Tran Vu Khanh, Ninh Van Thu
We prove here the Wolff-Denjoy-type theorem for a very large class of pseudoconvex domains in $\mathbb C^n$ that may contain many classes of pseudoconvex domains of finite type and infinite type.
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We study the boundary regularity of proper holomorphic mappings between strictly pseudoconvex domains with $C^2$-boundaries.
On Holomorphic L_{2} functions on Coverings of Strongly Pseudoconvex Manifolds
本文证明了Gromov，Henkin和Shubin [GHS]关于强伪凸流形覆盖上的全纯L_{2}函数的一个结果的推广.
On complete pseudoconvex Reinhardt domains in $\mathbb{C}^2$, we show that there is no nonzero Hankel operator with an anti-holomorphic symbol that is Hilbert-Schmidt. We also present examples of unbounded non-pseudoconvex domains that admit nonzero Hilbert-Schmidt Hankel operators with anti-holomorphic symbols.
We give a parameter version of Graham-Kerzman approximation theorem for bounded holomorphic functions on strictly pseudoconvex domains. As an application, we present some uniform estimates for the boundary behaviour of the Kobayashi and Carathéodory pseudodistences on such domains.
In this article we prove a global result in the spirit of Basener's theorem regarding the relation between q-pseudoconvexity and q-holomorphic convexity: we prove that any smoothly bounded strictly q-pseudoconvex open subset of the complex Euclidean space is (q + 1)-holomorphically convex; moreover, assuming that the given open set verifies an additional assumption, we prove that it is q-holomorphically convex. We also prove that any open subset of the complex n-dimensional Euclidean space is n-holomorphically convex.
We study the stable embedding problem for a CR family of 3-dimensional strongly pseudoconvex CR manifolds with each fiber bounding a stein manifold.
We show that there are strictly pseudoconvex, real algebraic hypersurfaces in $\bC^{n+1}$ that cannot be locally embedded into a sphere in $\bC^{N+1}$ for any $N$. In fact, we show that there are strictly pseudoconvex, real algebraic hypersurfaces in $\bC^{n+1}$ that cannot be locally embedded into any compact, strictly pseudoconvex, real algebraic hypersurface.
Let $D$ be a strictly pseudoconvex domain and $X$ be a singular analytic set of pure dimension $n-1$ in $C^n$ such that $X\cap D\neq \emptyset$ and $X\cap bD$ is transverse. We give sufficient conditions for a function holomorphic on $D\cap X$ to admit a holomorphic extension which belongs to $L^q(D),$ $q\in [1,+\infty[$, or to $BMO(D)$. The extension is given by mean of integral representation formulas and residue currents.
Application of the Complex Monge-Ampere equation to the study of proper holomorphic mappings of strictly pseudoconvex domains
对光滑有界的严格拟凸区域构造了一个特殊的多重次调和定义函数，使得复Hessian行列式在边界上消失到高阶。这种构造，结合复Monge-Ampere方程解的正则性和反射原理，使我们能够给出February man映射定理的一个新的证明.
We study regularity properties of CR maps in positive codimension valued in pseudoconvex manifolds which carry a nontrivial Levi foliation. We introduce an invariant which can be used to deduce that any sufficiently regular CR map from a minimal manifold into such a foliated target is either generically smooth or geometrically highly constrained, and to show generic smoothness of sufficiently regular CR transversal CR maps between pseudoconvex hypersurfaces. As an application, we discuss boundary regularity of proper holomorphic maps into bounded symmetric domains.
In this paper, we generalize a recent work of Liu et al. from the open unit ball $\mathbb B^n$ 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 domain…
Holomorphic L^{p}-functions on Coverings of Strongly Pseudoconvex Manifolds
在本文中，我们将讨论如何在强伪凸流形的无分支覆盖上构造全纯L^{p}-函数。同时，我们证明了这类函数的一些扩张定理和逼近定理。
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.
Holomorphic Functions of Slow Growth on Coverings of Pseudoconvex Domains in Stein Manifolds
本文应用文献[Br 1]中的方法研究Stein流形中伪凸域覆盖上的慢增长全纯函数。特别地，我们推广和加强了Gromov，Henkin和Shubin [GHS]关于伪凸流形覆盖上的全纯L^{2}函数在Stein流形覆盖情形下的某些结果.
