Alexander Brudnyi
In this paper we prove an extension of a result of Gromov, Henkin and Shubin [GHS] on holomorphic L_{2} functions on coverings of strongly pseudoconvex manifolds.
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We study the boundary regularity of proper holomorphic mappings between strictly pseudoconvex domains with $C^2$-boundaries.
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.
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.
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.
We construct a special plurisubharmonic defining function for a smoothly bounded strictly pseudoconvex domain so that the determinant of the complex Hessian vanishes to high order on the boundary. This construction, coupled with regularity of solutions of complex Monge-Ampere equation and the reflection principle, enables us to give a new proof of the Fefferman mapping theorem.
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…
We prove an analog of the classical Hartogs extension theorem for CR $L^{2}$ functions defined on boundaries of certain (possibly unbounded) domains on coverings of strongly pseudoconvex manifolds. Our result is related to a problem posed in the paper of Gromov, Henkin and Shubin [GHS] on holomorphic $L_{2}$ functions on coverings of pseudoconvex manifolds.
In this paper we will show how to construct holomorphic L^{p}-functions on unbranched coverings of strongly pseudoconvex manifolds. Also, we prove some extension and approximation theorems for such functions.
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.
We apply the methods developed in [Br1] to study holomorphic functions of slow growth on coverings of pseudoconvex domains in Stein manifolds. In particular, we extend and strengthen certain results of Gromov, Henkin and Shubin [GHS] on holomorphic L^{2} functions on coverings of pseudoconvex manifolds in the case of coverings of Stein manifolds.
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 construct similar formulas for cetain classes of holomorphic functions defined on coverings of such domains.
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.
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).
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.
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 $Ω$.
This paper studies complex cobordisms between compact, three dimensional, strictly pseudoconvex Cauchy-Riemann manifolds. Suppose the complex cobordism is given by a complex 2-manifold X with one pseudoconvex and one pseudoconcave end. We answer the following questions. What is the relation between the embeddability of the pseudoconvex end and the embeddability of the pseudoconcave end of X? Do all CR-functions on the pseudoconvex end of X extend to holomorphic functions on the interior of X? We prove that embeddability of a strictly pseudoconvex Cauchy-Riemann 3-manifold is not a complex-cobo…
We study jets of germs of holomorphic maps between two strongly pseudoconvex domains under the condition that the image of one domain is contained into the other and a given boundary point is (non-tangentially) mapped to a given boundary point. We completely characterize the (non-tangential) 1-jets. Moreover we give algebraic inequalities (in terms of Chern-Moser normal forms up to a certain low order) for the admissible germs with a given 1-jet. Also, we prove a rigidity result which says that there exists a germ tangent to the Identity (in some local charts) if and only if the two domains ar…
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 …
We prove that proper pseudo-holomorphic maps between strictly pseudoconvex regions in almost complex manifolds extend to the boundary. The key point is that the Jacobian is far from zero near the boundary, and the proof is mainly based on an almost complex analogue of the scaling method. We also establish a link between the regularity of the extension and the regularity of the almost complex structures, and we determine explicit estimates for the Hölderian norms. As a corollary, we get in the smooth case a necessary and sufficient condition on the almost complex structure of the target's space…
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…
We study proper holomorphic mappings between strictly pseudoconvex domains with low boundary regularity.
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.
We show, e.g., that a holomorphic Banach vector bundle over a pseudoconvex open subset of, say, Hilbert space is holomorphically trivial if it is continuously trivial. Some applications are also given.
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 $\overline M$.
It is proved that the germ of a holomorphic map from a real analytic hypersurface M in C^n into a strictly pseudoconvex compact real algebraic hypersurface M' in C^N, 1 < n < N extends holomorphically along any path on M.
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。
Let $G$ be a unimodular Lie group, $X$ a compact manifold with boundary, and $M$ the total space of a principal bundle $G\to M\to X$ so that $M$ is also a strongly pseudoconvex complex manifold. In this work, we show that if $G$ acts by holomorphic transformations satisfying a local property, then the space of square-integrable holomorphic functions on $M$ is infinite $G$-dimensional.
We prove the existence of global Bishop discs in a strictly pseudoconvex Stein domain in an almost complex manifold of complex dimension 2.
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 $f:\mathbb{D}\rightarrow bΩ$.
A complete characterization of proper holomorphic mappings between domains from the class of all pseudoconvex Reinhardt domains in $\C^2$ with the logarithmic image equal to a strip or a half-plane is given.
