An embedding theorem for pseudoconvex domains in Banach spaces

On the existence of holomorphic embeddings of strictly pseudoconvex algebraic hypersurfaces into spheres

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.

Complete proper holomorphic embeddings of strictly pseudoconvex domains into balls

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.

Analytic sheaves in Banach spaces

We introduce a class of analytic sheaves in a Banach space X, that we call cohesive sheaves. Cohesion is meant to generalize the notion of coherence from finite dimensional analysis. Accordingly, we prove the analog of Cartan's Theorems A and B for cohesive sheaves over pseudoconvex open subsets of X, provided X has an unconditional basis.

弱伪凸复空间的全局嵌入与精细Runge型逼近定理

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。

Equivariant embeddings of strongly pseudoconvex Cauchy-Riemann manifolds

Let $X$ be a CR manifold with transversal, proper CR $G$-action. We show that $X/G$ is a complex space such that the quotient map is a CR map. Moreover the quotient is universal, i.e. every invariant CR map into a complex manifold factorises uniquely over a holomorphic map on $X/G$. We then use this result and complex geometry to proof an embedding theorem for (non-compact) strongly pseudoconvex CR manifolds with transversal $G \rtimes S^1$-action. The methods of the proof are applied to obtain a projective embedding theorem for compact CR manifolds.

On CR embeddings of strictly pseudoconvex hypersurfaces into spheres in low dimensions

It follows from the 2004 work of the first author, X.Huang, and D. Zaitsev that any local CR embedding $f$ of a strictly psedoconvex hypersurface $M^{2n+1}\subset\bC^{n+1}$ into the sphere $\bS^{2N+1}\subset \bC^{N+1}$ is rigid, i.e.\ any other such local embedding is obtained from $f$ by composition by an automorphism of the target sphere $\bS^{2N+1}$, {\it provided} that the codimension $N-n<n/2$. In this paper, we consider the limit case $N-n=n/2$ in the simplest situation where $n=2$, i.e.\ we consider local CR embeddings $f\colon M^5\to \bS^7$. We show that there are at most two different…

Embeddability of some strongly pseudoconvex CR manifolds

We obtain an embedding theorem for compact strongly pseudoconvex CR manifolds which are bounadries of some complete Hermitian manifolds. We use this to compactify some negatively curved Kaehler manifolds with compact strongly pseudoconvex boundary. An embedding theorem for Sasakian manifolds is also derived.

Dyadic Carleson embedding and sparse domination of weighted composition operators on strictly pseudoconvex domains

In this paper, we study the behavior of the weighted composition operators acting on Bergman spaces defined on strictly pseudoconvex domains via the sparse domination technique from harmonic analysis. As a byproduct, we also prove a weighted type estimate for the weighted composition operators which is adapted to Sawyer-testing conditions. Our results extend the work by the first author, Li, Shi and Wick under a much more general setting.

On a Holomorphic Family of Stein Manifolds with Strongly Pseudoconvex Boundaries

We study the stable embedding problem for a CR family of 3-dimensional strongly pseudoconvex CR manifolds with each fiber bounding a stein manifold.

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…

带边黎曼曲面在强拟凸域中的嵌入

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。

An analytic Koszul complex in a Banach space

We show that the holomorphic ideal sheaf of a linear section of a pseudoconvex open subset $Ω$ of, say, a Hilbert space $X=\ell_2$ is acyclic. We also prove an analog of Hefer's lemma, i.e., if $f:Ω\timesΩ\to\CC$ is holomorphic and $f(x,x)=0$ for $x\inΩ$, then there is a holomorphic $g:Ω\timesΩ\to X^*$ with values in the dual space $X^*$ of $X$ such that $f(x,y)=g(x,y)(x-y)$

余线性子系统的大小与在弱伪凸流形上理想层的近似定理

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。

三维严格拟凸CR流形的形变与嵌入

Deformations and embeddings of three-dimensional strictly pseudoconvex CR manifolds

本文研究三维严格拟凸CR流形的形变与嵌入问题，证明标准CR三维球面的可嵌入形变空间在原点附近是Frechet子流形，建立修正的Cheng-Lee切片定理并用球谐函数精确刻画可嵌入形变，还构造了从无穷小可嵌入形变出发的规范可嵌入形变族。

On the stability of equivariant embedding of compact CR manifolds with circle action

We prove the stability of the equivariant embedding of compact strictly pseudoconvex CR manifolds with transversal CR circle action under circle invariant perturbations of the CR structures.

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…

Convergence of formal embeddings between real-analytic hypersurfaces in codimension one

We show that every formal embedding sending a real-analytic strongly pseudoconvex hypersurface in $M\subset \C^N$ into another such hypersurface in $M'\subset \C^{N+1}$ is convergent. More generally, if $M$ and $M'$ are merely Levi-nondegenerate, the same conclusion holds for any formal embedding provided either that the embedding is CR transversal or the target hypersurface does not contain any complex curves.

