Pseudoconvex Domains Research Articles

Exposing boundary points of strongly pseudoconvex subvarieties in complex spaces

We prove that all locally exposable points in a Stein compact in a complex space can be exposed along a given curve to a given real hypersurface. Moreover, the exposing map for a boundary point can be sufficiently close to the identity map outside any fixed neighborhood of the point. We also prove a parametric version of this result for bounded strongly pseudoconvex domains in $\mathbb C^n$. For a bounded strongly pseudoconvex domain in $\mathbb C^n$ and a given boundary point of it, we prove that there is a global coordinate change on the closure of the domain which is arbitrarily close to the identity map with respect to the $C^1$-norm and maps the boundary point to a strongly convex boundary point.

Geometry and Topology of the Space of Plurisubharmonic Functions

Let $$\Omega $$ be a strongly pseudoconvex domain. We introduce the Mabuchi space of strongly plurisubharmonic functions in $$\Omega $$ . We study the metric properties of this space using Mabuchi geodesics and establish regularity properties of the latter, especially in the ball. As an application, we study the existence of local Kähler–Einstein metrics.

Lp mapping properties for the Cauchy–Riemann equations on Lipschitz domains admitting subelliptic estimates

ABSTRACTWe show that on bounded Lipschitz pseudoconvex domains that admit good weight functions the -Neumann operators , and are bounded on spaces for some values of p greater than 2.

Multivariable isometries related to certain convex domains

Several interesting results exist in the literature on subnormal operator tuples having their spectral properties tied to the geometry of strictly pseudoconvex domains or to that of bounded symmetric domains in $\mathbb{C} ^n$. We introduce a class $\Omega ^{(n)}$ of convex domains in $\mathbb{C} ^n$ which, for $n \geq 2$, is distinct from the class of strictly pseudoconvex domains and the class of bounded symmetric domains and which lends itself to the application of theories related to the abstract inner function problem and the $\overline \partial $-Neumann problem, allowing us to make a number of interesting observations about certain subnormal operator tuples associated with the members of the class $\Omega ^{(n)}$.

A local weighted Axler–Zheng theorem in ℂn

The well-known Axler-Zheng theorem characterizes compactness of finite sums of finite products of Toeplitz operators on the unit disk in terms of the Berezin transform of these operators. Subsequently this theorem was generalized to other domains and appeared in different forms, including domains in $\mathbb{C}^n$ on which the $\overline{\partial}$-Neumann operator $N$ is compact. In this work we remove the assumption on $N$, and we study weighted Bergman spaces on smooth bounded pseudoconvex domains. We prove a local version of the Axler-Zheng theorem characterizing compactness of Toeplitz operators in the algebra generated by symbols continuous up to the boundary in terms of the behavior of the Berezin transform at strongly pseudoconvex points. We employ a Forelli-Rudin type inflation method to handle the weights.

Résolution du∂∂¯pour les formes différentielles ayant une valeur au bord au sens des courants dans un domaine étoilé strictement pseudoconvexe deℂn

We solve the ∂∂ ¯-problem for a form with distribution boundary value on a strictly pseudoconvex starry domain Ω of ℂ n and on ℂ n ∖Ω ¯ where Ω is a strictly pseudoconvex starry domain of ℂ n .

On the $L^2$-Dolbeault cohomology of annuli

For certain annuli in $\mathbb{C}^n$, $n\geq 2$, with non-smooth holes, we show that the $\bar{\partial}$-operator from $L^2$ functions to $L^2$ $(0,1)$-forms has closed range. The holes admitted include products of pseudoconvex domains and certain intersections of smoothly bounded pseudoconvex domains. As a consequence, we obtain estimates in the Sobolev space $W^1$ for the $\bar{\partial}$-equation on the non-smooth domains which are the holes of these annuli.

Boundary behavior of the Bergman metric

More precise estimates for the Bergman metric on strongly pseudoconvex domains are given, based on the use of the squeezing function.

The Log Term in the Bergman and Szegő Kernels in Strictly Pseudoconvex Domains in $\mathbb C^2$

In this paper, we consider bounded strictly pseudoconvex domains D\subset \mathbb C^2 with smooth boundary M=M^3:=\partial D , and the asymptotic expansion of the Bergman kernel on the diagonal K_B\sim \frac{\phi_{B}}{\rho^{n+1}}+\psi_B\log\rho, where \rho>0 is a Fefferman defining equation for D . The Ramadanov Conjecture states that if the log term \psi_B vanishes to infinite order on M , then M is locally spherical. In \mathbb C^2 , the validity of this conjecture is known and follows from work of Boutet de Monvel, Burns, and Graham; indeed, it suffices that \psi_B=O(\rho^2) locally on M to conclude that M is locally spherical. On the other hand, it is also known that the boundary values alone of the log term b\psi_B:=(\psi_B)|_M on M does not determine the CR geometry of M locally; e.g., the vanishing of b\psi_B on an open subset of M does not imply that M is locally spherical there. The main result in this paper, however, is that if D\subset \mathbb C^2 is assumed to have transverse symmetry, then the global vanishing of b\psi_B on M implies that M is locally spherical. A similar result is proved for the Szegő kernel.