Integral Representations of Holomorphic Functions on Coverings of Pseudoconvex Domains in Stein Manifolds
Stein流形中伪凸域上的全纯函数的经典整表示公式在多复变数的函数构造论中起着重要的作用。本文对定义在这类域的覆盖上的某些全纯函数类构造了类似的公式。
We construct a complete proper holomorphic embedding from any strictly pseudoconvex domain with $\mathcal{C}^2$-boundary in $\mathbb{C}^n$ into the unit ball of $\mathbb{C}^N$, for $N$ large enough, thereby answering a question of Alarcon and Forstneric.
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。
We use the method of pseudoanalytic continuation to obtain a characterization of spaces of holomorphic functions with boundary values in Besov spaces in terms of polynomial approximations.
Manifolds of holomorphic mappings from strongly pseudoconvex domains
设D是Stein流形S中的有界强伪凸区域，Y是复流形.证明了从D的闭包到Y的任意连续映射的图在D中是全纯的，在SxY中存在一个开Stein邻域基.利用这一点，我们表明，许多经典的空间的地图从封闭的D到Y是全纯的D是无限维的复流形的局部凸拓扑向量空间（Banach，希尔伯特或Frechet）。
Let $p:X\rightarrow Y$ be a surjective holomorphic mapping between Kähler manifolds. Let $D$ be a bounded smooth domain in $X$ such that every generic fiber $D_y:=D\cap p^{-1}(y)$ for $y\in Y$ is a strongly pseudoconvex domain in $X_y:=p^{-1}(y)$, which admits the complete Kähler-Einstein metric. This family of Kähler-Einstein metrics induces a smooth $(1,1)$-form $ρ$ on $D$. In this paper, we prove that $ρ$ is positive-definite on $D$ if $D$ is strongly pseudoconvex. We also discuss the extensioin of $ρ$ as a positive current across singular fibers.
We shall give a variational formula of the full Bergman kernels associated to a family of smoothly bounded strongly pseudoconvex domains. An equivalent criterion for the triviality of holomorphic motions of planar domains in terms of the Bergman kernel is given as an application.
If $V$ is an analytic set in a pseudoconvex domain $Ω$, we show there is always a pseudoconvex domain $G \subseteq Ω$ that contains $V$ and has the property that every bounded holomorphic function on $V$ extends to a bounded holomorphic function on $G$ with the same norm. We find such a $G$ for some particular analytic sets. When $Ω$ is an operhedron we show there is a norm on holomorphic functions on $V$ that can always be preserved by extensions to $Ω$.
On a local characterization of pseudoconvex domains
在$\Bbb C^n$中，定义域的伪凸性是通过在任意无界的边界点附近存在局部定义的多重次调和/全纯函数来描述的.
We give the parameter version of localization theorem for Bergman metric near the boundary points of strictly pseudoconvex domains. The approximation theorem for square integrable holomorphic functions on such domains in the spirit of Graham-Kerzman is proved in the hereby paper, as well.