CR Embeddings, Chains, and the Fefferman Bundle

In this paper it is shown that a CR embedding from one strictly pseudoconvex hypersurface into another (of strictly larger dimension) sends chains on the source to chains on the target if and only if the embedding has a lift to a conformal isometry of the associated Fefferman bundles with vanishing pseudo-Riemannian second fundamental form. When the target is a sphere these conditions simplify nicely.

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…

Szegő kernel expansion and equivariant embedding of CR manifolds with circle action

Let $X$ be a compact strongly pseudoconvex CR manifold with a transversal CR $S^1$-action. In this paper, we establish the asymptotic expansion of Szegő kernels of positive Fourier components and by using the asymptotics, we show that $X$ can be equivariant CR embedded into some $\mathbb C^N$ equipped with a simple $S^1$-action. An equivariant embedding of quasi-regular Sasakian manifold is also derived.

全纯映射的稠密像逼近与累积极限结果

Approximation and accumulation results of holomorphic mappings with dense image

本文给出四个关于流形值全纯映射的逼近定理，分别用稠密像的全纯嵌入或映射逼近原映射，并构造收敛到稠密像映射的序列。

Flatness of CR Submanifolds in a Sphere

Let $M$ be the image of a smooth CR embedding of a strictly pseudoconvex CR real hypersurface into a sphere. If the CR second fundamental form of $M$ vanishes, we show that $M$ is a totally geodesic submanifold.

Injectivity of the quotient Bers embedding of Teichmüller spaces

The Bers embedding of theTeichmüller space is a homeomorphism into the Banach space of certain holomorphic automorphic forms. For a subspace of the universal Teichmüller space and its corresponding Banach subspace, we consider whether the Bers embedding can project down between their quotient spaces. If this is the case, it is called the quotient Bers embedding. Injectivity of the quotient Bers embedding is the main problem in this paper. Alternatively, we can describe this situation as the universal Teichmüller space having an affine foliated structure induced by this subspace. We give severa…

Szegő kernels and equivariant embedding theorems for CR manifolds

We consider a compact connected CR manifold with a transversal CR locally free $\mathbb R$-action endowed with a rigid positive CR line bundle. We prove that a certain weighted Fourier-Szegő kernel of the CR sections in the high tensor powers admits a full asymptotic expansion and we establish $\mathbb R$-equivariant Kodaira embedding theorem for CR manifolds. Using similar methods we also establish an analytic proof of an $\mathbb R$-equivariant Boutet de Monvel embedding theorem for strongly pseudoconvex CR manifolds. In particular, we obtain equivariant embedding theorems for irregular Sasa…

$G$-equivariant embedding theorems for CR manifolds of high codimension

Let $(X,T^{1,0}X)$ be a $(2n+1+d)$-dimensional compact CR manifold with codimension $d+1$, $d\geq1$, and let $G$ be a $d$-dimensional compact Lie group with CR action on $X$ and $T$ be a globally defined vector field on $X$ such that $\mathbb C TX=T^{1,0}X\oplus T^{0,1}X\oplus\mathbb C T\oplus\mathbb C\underline{\mathfrak{g}}$, where $\underline{\mathfrak{g}}$ is the space of vector fields on $X$ induced by the Lie algebra of $G$. In this work, we show that if $X$ is strongly pseudoconvex in the direction of $T$ and $n\geq 2$, then there exists a $G$-equivariant CR embedding of $X$ into $\math…

关于 C^n 中极小有界齐性域的一些新嵌入

On some new embeddings in minimal bounded homogeneous domains in $C^n$

本文在极小有界齐性域中，借助 Yamaji 的 r-格性质，建立了 Bergman 型解析空间新的精确嵌入结果，并部分推广了 Yamaji 的嵌入定理。

CR流形上Toeplitz算子的半经典谱渐近

Semi-classical spectral asymptotics of Toeplitz operators on CR manifolds

本文研究紧致严格伪凸可嵌入 CR 流形上 Toeplitz 算子的重标截断泛函演算，证明其在半经典参数趋于无穷时具有完整渐近展开，并由此推出 Kodaira 型嵌入、Tian 收敛及扰动球面嵌入等定理。

螺旋状域的研究：多项式凸性、Loewner链与稠密全纯曲线

A study of spirallike domains: polynomial convexity, Loewner chains and dense holomorphic curves

本文研究螺旋状域的多项式凸性，证明有界伪凸螺旋状域的闭包多项式凸，并给出单叶函数嵌入过滤Loewner链的充要条件，进而构造稠密全纯曲线并联系复合算子的超循环性。