THE EXSISTENCE OF $\mu -HOLOMORPHIC$ SEPARATING FUNCTION ON BOUNDED SMOOTH DOMAINS

The Complex analysis of strongly pseudoconvex domains in C^n is rather well known.In this paper it is proved that for a bounded smoothly domain $\omega$ there is a new complex structure on it under which $\omega$ will locally become a strongly convex even though the point on b&\omega& is not a pseudoconvex point from the view of the original complex structure.Particularly if $\omega$ is a weakly pseudoconvex domain,the $\mu$ cam be ,ade siffocoemt;u c;pse tp tje progoma; cp,[;ex strictire.Therefore a lot of properties of strongly pseudoconvex domains will become true on weakly pseudoconvex domains,or general domains.For example,it is proved that there is a $\mu-holomorphic$ separatin function which is holomorphic under the new complex structure.     

A note on strong pseudoconvexity

A strongly pseudoconvex function is generalized to non-smooth settings. A complete characterization of the strongly pseudoconvex radially lower semicontinuous functions is obtained.      

Théorème de Poincaré–Alexander pour les domaines modèles

The Poincaré–Alexander Theorem states that holomorphic mappings defined on an open subset of the unit ball of ${\mathbb{C}}^n$ may, under certain conditions, be extended to a biholomorphism of the unit ball. In a complex manifold, every strongly pseudoconvex homogeneous domain is biholomorphic to the unit ball. In an almost complex manifold, the unit ball is not the only strongly pseudoconvex homogeneous domain. A strongly pseudoconvex homogeneous domain is biholomorphic to a model domain. The aim of this paper is to extend this theorem to model domains.     

Non-embeddable real algebraic hypersurfaces

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.     

Embeddability of some strongly pseudoconvex CR manifolds

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

     

Deformations of strongly pseudoconvex domains

We show that two smoothly bounded, strongly pseudoconvex domains which are diffeomorphic may be smoothly deformed into each other, with all intermediate domains being strongly pseudoconvex. This result relates to Lempert’s ideas about Kobayashi extremal discs, and also has intrinsic interest.     

拟凸域上沿某些方向的边界 Schwarz 引理*

本文直接利用全纯映照的性质研究边界Schwarz引理,建立了拟凸域上沿某些满足正定条件的方向的边界Schwarz引理. 文章推广了强拟凸域情形的主要结果,但是证明的方法是不一样的.     

On Unitary Invariant Strongly Pseudoconvex Complex Landsberg Metrics

In this paper,we prove that a unitary invariant strongly pseudoconvex complex Finsler metric is a complex Landsberg metric if and if only if it comes from a unitary invariant Hermitian metric. This implies that there does not exist unitary invariant complex Landsberg metric unless it comes from a unitary invariant Hermitian metric.     

On compactifiable strongly pseudoconvex threefolds

In contrast with the 2-dimensional case, we would like to exhibit here an example of a compactifiable strongly pseudoconvex threefold X such that a) the analytic Kodaira dimension κ(X)=3 and b) its exceptional subvariety S is projective algebraic; yet X is not quasi-projective.     

On conformal complex Finsler metrics

In this paper, we study conformal transformations in complex Finsler geometry. We first prove that two weakly Kähler Finsler metrics cannot be conformal. Moreover, we give a necessary and sufficient condition for a strongly pseudoconvex complex Finsler metric to be locally conformal weakly Kähler Finsler. Finally, we discuss conformal transformations of a strongly pseudoconvex complex Finsler metric, which preserve the geodesics, holomorphic S curvatures and mean Landsberg tensors.

     

On strongly pseudoconvex manifolds with one-dimensional exceptional set

Let X be a strongly pseudoconvex n-dimensional manifold with one-dimensional connected exceptional set S. Here we show that if S is reducible, then X is embeddable into C^N ×P ^{m for some N, m and in particular X is Kählerian with a possible exception for n = 3; we analyze this exceptional case but we do not know if it may occur. The case in which S is irreducible was previously analyzed by Tan [11], who proved that X is embeddable if either n ≠ 3 or S is not isomorphic to} P ¹. In the same paper, Tan gave an example of a non-embeddable and non-Kählerian strongly pseudoconvex three-dimensional manifold withP ¹ as an exceptional set.     

On certain non-Kahlerian strongly pseudoconvex manifolds

It has been conjectured that strongly pseudoconvex manifoldsX such that its exceptional setS is an irreducible curve can be embedded biholomorphically into some ? ^N ×P _m . In this paper we show that this is true, with one exception, namely when dim_? X = 3 and its first Chern classc ₁ (K _X ¦S) = 0 whereS ?P ₁ andK _X is the canonical bundle ofX. On the other hand, we explicitly exhibit such a 3-foldX that is not Kahlerian; also we construct non-Kahlerian strongly pseudoconvex 3-foldX whose exceptional setS is a ruled surface; those concrete examples naturally raise the possibility of classifying non-Kahlerian strongly pseudoconvex 3-folds.     

Schatten class weighted composition operators on Bergman spaces of bounded strongly pseudoconvex domains

We characterize the Schatten class weighted composition operators on Bergman spaces of bounded strongly pseudoconvex domains in terms of the Berezin transform.     

Automorphism Group of Toeplitz Algebras on Certain Pseudoconvex Domains

in this paper, we characterize the automorphism groups of Toeplitz algebras oncertain strongly pseudoconvex domains of Cn, and obtain a generalized BDF theorem for aspecial kind of essential normal operator tupies.     

Algebraicity of real analytic hypersurfaces and blowing-down

We show that a germ (M, 0) of real analytic strongly pseudoconvex hypersurface is holomorphically equivalent to a germ of real algebraic hypersurface if and only if there is a blowing-down from (M, 0) to a germ of real algebraic hypersurface.     

On the upper-semicontinuity of automorphism groups of model domains in almost complex manifolds

We will prove that the automorphism groups of the strongly pseudoconvex model domains in almost complex manifolds are isomorphically embedded into the automorphism group of the unit ball.     

On the automorphisms of Siegel domains

We prove that every holomorphic automorphism of a strongly pseudoconvex Siegel domain with no pole on the edge is an affine automorphism. Some related results are also given. This work was supported in part by a grant from the Ministry of Science and Technology of the Republic of Slovenia.     

A Direct Connection Between the Bergman and Szegő Projections

We use Stokes’s theorem to establish an explicit and concrete connection between the Bergman and Szeg? projections on the disc, the ball, and on strongly pseudoconvex domains.     

Horizontal -Laplacian on complex Finsler manifolds

A horizontal ■-Laplacian is defined on strongly pseudoconvex complex Finsler manifolds, first for functions and then for horizontal differential forms of type (p,q). The principal part of the (?)-Laplacian is computed in local coordinates. As an application, the (?)-Laplacian on strongly Kahler Finsler manifold is obtained explicitly in terms of the horizontal covariant derivatives of the Chern-Finsler conncetion.