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.     

RESEARCH ANNOUNCEMENTS：On the Fractional Calculus of a Type of Weierstrass Functions

I Introduction In recent years, fractals have shown important applications in many fields. [1, 2] and [3] havedone some excellent initial and conclusion work on fractal and it's mathematical foundations.However, a fractal function: a type of Weierstrass functions defined bybecause of it's special fractal properties, [1,2, 4, 5] have given some detailed discussion about it'sgraph, fractal dimension, etc.     

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.     

RESEARCH ANNOUNCEMENTS——An Asymptotic Order of Fourier Transform on SL（2,R）

Let SL(2,R)be the group of all real matrices of order2 with determinant 1.In this paper,weuse G to denote both SL(2.R) and the lie group SU(1,1)={(α/ββ/α):│α│~2-│β│~2=1,α,β=1,α,β∈C}because they are isomorphic to each other.     

第四类Caftan-Hartogs域上Bergman度量与Einstein-Kahler度量等价

In this paper,we discuss the invariaut complete metric on the Cartan-Hartogs domain of the fourth type.Firstly,we find a new invariant complete metric,and prove the equivalence between Bergman metric and the new metric;Secondly,the Ricci curvature of the new metric has the super bound and lower bound;Thirdly,we prove that the holomorphic sectional curvature of the new metric has the negative supper bound;Finally,we obtain the equivalence between Bergman metric and Einstein-Kahler metric on the Cartan-Hartogs domain of the fourth type.     

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.     

Caftan-Hartogs域上K(a)hler-Einstein度量与Bergman度量的等价

本文研究的是华罗庚域的特殊类型第二类Cartan-Hartogs域的不变Bergman度量与Kahler-Einstein度量的等价问题.引入一种与Bergman度量等价的新的完备的Kahler度量ωgλ,其Ricci曲率和全纯截取率具有负的上下界.然后应用丘成桐对Schwarz引理的推广证明ωgλ等价于Kahler-Einstein度量,从而得到了Bergman度量与Khhler-Einstein度量的等价,即丘成桐关于度量等价的猜想在第二类Cartan-Hartogs域上成立.     

The Comparison Theorem for the Bergman and Kobayashi Metrics on Cartan-Hartogs Domain of the First Type

TheRiemannmappingtheoremhaslongbeenthemaindrivingforceinthedevelopmentoftheclassicalgeometricfunctiontheory.Thelackofsuchatheoreminthehigherdimensionalspaceforcesonetolookforalternatives.Theuseofvariousbiholomorphicinvariantsseemstoaccomplishasimilargoaltoacertainextent.TheBergman,CaratheodoryandKobayashimetricsareimportantbiholomorphicinvari-ants.Theyplayveryimportantroleinstudyingtheboundarygeometryofthedomainandbiholomorphicmappingsextendingsmoothlytotheboundariesoftherelevantdomains.Ther…     

The Comparison Theorem on Cartan-Hartogs Domain of the Fourth Type

As is known to all, theory of invariant metric is very important in several complex analysis. The Bergman, Caratheodory and Kobayashi metrics are important biholomorphic invariants. They play very important role in studying the boundary geometry of the domain and biholomorphic mappings extending smoothly to the boundaries of the relevant domains.     

The Equivalence of the Complete Einstein-Kahler Metric and the Bergman Metric on YIII

In this paper we give the proof about the equivalence of the complete Einstein- Kahler metric and the Bergman metric on Cartan-Hartogs domain of the third type. And we obtain the method of getting the equivalence of two metrics.     

The Comparison Theorem for Bergman and Kobayashi Metrics on Cartan-Hartogs Domain of the Third Type

In this paper, we give the holomorphic sectional curvature under invariant Kähler metric on a Cartan-Hartogs domain of the third type Y _III (N,q,K) and construct an invariant Kähler metric, which is complete and not less than the Bergman metric, such that its holomorphic sectional curvature is bounded above by a negative constant. Hence we obtain a comparison theorem for the Bergman and Kobayashi metrics on Y _III (N,q,K).     

k＜1时Cartan-Hartogs域上的极值问题

本文讨论两种类型的极值问题,其中一种类型的极值问题可以认为是复平面上经典的Schwarz引理在高维的一个推广;另一种类型的极值是某空间上的度量,可以用来考虑域的双全纯等价分类问题．在本文中,k＜1时Cartan-Hartogs域与单位超球间的极值与极值映照被得到。