A family of compact strictly pseudoconvex hypersurfaces in $\mathbb{C}^2$ without umbilical points

We prove the following: For $\epsilon>0$, let $D_\epsilon$ be the bounded strictly pseudoconvex domain in $\mathbb C^2$ given by \begin{equation*} (\log|z|)^2+(\log|w|)^2<\epsilon^2. \end{equation*} The boundary $M_\epsilon:=\partial D_\epsilon\subset \mathbb C^2$ 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.

Smoothing properties of the Friedrichs operator on Lp spaces

We show that the Friedrichs operator exhibits smoothing properties in the [Formula: see text] scale. In particular, we prove that on any smoothly bounded pseudoconvex domain the Friedrichs operator maps [Formula: see text] to [Formula: see text] for some [Formula: see text].

Asymptotic boundary behavior of the Bergman curvatures of a pseudoconvex domain

We present a method of obtaining a lower bound estimate of the curvatures of the Bergman metric without using the regularity of the kernel function on the boundary. As an application, we prove the existence of an uniform lower bound of the bisectional curvatures of the Bergman metric of a smooth bounded pseudoconvex domain near the boundary with constant Levi rank.

Conformality and Q-harmonicity in sub-Riemannian manifolds

We establish regularity of conformal maps between sub-Riemannian manifolds from regularity of Q-harmonic functions, and in particular we prove a Liouville-type theorem, i.e., 1-quasiconformal maps are smooth in all contact sub-Riemannian manifolds. Together with the recent results in [15], our work yields a new proof of the smoothness of boundary extensions of biholomorphims between strictly pseudoconvex smooth domains [29].

The $$\overline{\partial }$$ ∂ ¯ -equation on variable strictly pseudoconvex domains

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.

The Schwarz lemma at the boundary of the symmetrized bidisc

The symmetrized bidisc G2 is defined byG2:={(z1+z2,z1z2)∈C2:|z1|<1,|z2|<1,z1,z2∈C}. It is a bounded inhomogeneous pseudoconvex domain without C1 boundary, and especially the symmetrized bidisc hasn't any strongly pseudoconvex boundary point and the boundary behavior of both Carathéodory and Kobayashi metrics over the symmetrized bidisc is hard to describe precisely. In this paper, we study the boundary Schwarz lemma for holomorphic self-mappings of the symmetrized bidisc G2, and our boundary Schwarz lemma in the paper differs greatly from the earlier related results.

Schatten class Hankel and $$\overline{\partial }$$ ∂ ¯ -Neumann operators on pseudoconvex domains in $$\mathbb {C}^{n}$$ C n

Let $\Omega$ be a $C^2$-smooth bounded pseudoconvex domain in $\mathbb{C}^n$ for $n\geq 2$ and let $\varphi$ be a holomorphic function on $\Omega$ that is $C^2$-smooth on the closure of $\Omega$. We prove that if $H_{\overline{\varphi}}$ is in Schatten $p$-class for $p\leq 2n$ then $\varphi$ is a constant function. As a corollary, we show that the $\overline{\partial}$-Neumann operator on $\Omega$ is not Hilbert-Schmidt.

Asymptotics of invariant metrics in the normal direction and a new characterisation of the unit disk

We give improvements of estimates of invariant metrics in the normal direction on strictly pseudoconvex domains. Specifically we will give the second term in the expansion of the metrics. This depends on an improved localisation result and estimates in the one variable case. Finally we will give a new characterisation of the unit disk in \({\mathbb {C}}\) in terms of the asymptotic behaviour of quotients of invariant metrics.

Families of Strictly Pseudoconvex Domains and Peak Functions

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,zeta } for all G_t at every zeta in partial G_t.

On Boundary Points at Which the Squeezing Function Tends to One

J. E. Fornaess posed the question whether the boundary point of smoothly bounded pseudoconvex domain is strictly pseudoconvex, when the asymptotic limit value of the squeezing function is 1. The purpose of this paper is to give an affirmative answer if the domain is in $$\mathbb {C}^{2}$$ with smooth boundary of finite type in the sense of D’Angelo (Ann Math (2) 115(3):615–637, 1982).

On noncompactness of the [formula omitted]-Neumann problem on pseudoconvex domains in [formula omitted

We prove that a smooth bounded pseudoconvex domain Ω⊆C3 with a one-dimensional complex manifold M in its boundary has a noncompact ∂‾-Neumann operator on (0,1)-forms, under the additional assumption that bΩ has finite regular D'Angelo 2-type at a point of M.