Boundary jets of holomorphic maps between strongly pseudoconvex domains
研究了两个强伪凸区域之间的全纯映射的芽喷，条件是一个区域的象包含在另一个区域中，且给定的边界点（非切）映射到给定的边界点.我们完全表征（非切向）1-射流。此外，我们给出了代数不等式（在Chern-Moser正规形到一定的低阶）的容许芽与给定的1-jet。此外，我们证明了一个刚性的结果说，存在一个胚芽切线的身份（在某些地方的图表），当且仅当两个域是正切加权阶五。
On a bounded strictly pseudoconvex domain in $\mathbb{C}^n$, $n>1$, the smoothness of the Cheng-Yau solution to Fefferman's complex Monge-Ampere equation up to the boundary is obstructed by a local CR invariant of the boundary. For a bounded strictly pseudoconvex domain $Ω\subset \mathbb{C}^2$ diffeomorphic to the ball, we prove that the global vanishing of this obstruction implies biholomorphic equivalence to the unit ball, subject to the existence of a holomorphic vector field satisfying a mild approximate tangency condition along the boundary. In particular, by considering the Euler vector …
Proper pseudo-holomorphic maps between strictly pseudoconvex regions
证明了殆复流形中严格伪凸区域之间的真伪全纯映射延伸到边界。关键是雅可比矩阵在边界附近远离零，证明主要是基于尺度方法的几乎复杂的模拟。我们还建立了一个链接的正则性的扩展和正则性的殆复结构，我们确定明确的估计的Hölderian规范。作为推论，在光滑情形下，得到了真伪全纯映射光滑扩张的目标空间具有殆复结构的一个充要条件。
Hartogs Type Theorems for CR L^{2} functions on Coverings of Strongly Pseudoconvex Manifolds
我们证明了一个模拟的经典Hartogs扩张定理CR $L^{2}$功能定义在边界上的某些（可能是无界的）域覆盖的强伪凸流形。我们的结果是有关的一个问题中提出的文件Gromov，Henkin和Shubin [GHS]的全纯$L_{2}$功能覆盖的伪凸流形。
Approximation of holomorphic mappings on strongly pseudoconvex domains
设D是Stein流形中相对紧的强伪凸区域，Y是复流形.证明了由D的闭包到Y的全纯连续映射组成的集合A（D，Y）是复巴拿赫流形.当D是C（或Stein流形中任何其他拓扑平凡强伪凸区域）中的单位圆盘时，A（D，Y）局部模化为巴拿赫空间A（D，C^n）=A（D）^n，n= dimY.类似的结果也适用于在D中全纯的映射，并且对于任意正整数r，直到边界属于C^r类。我们还建立了D的闭包上的连续或光滑纤维丛的截面的Oka性质，这些纤维丛在D上是全纯的，并且其纤维具有凸逼近性质。本文中使用的主要分析技术是在Stein流形中的Cartan对上粘合全纯喷雾的方法，控制直到边界，这是在我们的论文“复流形中的全纯曲线“（Duke Math.J.139（2007），no.2，203- 253）中发展的。
We study proper holomorphic mappings between strictly pseudoconvex domains with low boundary regularity.
We show that if a bounded pseudoconvex domain satisfies the solvability of the bounded $\bar{\partial}$ problem, then the ideal of bounded holomorphic functions vanishing at a point in the domain is finitely generated. We also prove a smooth analog of the main result for bounded pseudoconvex domains with a sufficiently smooth boundary and also consider the Bergman space case.
A remark on Oka's lemma and a geometric property of pseudoconvex domains
该研究给出Oka引理的直接证明，探讨距离函数的次调和性相关问题，还提出拟凸域的一项新几何性质。
The purpose of this article is to prove that there exists a real smooth pseudoconvex hypersurface germ $(M,p)$ of D'Angelo infinite type in $\mathbb C^{n+1}$ such that it does not admit any (singular) holomorphic curve in $\mathbb C^{n+1}$ tangent to $M$ at $p$ to infinite order.
Let $Ω$ be a bounded pseudoconvex Hartogs domain. There exists a natural complete Kähler metric $g^Ω$ in terms of its defining function. In this paper, we study two problems. The first one is determining when $g^Ω$ is Einstein or extremal. The second one is the existence of holomorphic isometric immersions of $(Ω, g^Ω)$ into finite or infinite dimensional complex space forms.
On holomorphic Banach vector bundles over Banach spaces
我们显示，例如，在希尔伯特空间的伪凸开子集上的全纯Banach向量丛是全纯平凡的，如果它是连续平凡的。并给出了一些应用实例。
We study the compactness of composition operators on the Bergman spaces of certain bounded pseudoconvex domains in $\mathbb{C}^n$ with non-trivial analytic disks contained in the boundary. As a consequence we characterize that compactness of the composition operator with a holomorphic, continuous symbol (up to the closure) on the Bergman space of the polydisk.
A new explicit construction of Cauchy-Fantappié kernels is introduced for an arbitrary weakly pseudoconvex domain with smooth boundary. While not holomorphic in the parameter, the new kernel reflects the complex geometry and the Levi form of the boundary. Some estimates are obtained for the corresponding integral operator, which provide evidence that this kernel and related constructions give useful new tools for complex analysis on this general class of domains.
