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.
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 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 prove an extension of a result of Gromov, Henkin and Shubin [GHS] on holomorphic L_{2} functions on coverings of strongly pseudoconvex manifolds.
The singular and regular type of a point on a real hypersurface $\mathcal H$ in $\mathbb C^n$ are shown to agree when the regular type is strictly less than 4. If $\mathcal H$ is pseudoconvex, we show they agree when the regular type is 4. A non-pseudoconvex example is given where the regular type is 4 and the singular type is infinite.
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.
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 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 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.
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.
Given a pseudoconvex hypersurface in C^n and an arbitrary weight, we show the existence of local coordinates in which the polynomial model contains a particularly simple sum of squares of monomials. Our second main result provides a normalization of a part of any Catlin boundary system. We illustrate by an example that this normalization cannot be extended to the rest of the boundary system due to the existence of what we refer to as torsion.
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.
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.
We explicitly describe germs of strongly pseudoconvex non-spherical real-analytic hypersurfaces $M$ at the origin in $\CC^{n+1}$ for which the group of local CR-automorphisms preserving the origin has dimension $d_0(M)$ equal to either $n^2-2n+1$ with $n\ge 2$, or $n^2-2n$ with $n\ge 3$. The description is given in terms of equations defining hypersurfaces near the origin, written in the Chern-Moser normal form. These results are motivated by the classification of locally homogeneous Levi non-degenerate hypersurfaces in $\CC^3$ with $d_0(M)=1,2$ due to A. Loboda, and complement earlier joint w…
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.
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 study various classes of real hypersurfaces that are not embeddable into more special hypersurfaces in higher dimension, such as spheres, real algebraic compact strongly pseudoconvex hypersurfaces or compact pseudoconvex hypersurfaces of finite type. We conclude by stating some open problems.
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 introduce analogues of a map due to Rossi and show how they can be used to explicitly determine all covers of certain homogeneous strongly pseudoconvex 3-dimensional hypersurfaces that appear in the classification obtained by E. Cartan in 1932.
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…
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.
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.
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.
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 give an explicit verifiable characterization of weakly pseudoconvex but locally nonconvexifiable hypersurfaces of finite type in dimension two. It is expressed in terms of a generalized model, which captures local geometry of the hypersurface both in the complex tangential and nontangential directions. As an application we obtain a new class of nonconvexifiable pseudoconvex hypersurfaces with convex models.
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 classify the Lie algebras of infinitesimal CR automorphisms of weakly pseudoconvex hypersurfaces of finite multitype in $\mathbb C^N$. In particular, we prove that such manifolds admit neither nonlinear rigid automorphisms, nor real or nilpotent rotations. As a consequence, this leads to a proof of a sharp 2-jet determination result for local automorphisms. Moreover, for hypersurfaces which are not balanced, CR automorphisms are uniquely determined by their 1-jets. The same classification is derived also for special models, given by sums of squares of polynomials. In particular, in the case…
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 $\mathbb{C}^N.$
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 $M$ be a pseudoconvex, oriented, bounded and closed CR submanifold of $\mathbb{C}^{n}$ of hypersurface type. Our main result says that when a certain $1$-form on $M$ is exact on the null space of the Levi form, then the complex Green operator on $M$ satisfies Sobolev estimates. This happens in particular when $M$ admits a set of plurisubharmonic defining functions or when $M$ is strictly pseudoconvex except for the points on a simply connected complex submanifold.
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.
We shall give a variational formula of the full Bergman kernels associated to a family of smoothly bounded strongly pseudoconvex domains. An equivalent criterion for the triviality of holomorphic motions of planar domains in terms of the Bergman kernel is given as an application.
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 $Ω$.
Pseudoconvexity of a domain in $\Bbb C^n$ is described in terms of the existence of a locally defined plurisubharmonic/holomorphic function near any boundary point that is unbounded at the point.
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.
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 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.
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…
We show that if a bounded pseudoconvex domain satisfies the solvability of the bounded $\bar{\partial}$ problem, then the ideal of bounded holomorphic functions vanishing at a point in the domain is finitely generated. We also prove a smooth analog of the main result for bounded pseudoconvex domains with a sufficiently smooth boundary and also consider the Bergman space case.
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…
Let $Ω$ be a bounded pseudoconvex Hartogs domain. There exists a natural complete Kähler metric $g^Ω$ in terms of its defining function. In this paper, we study two problems. The first one is determining when $g^Ω$ is Einstein or extremal. The second one is the existence of holomorphic isometric immersions of $(Ω, g^Ω)$ into finite or infinite dimensional complex space forms.
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.
We study the compactness of composition operators on the Bergman spaces of certain bounded pseudoconvex domains in $\mathbb{C}^n$ with non-trivial analytic disks contained in the boundary. As a consequence we characterize that compactness of the composition operator with a holomorphic, continuous symbol (up to the closure) on the Bergman space of the polydisk.
In the works of Darboux and Walsh it was remarked that a one to one self mapping of $\rr^3$ which sends convex sets to convex ones is affine. It can be remarked also that a $\calc^2$-diffeomorphism $F:U\to U^{'}$ between two domains in $\cc^n$, $n\ge 2$, which sends pseudoconvex hypersurfaces to pseudoconvex ones is either holomorphic or antiholomorphic. \smallskip In this note we are interested in the self mappings of $\cc^n$ which send convex hypersurfaces to pseudoconvex ones. Their characterization is the following: {\it A $\calc^2$ - diffeomorphism $F:U'\to U$ (where $U', U\subset \cc^n$ …
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.
A new explicit construction of Cauchy-Fantappié kernels is introduced for an arbitrary weakly pseudoconvex domain with smooth boundary. While not holomorphic in the parameter, the new kernel reflects the complex geometry and the Levi form of the boundary. Some estimates are obtained for the corresponding integral operator, which provide evidence that this kernel and related constructions give useful new tools for complex analysis on this general class of 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.
