Salomon Sambou, Souhaibou Sambou
We solve the $\partial \bar{\partial}$-problem for the differential forms of class $C^\infty$ with boundary value in currents sense defined on a contractible completely strictly pseudoconvex domain of a complex manifold.
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We solve the $\partial\bar{\partial}$-problem for extensible currents defined on a strongly pseudoconvex domain of a contractible manifold.
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…
We solve the $\partial \bar{\partial}$-problem for a form with distribution boundary value on a strongly pseudoconvex contractible domain of a complex manifold.
Locally homogeneous strictly pseudoconvex hypersurfaces in $\mathbb C^2$ were classified by E.\,Cartan in 1932. In this work, we complete the classification of locally homogeneous strictly pseudoconvex hypersurfaces in $\mathbb C^3$.
On the existence of holomorphic embeddings of strictly pseudoconvex algebraic hypersurfaces into spheres
我们证明了在$\bC^{n+1}$中存在严格的伪凸的实代数超曲面，对于任何$N$，它们不能局部嵌入到$\bC^{N+1}$中的球面中。事实上，我们表明，有严格的伪凸，真正的代数超曲面在$\bC^{n+1}$，不能局部嵌入到任何紧凑的，严格的伪凸，真正的代数超曲面。
We study the boundary regularity of proper holomorphic mappings between strictly pseudoconvex domains with $C^2$-boundaries.
We construct a class of strictly pseudoconvex domains in Cdwhose core has non-empty interior. Consequently these cores are not pluripolar. This answers a question posed by Harz, Shcherbina and Tomassini.
We prove that given a family of strictly pseudoconvex domains varying in C2 topology on domains, there exists a continuously varying family of exposing maps for all boundary points of all domains.
Boundary behaviour of the Fefferman--Szegö metric in strictly pseudoconvex domains
本文研究严格伪凸域中 Fefferman--Szegö 度量及相关不变量的边界行为。
We prove that given a family $(G_t)$ of strictly pseudoconvex domains varying in $\mathcal{C}^2$ topology on domains, there exists a continuously varying family of peak functions $h_{t,ζ}$ for all $G_t$ at every $ζ\in\partial G_t.$
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.
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.
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.
Uniformization of strictly pseudoconvex domains
证明了两个具有实解析边界的严格伪凸Stein域具有双全纯泛覆盖，只要它们的边界局部双全纯等价.这个陈述可以被看作是黎曼单值化定理的高维模拟。
We study the complex Monge-Ampère equation $(dd^c u)^n=μ$ in a strictly pseudoconvex domain $Ω$ with the boundary condition $u=φ$, where $φ\in C(\partialΩ)$. We provide a non-trivial sufficient condition for continuity of the solution $u$ outside "small sets".
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.
We obtain weighted estimates for the $\bar{\partial}$-Neumann operator on intersections of two smooth strictly pseudoconvex domains in $\mathbb{C}^2$. The regularity estimates are described with the use of Sobolev norms with weights which are powers of the defining functions of the two domains.
An extension of the estimates for the squeezing function of strictly pseudoconvex domains obtained recently by J. E. Fornæss and E. Wold in \cite{FW1} is applied to derive a sharp boundary behaviour of invariant metrics and Bergman curvatures.
The main objective of this paper is to survey some recent results on the Chern--Moser question concerning existence of umbilical points on three dimensional CR submanifolds in $\mathbb C^2$.
We investigate regularity properties of the $\overline{\partial}$-equation on domains in a complex euclidean space that depend on a parameter. Both the interior regularity and the regularity in the parameter are obtained for a continuous family of pseudoconvex domains. The boundary regularity and the regularity in the parameter are also obtained for smoothly bounded strongly pseudoconvex domains.