Global embeddings of weakly pseudoconvex complex spaces and refined Runge-type approximation theorems
本文通过典范奇点解消构造奇异Hermite度量，利用强开性控制乘子理想层，建立了适用于复空间的精细Runge型逼近定理，并证明了弱伪凸复空间正则点的全局全纯嵌入，进而解决了弱伪凸复流形的并集问题。
We build in a given pseudoconvex (Runge) domain $D$ of $\mathbb{C}^N$ a $\mathcal O(D)$ convex set $Γ$, every connected component of which is a holomorphically contractible (convex) compact set, enjoying the property that any continuous path $γ:[0,1)\rightarrow D$ with $\lim _{r\rightarrow 1}γ(r)\in \partial D$ and omitting $Γ$ has infinite length. This solves a problem left open in a recent paper by Alarcón and Forstnerič.
Bigness of adjoint linear subsystem and approximation theorems with ideal sheaves on weakly pseudoconvex manifolds
本文研究了弱伪凸流形上余线性子系统的大小及其与理想层相关的近似定理，推广了德马尔在复几何中的正性特征。
In this paper, we obtain optimal $L^2$ extension of holomorphic sections of a holomorphic vector bundle from subvarieties in weakly pseudoconvex Kähler manifolds. Moreover, in the case of line bundle the Hermitian metric is allowed to be singular.
This work consists of two parts. In the first part, we consider a compact connected strongly pseudoconvex CR manifold $X$ with a transversal CR $S^{1}$ action. We establish an equidistribution theorem on zeros of CR functions. The main techniques involve a uniform estimate of Szegő kernel on $X$. In the second part, we consider a general complex manifold $M$ with a strongly pseudoconvex boundary $X$. By using classical result of Boutet de Monvel-Sjöstrand about Bergman kernel asymptotics, we establish an equidistribution theorem on zeros of holomorphic functions on $\overline M$.
Extension of holomorphic maps between real hypersurfaces of different dimension
证明了从C^n中的实解析超曲面M到C^N中的严格伪凸紧致实代数超曲面M'（1 < n < N）的全纯映射的芽沿M上的任意路径全纯扩张.
Geometry of holomorphic invariant strongly pseudoconvex complex Finsler metrics on the classical domains
本文在I-IV型不可约经典域上通过对Bergman度量形变构造了一类全纯不变强伪凸复Finsler度量，证明了其完备性及与Bergman度量相似的负曲率性质。
The Levi Problem On Strongly Pseudoconvex $G$-Bundles
设$G$是幺模李群，$X$是带边界的紧致流形，$M$是主丛$G\to M\to X$的全空间，使得$M$也是强伪凸复流形.本文证明了如果$G$作用于满足局部性质的全纯变换，则$M$上的平方可积全纯函数空间是无限$G$维的.
Proper J-holomorphic discs in Stein domains of dimension 2
证明了复维数为2的殆复流形中严格伪凸Stein域上整体Bishop圆的存在性.
Let $D$ be a strictly pseudoconvex domain in $\C^N$ and $X$ a pure-dimensional non-reduced subvariety that behaves well at $\partial D$. We provide $L^p$-estimates of extensions of holomorphic functions defined on $X$.
Let $Ω$ be a bounded pseudoconvex domain in $\mathbb{C}^2$ with Lipschitz boundary or a bounded convex domain in $\mathbb{C}^n$ and $ϕ\in C(\overlineΩ)$ such that $H_ϕ$ is compact on $A^2(Ω)$. Then $ϕ\circ f$ is holomorphic for any holomorphic $f:\mathbb{D}\rightarrow bΩ$.
On the Boundary Behaviour of Invariants and Curvatures of the Kobayashi--Fuks Metric in Strictly Pseudoconvex Domains
本文利用缩放方法研究严格伪凸域上Kobayashi--Fuks度量的边界行为，考察其典范不变量、全纯截面曲率及Ricci曲率，扩展并改进了现有分析。
Proper holomorphic mappings in the special class of Reinhardt domains
给出了$\C^2$中所有伪凸Reinhardt域类中域之间的真全纯映射的一个完整刻画，其对数象为一条或一个半平面.
