RESEARCH ANNOUNCEMENTS Einstein-Khler Metric on Cartan-Hartogs Domain of the First Type

S. Y. Cheng and S. T. Yau showed in [CY] that any C2 bounded pseudoconvex domain in C?has a complete Einstein-Kahler metric with constant negative Ricci curvature. N. Mok and S. T. Yau[MY] have extended this result to arbitrary bounded pseudoconvex domain in Cn. Complete Einstein-Kahler metric with Explicit form, however, is only known in the case of homogeneous domain.     

On a Generalized Dirichlet Problem for Analytic Family

In this paper, we give the Silov boundary for an analytic family on a bounded strictly pseudoconvex domain or an analytic polyhedron in Cn, and get a necessary and sufficient condition for a generalized Dirichlet problem to be solvable for an analytic family on a bounded holomorphic domain. Especially, we derive that this condition is just that the continuous real boundary value is prescribed on and only on the Silov boundary for an analytic family on a bounded strictly pseudoconvex domain or an analytic polyhedron.     

具逐片C^1边界条件开集上的C—R方程的积分解算子

The methods of Niles ψ vrelid which he used in strictly pseudoconvex domains are cosulted,in this paper,and not only for imbomogeneous cauchy-Riemann equaitons on some cases that for bounded open sets with C^-boundary,bounded open sets with piecewise C^1-boundary,and peudoconvex domains with C^1-boundary are discussed,but also their linear integral operators solutions and elementary estimated are corresponding given.     

调和Bergman度量

We have constructed the positive definite metric matrixes for the bounded domains of Rn and proved an inequality which is about the Jacobi matrix of a harmonic mapping on a bounded domain of Rn and the metric matrix of the same bounded domain.     

Singular integral on bounded strictly pseudoconvex domain

Kytmanov and Myslivets gave a special Cauchy principal value of the singular integral on the bounded strictly pseudoconvex domain with smooth boundary. By means of this Cauchy integral principal value, the corresponding singular integral and a composition formula are obtained. This composition formula is quite different from usual ones in form. As an application, the corresponding singular integral equation and the system of singular integral equations are discussed as well.     

局部对称Lorentz空间中的具有常数量曲率的完备类空超曲面

The complete space-like hypersurfaces with constant normal saclar curvature is discussed in a locally symmetric Lorentz space. A classified theorem is obtained by the operator L_1 introduced by S Y Cheng and S T Yau [3].     

本刊英文版2015年58卷第1期摘要(英文)

<正>On the lower bounds of the curvatures in a bounded domain LU Qi Keng Abstract Let KD(z,z)be the Bergman kernel of a bounded domain D in Cnand Sect D(z,ξ)and Ricci D(z,ξ)be the holomorphic sectional curvature and Ricci curvature of the Bergman metric ds2=TD(z,z)dzαdzβrespectivelyαβat the point z∈D with tangent vectorξ.It is proved by constructing suitable minimal functions that     

Submanifolds with Parallel Mean Curvature Vector

In a previous paper[1],S.T.Yau proved the following theorem:Theorem A Let M~n be an n-dimensional compact submanifold with parallel meancurvature in S~n p with p>1.If(3 n~(1/2)-1/p-1)S≤n,then M~n lies in a totally geodesicS~n 1.Lemma1[2]If a given set of n 1(n≥2)real numbers a_1,…,a_n and k satisfy thein(?)ality     

Proper Holomorphic Maps from Domains in C2 with Transverse Circle Action

The authors consider proper holomorphic mappings between smoothly bounded pseudoconvex regions in complex 2-space,where the domain is of finite type and admits a transverse circle action.The main result is that the closure of each irreducible component of the branch locus of such a map intersects the boundary of the domain in the union of finitely many orbits of the group action.     

SEMIDISCRETIZATION IN SPACE OF NONLINEAR DEGENERATE MRABOLIC EQUATIONS WITH BLOW-UP OF THE SOLUTIONS

1. IntroductionLet n be a bounded domain in AN with smooth boundary Off. We consider thefollowing initial boundary value problem:where 6, p are positive constants and "o(x) is a nonnegative bounded continuous function on fi.When N = 1 and 5 ~ 2, the problem arises in a model for the resistive diffusion of aforce--free magnetic field in a plasma confined between two walls in one dimension (see[5], [8], [9], [10] and [14]). Equation (1) also describes the evolution of the curvatureof a locally…     