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 prove the following: For $ε>0$, let $D_ε$ be the bounded strictly pseudoconvex domain in $\mathbb C^2$ given by \begin{equation*} (\log|z|)^2+(\log|w|)^2<ε^2. \end{equation*} The boundary $M_ε:=\partial D_ε\subset \mathbb C^2$ is a compact strictly pseudoconvex CR manifold without umbilical points. This resolves a long-standing question in complex analysis that goes back to the work of S.-S. Chern and J. K. Moser in 1974.
In recent work by Zimmer it was proved that if $Ω\subset\mathbb C^n$ is a bounded convex domain with $C^\infty$-smooth boundary, then $Ω$ is strictly pseudoconvex provided that the squeezing function approaches one as one approaches the boundary. We show that this result fails if $Ω$ is only assumed to be $C^2$-smooth.
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。
A sphericity criterion for strictly pseudoconvex hyper surfaces in $\mathbb{C}^2$ via invariant curves
本研究针对复二维空间中严格伪凸超曲面的球面性判定问题，通过建立链与平稳圆盘边界的对应关系，证明了满足该对应条件的超曲面具有局部球面性，为复几何中相关超曲面的几何性质判定提供了新的判据。
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 give sharp and explicit upper bounds for the first positive eigenvalue $λ_1(\Box_b)$ of the Kohn-Laplacian on compact strictly pseudoconvex hypersurfaces in $\mathbb{C}^{n+1}$ in terms of their defining functions. As an application, we show that in the family of real ellipsoids, $λ_1(\Box_b)$ has a unique maximum value at the CR sphere.
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 $\sum\limits_{k=1}^{\infty}2^{2lk}\left\lvert f(z)-P_{2^k}(z) \right\rvert^2\in L^p(\partialΩ).$
Principal parts of operators in the dbar-Neumann problem on strictly pseudoconvex non-smooth domains
我们的主要目标是描述积分核的dbar-Neumann算子的某些非光滑域，所谓的Henkin-Leiterer域。我们这样做是通过明确构建一个积分核占主要映射性质的运营商。我们使用我们的计算得到$L^p$估计。
Logarithmic singularity of the Szegö kernel and a global invariant of strictly pseudoconvex domains
严格拟凸区域的Szego核在边界对角线上存在奇点，它由极点和对数型奇点组成。本文证明了对数奇性系数边界上的积分给出区域的双全纯不变量或边界的CR不变量。我们还表明，相同的不变量出现的系数的对数项的体积膨胀的域相对于Bergman体积元。
Umbilical points on three dimensional strictly pseudoconvex CR manifolds. I. Manifolds with $U(1)$-action
在紧的、三维的、严格伪凸的CR流形上，CR意义下的脐点的存在性问题在S.- S.陈省身和J. K. 1974年的莫泽。本文考虑紧致的、三维的、严格伪凸的CR流形，它具有自由的、横截于圆群$U(1)$的作用。我们证明了每一个这样的CR流形$M$至少有一个脐点轨道，{ \it提供}黎曼曲面$X:=M/U(1)$不是环面。特别地，$\mathbb C^2$中的每个紧致的、圆的和严格伪凸的超曲面至少有一圈脐点。脐点的存在的情况下，$X$是一个环面是开放的一般，但它表明，如果这样的$M$有额外的对称性，在某种意义上，那么它必须拥有脐点以及。
Exposing points on the boundary of a strictly pseudoconvex or a locally convexifiable domain of finite 1-type
证明了对于1-型$2k $的任何有界域$Ω\subset\Cp ^n$，若$Ω\subset\Cp ^n$在$p\in bΩ$上局部可凸，且具有Stein邻域基，则存在一个双全纯映射$f:\barΩ\rightarrow \Cp ^n $，使得$f(p)$是$f{(\barΩ)}$的$2k$型的全局极值点.
