Stefan Nemirovski, Rasul Shafikov
It is shown that two strictly pseudoconvex Stein domains with real analytic boundaries have biholomorphic universal coverings provided that their boundaries are locally biholomorphically equivalent. This statement can be regarded as a higher dimensional analogue of the Riemann uniformization theorem.
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It is shown that the unit ball in ${\mathbb C}^n$ is the only complex manifold that can universally cover both Stein and non-Stein strictly pseudoconvex domains.
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 $\mathbb C^3$.
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 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 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.
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 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.
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.
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。
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.
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 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 $\sum\limits_{k=1}^{\infty}2^{2lk}\left\lvert f(z)-P_{2^k}(z) \right\rvert^2\in L^p(\partialΩ).$
Our main goal is to describe the integral kernel of the dbar-Neumann operator on certain non-smooth domains, the so-called Henkin-Leiterer domains. We do so by explicitly constructing an integral kernel which accounts for the main mapping properties of the operator. We use our calculations to derive $L^p$ estimates.
The Szego kernel of a strictly pseudoconvex domain admits a singularity on the boundary diagonal, which consists of a pole and logarithmic type singularity. In this paper, we prove that the integral over the boundary of the coefficient of the logarithmic singularity gives a biholomorphic invariant of a domain, or a CR invariant of the boundary. We also show that the same invariant appears as the coefficient of the logarithmic term of the volume expansion of the domain with respect to the Bergman volume element.
The question of existence of umbilical points, in the CR sense, on compact, three dimensional, strictly pseudoconvex CR manifolds was raised in the seminal paper by S.-S. Chern and J. K. Moser in 1974. In the present paper, we consider compact, three dimensional, strictly pseudoconvex CR manifolds that possess a free, transverse action by the circle group $U(1)$. We show that every such CR manifold $M$ has at least one orbit of umbilical points, {\it provided} that the Riemann surface $X:=M/U(1)$ is not a torus. In particular, every compact, circular and strictly pseudoconvex hypersurface in $…
We show that for any bounded domain $Ω\subset\Cp ^n$ of 1-type $2k $ which is locally convexifiable at $p\in bΩ$, having a Stein neighborhood basis, there is a biholomorphic map $f:\barΩ\rightarrow \Cp ^n $ such that $f(p)$ is a global extreme point of type $2k$ for $f{(\barΩ)}$.
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…
In this paper we denote a type of affine homogeneous real hypersurface of $\mathbb{C}^3$ and present a classification of homogeneous surfaces of the type (1/2,0). The result was obtained by reducing the classification problem mentioned above to the problem of solving a system of nonlinear (quadratic) equations. Solutions of such system represent all the Lie algebras corresponding to the homogeneous surfaces under consideration. Some details of integration procedure are discussed for obtained algebras.
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 study the equation $\dot{u}=\log\det (u_{α\barβ})+f(t,z,u)$ in domains of $\mathbb C^n$. This equation has a close connection with the Kähler-Ricci flow. In this paper, we consider the case of the boundary conditions are smooth and the initial conditions are bounded.
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 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…
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…
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.
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.
$\bar\partial$ 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。
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…
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.
We show that every strictly pseudoconvex domain $Ω$ with smooth boundary in a complex manifold $\mathcal{M}$ admits a global defining function, i.e., a smooth plurisubharmonic function $φ\colon U \to \mathbb R$ defined on an open neighbourhood $U \subset \mathcal{M}$ of $\overlineΩ$ such that $Ω= \{φ< 0\}$, $dφ\neq 0$ on $bΩ$ and $φ$ is strictly plurisubharmonic near $bΩ$. We then introduce the notion of the core $\mathfrak{c}(Ω)$ of an arbitrary domain $Ω\subset \mathcal{M}$ as the set of all points where every smooth and bounded from above plurisubharmonic function on $Ω$ fails to be strictl…
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
It is shown that the Ramadanov conjecture implies the Cheng conjecture. In particular it follows that the Cheng conjecture holds in dimension two.
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…
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.
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 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}$.
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 $…
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.
We propose a proof of the characterization of spherical hypersurfaces as the only strictly pseudoconvex hypersurfaces which have contracting germs of diffeomorphisms. Unlike previous proofs, our approach does not use Chern-Moser theory. It only relies on Pinchuk dilation techniques. We also give some applications of this result.
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 …
The metrics of S. Y. Cheng and S.-T. Yau are considered on a strictly pseudoconvex domains in a complex manifold. Such a manifold carries a complete Kähler-Einstein metric if and only if its canonical bundle is positive. We consider the restricted case in which the CR structure on $\partial M$ is normal. In this case M must be a domain in a resolution of the Sasaki cone over $\partial M$. We give a condition on a normal CR manifold which it cannot satisfy if it is a CR infinity of a Kähler-Einstein manifold. We are able to mostly determine those normal CR 3-manifolds which can be CR infinities…
We construct new complete Einstein metrics on smoothly bounded strictly pseudoconvex domains in Stein manifolds. This is done by deforming the Kähler-Einstein metric of Cheng and Yau, the approach that generalizes the works of Roth and Biquard on the deformations of the complex hyperbolic metric on the unit ball. Recasting the problem into the question of vanishing of an $L^2$ cohomology and taking advantage of the asymptotic complex hyperbolicity of the Cheng-Yau metric, we establish the possibility of such a deformation when the dimension of the domain is larger than or equal to three.
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。
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 curvature invariant of the boundary. For bounded strictly pseudoconvex domains in $\mathbb{C}^2$ which are diffeomorphic to the ball, we motivate and consider the problem of determining whether the global vanishing of this obstruction implies biholomorphic equivalence to the unit ball. In particular we observe that, up to biholomorphism, the unit ball in $\mathbb{C}^2$ is rigid with respect to defor…
Let $D=\{ρ< 0\}$ be a smooth relatively compact domain in an almost complex manifold $(M,J)$, where $ρ$ is a smooth defining function of $D$, strictly $J$-plurisubharmonic in a neighborhood of the closure $\overline{D}$ of $D$. We prove that $D$ has a connected boundary and is Gromov hyperbolic.
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