prove that every continuous map from a Stein manifold X to a complex manifold Y can be made holomorphic by a homotopic deformation of both the map and the Stein structure on X. In the absence of topological obstructions the holomorphic map may be chosen to have pointwise maximal rank. The analogous result holds for any compact Hausdorff family of maps, but it fails in general for a noncompact family. Our main results are actually proved for smooth almost complex source manifolds (X,J) with the correct handlebody structure. The paper contains another proof of Eliashberg's (Int J Math 1:29--4…
An open subset U of a complex surface can be topologically perturbed to yield an open subset whose inherited complex structure is Stein, if and only if U is homeomorphic to the interior of a handlebody whose handles all have index equal or less than 2.
We provide sufficient conditions assuring that a suitably decorated 2-polyhedron can be thickened to a compact 4-dimensional Stein domain. We also study a class of flat polyhedra in 4-manifolds and find conditions assuring that they admit Stein, compact neighborhoods. We base our calculations on Turaev's shadows suitably "smoothed"; the conditions we find are purely algebraic and combinatorial. Applying our results, we provide examples of hyperbolic 3-manifolds admitting "many" positive and negative Stein fillable contact structures, and prove a 4-dimensional analogue of Oertel's result on inc…
We study pseudonorms on pluricanonical bundles over Stein manifolds. We prove that the pseudonorms determine holomorphic structures of Stein manifolds under certain assumptions. This theorem is based on and a generalization of the result obtained by Deng, Wang, Zhang and Zhou \cite{DWZZ} for bounded domains in $\mathbb{C}^n$. We also investigate Stein morphisms and the pseudonorms on direct images of pluricanonical bundles. Our main goal in this paper is to show that the pseudonorms also determine holomorphic structures of Stein morphisms. One important technique is an $L^{2/m}$-variant of the…
Non-Affine Stein Manifolds and Normal Crossing Divisors
该研究证明存在既非仿射也非拟射影却容许正规交叉除子紧化的Stein流形，借助接触几何类比的管道微积分研究相关邻域边界，并给出邻域由边界接触结构决定的条件。
In this paper we survey results on the existence of holomorphic embeddings and immersions of Stein manifolds into complex manifolds. Most results pertain to proper maps into Stein manifolds. We include a new result saying that every continuous map $X\to Y$ between Stein manifolds is homotopic to a proper holomorphic embedding provided that $\mathrm{dim} Y > 2\mathrm{dim}\, X$ and we allow a homotopic deformation of the Stein structure on $X$.
The topology of Stein surfaces and contact 3-manifolds is studied by means of handle decompositions. A simple characterization of homeomorphism types of Stein surfaces is obtained --- they correspond to open handlebodies with all handles of index lessthan or = 2. An uncountable collection of exotic R^4's is shown to admit Stein structures. New invariants of contact 3-manifolds are produced, including a complete (and computable) set of invariants for determining the homotopy class of a 2-plane field on a 3-manifold. These invariants are applicable to Seiberg-Witten theory. Several families of o…
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