Holomorphic Bergman isometries between bounded domains and the extension of Lu's theorem
该研究探讨复流形间保持Bergman度量的全纯映射，得出等维情形下全纯Bergman等距映射的性质，推广Lu定理并给出相关应用。
We show that each pseudoconvex domain $Ω\subset {\mathbb C}^n$ admits a holomorphic map $F$ to ${\mathbb C}^m$ with $|F|\le C_1 e^{C_2 \hatδ^{-6}}$, where $\hatδ$ is the minimum of the boundary distance and $(1+|z|^2)^{-1/2}$, such that every boundary point is a Casorati-Weierstrass point of $F$. Based on this fact, we introduce a new anti-hyperbolic concept --- universal dominability. We also show that for each $α>6$ and each pseudoconvex domain $Ω\subset {\mathbb C}^n$, there is a holomorphic function $f$ on $Ω$ with $|f|\le C_αe^{C_α' \hatδ^{-α}}$, such that every boundary point is a Picard…
We prove that every $J$-holomorphic variational vector field can be realized as derivation $\frac{d}{dt}_{|t=0}f_t$ where $(f_t)$ is a one parametric family of $J$-holomorphic discs. Furthermore, we discuss properness of an extremal $J$-holomorphic disc in a bounded pseudoconvex domain.
Let $Ω\subset\mathbb{C}^n$ be a strictly pseudoconvex Runge domain with $C^2$-smooth defining function, $l\in\mathbb{N},$ $p\in(1,\infty).$ We prove that the holomorphic function $f$ has derivatives of order $l$ in $H^p(Ω)$ if and only if there exists a sequence on polynomials $P_n$ of degree $n$ such that $\sum\limits_{k=1}^{\infty}2^{2lk}\left\lvert f(z)-P_{2^k}(z) \right\rvert^2\in L^p(\partialΩ).$
We prove a local boundary regularity result for the complete Kahler-Einstein metrics of negative Ricci curvature near strictly pseudoconvex boundary point. We also study the asymptotic behaviour of their holomorphic bisectional curvatures near such points.
K$\ddot{\operatorname{a}}$hlerity of invariant metrics on pseudoconvex domain of dimension two
本文针对二维有界有限型拟凸域，利用三阶拟有限几何的Kähler-Kobayashi度量与Kähler-Carathéodory度量证明单值化定理，并给出Reinhardt域为单位球的Bergman度量标量倍数判据及全纯截面曲率与Lu常数的刚性定理。
A Wong-Rosay type theorem for proper holomorphic self-maps
我们证明了C^k中有界域的动力学逃逸到边界的严格伪凸点的唯一的全纯自映射是欧氏球的自同构。这是一个Wong-Rosay型结果，其映射序列的度是先验无界的。
In this paper we construct a Stein neighborhood basis for any compact subvariety $A$ with strongly pseudoconvex boundary $bA$ and Stein interior $A\backslash bA$ in a complex space $X$. This is an extension of a well known theorem of Siu. When $A$ is a complex curve, our result coincides with the result proved by Drinovec-Drnovšek and Forstnerič. We shall adapt their proof to the higher dimensional case, using also some ideas of Demailly's proof of Siu's theorem. For embedded strongly pseudoconvex domain in a complex manifold we also find a basis of tubular Stein neighborhoods. These results a…
We show that biholomorphic mappings between two bounded, pseudoconvex domains with smooth boundary extend smoothly to the boundaries of the domains, under a regularity condition on a family of twisted Bergman-like projections. This result is inspired by and generalizes a sufficient condition for extension due to Bell and Ligocka.
Embedding bordered Riemann surfaces in strongly pseudoconvex domains
本文证明每个带光滑边界的带边黎曼曲面$M$均存在到任意有界强拟凸域的真全纯映射，且该映射在$n\ge3$时可延拓为浸入，在$n\ge4$时为嵌入，同时支持在紧集上逼近及在有限点插值。
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