We derive a new homotopy formula for a strictly pseudoconvex domain of $C^2$ boundary in ${\mathbf C}^n$ by using a method of Lieb and Range and obtain estimates in Lipschitz spaces for the homotopy operators. For $r>1$ and $q>0$, we obtain a $Λ_{r+{1}/{2}}$ solution $u$ to $\overline\partial u=f$ for $\overline\partial$-closed $(0,q)$ forms $f$ of class $Λ_{r}$ on the domain. We apply the estimates to obtain boundary regularities of $\mathcal D$-solutions for a domain in the Levi-flat Euclidean space.
In this paper we prove generic results concerning Hardy spaces in one or several complex variables. More precisely, we show that the generic function in certain Hardy type spaces is totally unbounded and hence non-extentable, despite the fact that these functions have non tangential limits at the boundary of the domain. We also consider local Hardy spaces and show that generically these functions do not belong, not even locally, to Hardy spaces of higher order. We work first in the case of the unit ball of Cn where the calculations are easier and the results are somehow better, and then we ext…
Affine homogeneous strictly pseudoconvex hypersurfaces of the type (1/2,0) in $\mathbb{C}^3$
本文给出了$\mathbb{C}^3$的一类仿射齐次实超曲面，并给出了（1/2，0）型齐次曲面的一个分类.通过将上述分类问题简化为求解非线性（二次）方程组的问题，获得了结果。这样的系统的解代表了所有对应于所考虑的齐次曲面的李代数。对所得代数讨论了积分过程的一些细节。
Degenerate complex Monge-Ampère flows on strictly pseudoconvex domains
本文研究了$\mathbb C^n$中定义域上的方程$\dot{u}=\log\det (u_{α\barβ})+f(t,z,u)$.该方程与Kähler-里奇流有密切的联系。在本文中，我们考虑的情况下，边界条件是光滑的，初始条件是有界的。
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 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.
We characterise in this work the $q$-plurisubharmonic functions in terms of the theory of viscosity solutions. We show that an upper semicontinuous function is $q$-plurisubharmonic if and only if its complex Hessian has at most $q$ strictly negative eigenvalues in the viscosity sense. This characterisation is then used to prove that the supremum convolution of a (strictly) $q$-plurisubharmonic function is again (strictly) $q$-plurisubharmonic on a maybe different set of definition. Finally, we use the supremum convolution to deduce a new characterisation for the $q$-pseudoconvex subsets in $\m…
On CR embeddings of strictly pseudoconvex hypersurfaces into spheres in low dimensions
它遵循第一作者X.Huang和D. Zaitsev证明了严格拟凸超曲面$M^{2n+1}\subset\bC^{n+1}$的$f$嵌入球面$\bS^{2N+1}\subset \bC^{N+1}$的局部CR是刚性的，即任何其它这样的局部嵌入通过由目标球面$\bS^{2N+1}$的自同构（假设\it）的余维$N-n<n/2$的合成而从$f$获得。在本文中，我们考虑极限情况$N-n=n/2$在最简单的情况下，其中$n=2$，即我们考虑局部CR嵌入$f\colon M^5\to \bS^7$。我们发现，最多有两个不同的本地嵌入，直到组成的自同构$\bS^7$。我们还确定了一个子类的5维，严格pseudoconvex超曲面$M^5$在其CR曲率，使刚性保持局部CR嵌入$f\colon M^5\to \bS^7$。
$\bar\partial$ Sobolev-type inequality and an improved $L^2$-estimate of $\bar\partial$ on bounded strictly pseudoconvex domains
研究有界严格伪凸域上的∂̄算子相关Sobolev型不等式，推导带迹Sobolev不等式推广，得到全纯函数最大模原理积分形式及∂̄的Hörmander L²估计改进结果。
Complete proper holomorphic embeddings of strictly pseudoconvex domains into balls
当$N$足够大时，我们构造了从$\mathbb{C}^n$中任意具有$\mathcal{C}^2$ -边界的严格伪凸域到$\mathbb{C}^N$的单位球的完全真全纯嵌入，从而回答了Alarcon和Forstneric的一个问题.
