RESEARCH ANNOUNCEMENTS——Einstein—Kahler 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 Cnhas 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. CompleteEinstein-Kahler metric with Explicit form, however, is only known in the case of homogeneousdomain.     

The computations of Einstein-Khler metric of Cartan-Hartogs domain

In this paper, we compute the complete Einstein-Kahler metric with explicit formula for the Cartan-Hartogs domain of the fourth type in some cases. Under this metric the holomorphic sectional curvature is given, which intervenes between -2k and -1. This is the sharp estimate.     

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.     

The Einstein-Kähler metric on the third Cartan-Hartogs domain

In this paper we discuss the Einstein-Kahler metric on the third Cartan-Hartogs domain Y111（n, q; K）. Firstly we get the complete Einstein Kahler metric with explicit form on Y111（n, q; K） in the case of K=q/2 ＋ 1/q-1. Secondly we obtain the holomorphic sectional curvature under this metric and get the sharp estimate for this holomorphic curvature. Finally we prove that the complete Einstein-Kahler metric is equivalent to the Bergman metric on Y111（n, q; K） in case of K=q/2＋1/q-1.     

The Einstein-Kahler metrics on Cartan-Hartogs domain of the first type

Let YI be the Cartan-Hartogs domain of the first type. We give the generating function of the Einstein-Kahler metrics on YI, the holomorphic sectional curvature of the invariant Einstein-Kahler metrics on YI. The comparison theorem of complete Einstein-Kahler metric and Kobayashi metric on YI is provided for some cases. For the non-homogeneous domain YI, when K =mn+1/m+n,m>1, the explicit forms of the complete Einstein-Kahler metrics are obtained.     

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.     

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.     

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].     

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.     

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     

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.     

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     

A Characterization of Complete Bounded Domain

1 IntroductionThis paper is concerned with biholomorphic mappings between two bounded domains D andG both in C". Consequently, an important question is whether the domain D is biholomorphicto G? We give an answer for this question under a very weak condition.Let D be a bounded domain and Bn the unit ball in Cn. Let T(D) be the holomorphictangent bundle of D. We will identify T(D) with D × Cn. Let H(D1, D2) be the family ofholomorphic mappings from D1 to D2. We introduce the followin…     

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…     

Bounded Plurisubharmonic Exhaustion Functions and Levi-flat Hypersurfaces

In this paper, we survey some recent results on the existence of bounded plurisubharmonic functions on pseudoconvex domains, the Diederich–Forn?ss exponent and its relations with existence of domains with Levi-flat boundary in complex manifolds.     

Hypersurface With Constant Mean Curvature in Hyperbolic Space Form

In [1], S. T. Yau proved the following theorems: (1) If M is compact hyper-surface with constant mean curvature and non-negative Ricci curvature in the Eucli-dean space, then M is umbilical. (2) If M is compact hypersurface with constantscalar curvature in hyperbolic space form and M has positive sectional curvature, thenM is totally umbilical. In this paper, we shall generalize the theorems as follows     

Einstein-Kahler Metric with Explicit Formula on Super-Cartan Domain of the Fourth Type

Let YIV be the Super-Cartan domain of the fourth type, We reduce the Monge-Ampere equation for the metric to an ordinary differential equation in the auxiliary function X = X（Z, W）. This differential equation can be solved to give an implicit function in X. We give the generating function of the Einstein Kahler metric on YIV. We obtain the explicit form of the complete Einstein-Kahler metric on YIV for a special case.     

THE PARAMETRIC REPRESENTATION FOR SPIRAL-LIKE MAPPINGS OF TYPE α ON BOUNDED BALANCED PSEUDOCONVEX DOMAINS

LetΩbe a bounded balanced pseudoconvex domain in Cn that admits a plurisubhamonic defining function of class C2. The authors prove that if f is a spiral-like mapping of typeαonΩ, then f can be expressed as a limiting form concerning a Schwarz mapping. Moreover, the characteristics for star-like mappings and spiral-like mappings of typeαare obtained. As a direct application, the growth theorem for spiral-like mappings of typeαis set up.     

The eigenvalue problem for the Laplacian equations

This article studies the Dirichlet eigenvalue problem for the Laplacian equations △u = -λu, x ∈Ω, u = 0, x ∈ (δ)Ω, where Ω (∩) Rn is a smooth bounded convex domain. By using the method of appropriate barrier function combined with the maximum principle, authors obtain a sharp lower bound of the difference of the first two eigenvalues for the Dirichlet eigenvalue problem. This study improves the result of S.T.Yau et al.     

一类有界Reinhardt域上的函数论

In this paper, we consider a class of bounded Reinhardt domains Dα(m, n1,…,nm). The Bergman kernel function K(z,z), the Bergman metric matrix T(z,z), the Cauchy-Szego kernel function S(z,ζ) are obtained. Then we prove that the formal Poisson kernel function is not a Poisson kernel function. At last, we prove that Dαis a quasiconvex domain and Dαis a stronger quasiconvex domain if and only if Dαis a hypersphere.