On Bergman completeness and Bergman stability

It is proved that any bounded pseudoconvex domain in is complete w.r.t. the Bergman metric if its boundary can be described locally as the graph of a continuous function in suitable coordinates for . Further arguments are given concerning the stability problems of the Bergman kernel on non-smooth pseudoconvex domains. Received November 30, 1998 / Revised December 21, 1999 / Published online September 5, 2000     

Nontangential Weighted Limit of the Infinitesimal Carathéodory Metric in an h-Extendible Boundary Point of a Smooth Bounded Pseudoconvex Domain in Cn

A boundary point of a domain in Cⁿ is said to be h-extendible if its Catlin's multitype coincides with its D'Angelo's type. The main purpose of this paper is to study the existence of nontangential weighted limit of the infinitesimal Carathéodory metric in such a point of a smooth bounded pseudoconvex domain in Cⁿ.     

On the existence of complete Kähler metrics of negative Riemannian curvature bounded away from zero on ellipsoidal domains inC n

Summary The purpose of this paper is to prove that every ellipsoidal domain in Cⁿ admits a complete Kähler metric whose Riemannian sectional curvature is bounded from above by a negative constant (Theorem 1). We construct a Kähler metric, in a natural way, as potential of a suitable function defining the boundary (§2). Directly we compute the curvature tensor and we find upper and lower bounds for the holomorphic sectional curvature (§ 4, § 5). In order to prove the boundness of Riemannian sectional curvature we use finally a classical pinching argument (§ 6). We also obtain that for certain ellipsoidal domains the curvature tensor is very strongly negative in the sense of [15] (§ 3). Finally we prove that the metric constructed on ellipsoidal domains in Cⁿ is the Bergman metric if and only if the domain is biholomorphic to the ball (Theorem 2). In [8], [9] R. E. Greene and S. G. Krantz gave large families of examples of complete Kähler manifolds with Riemannian sectional curvature bounded from above by a negative constant; they are sufficiently small deformations of the ball in Cⁿ, with the Bergman metric. Before the only known example of complete simply-connected Kähler manifold with Riemannian sectional curvature upper bounded by a negative constant, not biholomorphic to the ball, was the surface constructed by G. D. Mostow and Y. T. Siu in [14], to the best of the author's knowledge, is not known at present if this example is biholomorphic to a domain in Cⁿ.     

The Wu metric is not upper semicontinuous

We give an example of a bounded pseudoconvex domain in , the Wu metric of which (associated to the Kobayashi-Royden or the Azukawa metric) is not upper semicontinuous.

     

Continuity and boundary behavior of the Carathéodory metric

The boundary behavior of the higher-order Carathéodory metrics, the singular Carathéodory metric, and the Azukawa metric near anh-extendable boundary point of a bounded smooth pseudoconvex domain in ℂⁿ are studied. Translated fromMathematicheskie Zametski, Vol. 67, No. 2, pp. 230–240, February, 2000.     

Pseudoconvexity and Gromov hyperbolicity

We give an estimate for the distance functions related to the Bergman, Carathéodory, and Kobayashi metrics on a bounded strictly pseudoconvex domain with C²-smooth boundary. Our formula relates the distance function on the domain with the Carnot- Carathéodory metric on the boundary. As a corollary we conclude that the domain equipped with the any of the standard invariant distances is hyperbolic in the sense of Gromov. When the boundary of the domain is C³-smooth, our estimate is exact up to a fixed additive term.     

复Finsler流形上的Koppelman-Leray-Norguet公式

利用不变积分核(Berndtsson核),复Finsler度量和联系于Chern-Finsler联络的非线性联络,研究复Finsler流形上具有逐块光滑C~((1))边界的有界域上(p,q)型微分形式的积分表示,得到了(p,q)型微分形式的Koppelman-Leray-Norguet公式和■-方程的解．作为应用,利用复Finsler度量和联系于Chern-Finsler联络的非线性联络,给出了Stein流形上具有逐块光滑C~((1))边界的有界域上(p,q)型微分形式的Koppelman- Leray-Norguet公式以及■-方程的解,并且得到了Stein流形上实非退化强拟凸多面体上(p,q)型微分形式的积分表示式和■-方程的解．     

On hyperbolicity of pseudoconvex Reinhardt domains

We give a characterization of Kobayashi hyperbolicity of pseudoconvex Reinhardt domains. All such domains turn out to be biholomorphic to a bounded Reinhardt domain. In particular, any Kobayashi hyperbolic pseudoconvex Reinhardt domain is Kobayashi complete.