In this paper, we consider bounded strictly pseudoconvex domains $D\subset \mathbb C^2$ with smooth boundary $M=M^3:=\partial D$. If we consider the asymptotic expansion of the Bergman kernel on the diagonal $$ K_B\sim \frac{ϕ_B}{ρ^{n+1}}+ψ_B\logρ, $$ where $ρ>0$ is a Fefferman defining equation for $D$, then it is well known that the trace of the log term $bψ_B:=(ψ_B)|_M$ on $M$ does not determine the CR geometry of $M$ locally; e.g., the vanishing of $bψ_B$ on an open subset of $M$ does not imply that $M$ is locally spherical there. Nevertheless, the main result in this paper is that if $D\s…
We study the plurisubharmonic envelopes of functions in the setting of domains in $\mathbb C^n$. In particular we prove a complex analogue of a result of De Philippis and Figalli concerning the optimal regularity of such envelopes in smooth strictly pseudoconvex domains.
Given a pseudoconvex domain U with C^1-boundary in P^n, n>2, we show that if H^{2n-2}_\dR}(U)\not=0, then there is a strictly psh function in a neighborhood of boundary U. We also solve the \dbar-equation in X=P^n\ U, for data smooth (0,1) forms on X. We also discuss Levi-flat domains in surfaces. If Z is a real algebraic hypersurface in P^2, (resp a real-analytic hypersurface with a point of strict pseudoconvexity), then there is a strictly psh function in a neighborhood of Z.
On defining functions for unbounded pseudoconvex domains
证明了复流形$\mathcal{M}$中每个具有光滑边界的严格伪凸域$Ω$都存在一个全局定义函数，即，在$\overlineΩ$的开邻域$U \subset \mathcal{M}$上定义的光滑多次调和函数$φ\colon U \to \mathbb R$，使得$Ω= \{φ< 0\}$、$dφ\neq 0$在$bΩ$上和$φ$在$bΩ$附近严格多次调和。然后，我们引入的概念的核心$\mathfrak{c}(Ω)$的任意域$Ω\subset \mathcal{M}$的所有点的集合，其中每个光滑和有界从上述pluissubharmonic函数$Ω$上不严格pluissubharmonic。如果$Ω$在$\mathcal{M}$中不是相对紧凑的，则通常$\mathfrak{c}(Ω)$是非空的，即使在$\mathcal{M}$是Stein的情况下也是如此。证明了每一个具有光滑边界的严格拟凸域$Ω\subset \mathcal{M}$都存在一个严格多重次调和的整体定义函数精确地在$\mathfrak{c}(Ω)$的补域中.然后，我们研究核心的属性。在其他结果中，我们证明了1-伪核，我们表明，在一般情况下，核心不具有解析结构，我们调查刘维尔型性能的核心。
We consider the curvature strict positivity of the direct image bundle associated to a pseudoconvex family of bounded domains. The main result is that the curvature of the direct image bundle associated to a strictly pseudoconvex family of bounded circular domains or Reinhardut domains are strictly positive in the sense of Nakano, even if the weight functions are not strictly plurisubharmonic. This result gives a new geometric insight about the property of strict pseudoconvexity, and has some applications in complex analysis and convex analysis. We investigate that the main result implies a re…
For a smooth strictly pseudoconvex hypersurface in a complex manifold, we give a necessary and sufficient condition for being CR-diffeomorphic to a real-analytic CR manifold. Our condition amounts to a holomorphic extension property for the canonically associated function expressing $2$-jets of the formal Segre varieties in terms of their $1$-jets. We also express this condition in equivalent terms for a Fefferman type determinant
Note on the paper of Fu and Wong on strictly pseudoconvex domains with Kähler--Einstein Bergman metrics
证明了Ramadanov猜想蕴涵Cheng猜想。特别是它遵循程猜想举行的二维。
On the Gromov hyperbolicity of the Kobayashi metric on strictly pseudoconvex regions in the almost complex case
我们证明了每一个有界严格$J$ -凸区域配备Kobayashi度量在Gromov意义下是双曲的。我们将这一结果应用于伪全纯映射动力学的研究。
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.