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.
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。
Let $B_n$ and $P_n$ be the unit ball and the unit polydisk in $\mathbb{C}^n$ with $n\geq 2$ respectively. Denote $\mbox{Aut}(B_n)$ and $\mbox{Aut}(P_n)$ the holomorphic automorphism group of $B_n$ and $P_n$ respectively. In this paper, we prove that $B_n$ admits no $\mbox{Aut}(B_n)$-invariant strongly pseudoconvex complex Finsler metric other than a constant multiple of the Poincar$\acute{\mbox{e}}$-Bergman metric, while $P_n$ admits infinite many $\mbox{Aut}(P_n)$-invariant complete strongly convex complex Finsler metrics other than the Bergman metric. The $\mbox{Aut}(P_n)$-invariant complex …
We study the holomorphic embedding problem from a compact strongly pseudoconvex real algebraic hypersurface into a sphere of higher dimension. We construct a family of compact strongly pseudoconvex hypersurfaces $M_ε$ in $\mathbb{C}^2,$ and prove that for any integer $N$, there is a number $ε(N)$ with $0<ε(N)<1$ such that for any $ε$ with $0<ε<ε(N)$, $M_ε$ can not be locally holomorphically embedded into the unit sphere $\mathbb{S}^{2N-1}$ in $\mathbb{C}^N.$
We study on the biholomorphic equivalence of a strongly pseudoconvex bounded domain with differentiable spherical boundary to an open ball, and we study on the biholomorphicity of a proper holomorphic self mapping of a strongly pseudoconvex domain with real analytic boundary.
The Ricci curvature of the Bergman metric on a bounded domain $D\subset \mathbb{C}^n$ is strictly bounded above by $n+1$ and consequently $\log (K_D^{n+1}g_{B,D})$, where $K_D$ is the Bergman kernel for $D$ on the diagonal and $g_{B, D}$ is the Riemannian volume element of the Bergman metric on $D$, is the potential for a Kähler metric on $D$ known as the Kobayashi--Fuks metric. In this note we study the localization of this metric near holomorphic peak points and also show that this metric shares several properties with the Bergman metric on strongly pseudoconvex domains.
It is shown that if a proper holomorphic map $f: \mathbb C^n \to \mathbb C^N$, $1<n\le N$, sends a pseudoconvex real analytic hypersurface of finite type into another such hypersurface, then any $n-1$ dimensional component of the critical locus of $f$ intersects both sides of $M$. We apply this result to the problem of boundary regularity of proper holomorphic mappings between bounded domains in $\mathbb C^n$.
Hilbert-Schmidt Hankel operators with harmonic symbols on the Bergman space of strongly pseudoconvex domains in $\mathbb{C}^n$
研究C^n中光滑有界强伪凸域上Bergman空间的Hilbert-Schmidt Hankel算子，证明对于直到闭包的C^3类调和符号，H_φ是Hilbert-Schmidt当且仅当φ在域上全纯。
We show that the graph of a holomorphic motion of the unit disc cannot be biholomorphic to a strongly pseudoconvex domain in C n .
We prove a boundary version of the open mapping theorem for holomorphic maps between strongly pseudoconvex domains. That is, we prove that the local image of a holomorphic map $f:D\to D'$ close to a boundary regular contact point $p\in \de D$ where the Jacobian is bounded from zero along normal non-tangential directions has to eventually contain every cone (and more generally every admissible region) with vertex at $f(p)$.
We prove that for a strongly pseudoconvex domain $D\subset\mathbb C^n$, the infinitesimal Carathéodory metric $g_C(z,v)$ and the infinitesimal Kobayashi metric $g_K(z,v)$ coincide if $z$ is sufficiently close to $bD$ and if $v$ is sufficiently close to being tangential to $bD$. Also, we show that every two close points of $D$ sufficiently close to the boundary and whose difference is almost tangential to $bD$ can be joined by a (unique up to reparameterization) complex geodesic of $D$ which is also a holomorphic retract of $D$. The same continues to hold if $D$ is a worm domain, as long as the…
This short paper gives a constraint on Chern classes of closed strictly pseudoconvex CR manifolds (or equivalently, closed holomorphically fillable contact manifolds) of dimension at least five. We also see that our result is ''optimal'' through some examples.
We prove that every bounded strictly $J$-convex region equipped with the Kobayashi metric is hyperbolic in the sense of Gromov. We apply this result to the study of the dynamics of pseudo-holomorphic maps.
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