Having been unclear how to define that a domain is strictly pseudoconvex in the infinite-dimensional setting, we develop a general theory having Banach spaces in mind. We first focus on finite dimension and eliminate the need of two degrees of differentiability of the boundary of a domain, since differentiable functions are difficult to find in infinite dimension. We introduce $\ell$-strict pseudoconvexity for $\ell\geq 1$, $1$-strict pseudoconvexity at the boundary, $\ell$-uniform pseudoconvexity for $\ell\geq 0$ and finally strong pseudoconvexity. Defining $\ell$-strict pseudoconvexity and $…
Complex cobordism and embeddability of CR-manifolds
本文研究了紧的、三维的、严格伪凸的Cauchy-Riemann流形之间的复配边问题。设复配边由复2-流形X给出，X具有一个伪凸和一个伪凹端。我们回答以下问题。X的伪凸端的可嵌入性和伪凹端的可嵌入性之间的关系是什么？X的伪凸端上的CR-函数是否都扩张为X内部的全纯函数？证明了严格伪凸Cauchy-Riemann三维流形的可嵌入性不是复协边不变量。我们证明了一个新的现象：在X上的伪凸端存在CR-函数，它们不延伸到全纯函数。我们还证明了CR-函数从伪凸端的可拓性是必要的，但不足以在复配边下保持可嵌入性。
Sphericite et contractibilite des hypersurfaces strictement pseudoconvexes
本文证明了球面超曲面是唯一具有超同态收缩芽的严格伪凸超曲面。与以前的证明不同，我们的方法不使用Chern-Moser理论。它只依赖于Pinchuk扩张技术。我们也给出了这个结果的一些应用。
We construct a solution operator for $\overline{\partial}$ equation that gains $\frac{1}{2}$ derivative in the fractional Sobolev space $H^{s,p}$ on bounded strictly pseudoconvex domains in $\mathbb{C}^n$ with $C^2$ boundary, for all $1 < p < \infty$ and $ s > \frac{1}{p}$.
This is a survey on the correspondence between asymptotically complex hyperbolic Einstein metrics and CR structures on the boundary at infinity, which is the complex version of that between Poincaré-Einstein metrics and conformal structures. We mainly discuss existence theorems, and propose several open problems.
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.
In this note, we affirm the partial answer to the long open Conjecture which states that any closed embeddable strictly pseudoconvex CR $3$-manifold admits a contact form $θ$ with the vanishing CR $Q$-curvature. More precisely, we deform the contact form according to an CR analogue of $Q$%-curvature flow in a closed strictly pseudoconvex CR $3$-manifold $(M,\ J,[θ_{0}])$ of the vanishing first Chern class $c_{1}(T_{1,0}M)$. Suppose that $M$ is embeddable and the CR Paneitz operator $P_{0}$ is nonnegative with kernel consisting of the CR pluriharmonic functions. We show that the solution of CR …
Curvature formulas and curvature strict positivity of direct image bundles
本文证明严格伪凸有界域族的直接像丛曲率在Nakano意义下严格正，并给出曲率公式及Prékopa定理的对应版本。
Kähler-Einstein metrics on strictly pseudoconvex domains
S. Y. Cheng和S.- T.在复流形中的严格伪凸区域上考虑了Yau的一些性质。这样的流形带有完备的凯勒-爱因斯坦度量当且仅当它的典范丛是正的。我们考虑的限制情况下，CR结构的$\partial M$是正常的。在这种情况下，M必须是$\partial M$上的Sasaki锥的分解中的域。给出了正规CR流形上的一个条件，如果它是Kähler-Einstein流形的CR无穷远，则它不能满足这个条件。我们基本上能够确定那些正规的CR 3-流形，它们可以是CR无穷大。许多例子给出了Kähler-Einstein严格伪凸流形上的丛和决议。
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