第一类超 Cartan域上的比较定理

给出了第一类超Cartan域上不变Kähler 度量下的全纯截曲率的表达式.利用其Bergman度量的完备性,构造了一个不比Bergman度量小的完备的不变Kähler度量.证明了在此Kähler度量下的全纯截曲率有一个负上界, 从而证明了第一类超Cartan域的Bergman度量和Kobayashi度量的比较定理.     

Cartan-Hartogs域经典度量的等价

研究华罗庚域的特殊类型即第1类Cartan-Hartogs域的不变完备度量.首先找到了一种新的不变完备度量, 证明它们与Bergman度量等价; 第2,证明这些新的度量的Ricci曲率具有负的上下界;第3,我们证明了新的度量的全纯截曲率有 负的上下界; 最后,通过新的完备度量作为过渡, 并利用丘成桐的Schwarz引理,证明了第1类Cartan-Hartogs域的Bergman度量和Einstein-Kähler度量是等价的,也就是说丘成桐猜想在第1类Cartan-Hartogs域上成立.对其他几类的Cartan-Hartogs也有类似的结果.     

一类Reinhardt域的全纯截曲率

本文对Reinhardt域D(k)在不变Kahler度量下的全纯截曲率的具体表达式给出详细证明.并构造了一个不变的完备的不小于Bergman 度量的D(k)的Kahler度量,使得其全纯截曲率的上界是一个负常数,从而得到域D(k)的关于Bergman 度量和 Kobayashi度量的比较定理.     

超Cartan域的Einstein-K(a^)hler度量

给出了第1类超Cartan域的Einstein-K(a^)hler度量生成函数的隐函数表达式;给出了第1类超Cartan域的全纯截曲率及其估计,并由此对K＞mn-1时的第1类超Cartan域给出了Einstein-Kaihler度量和Kobayashi度量的比较定理;对一种特殊的超Cartan域给出了其完备的Einstein-K(a^)hler度量的显表达式,这在非齐性域中还是首次得到.     

超Cartan域的Einstein-K(?)hler度量

给出了第1类超Cartan域的Einstein-Khler度量生成函数的隐函数表达式;给出了第1类超Cartan域的全纯截曲率及其估计,并由此对K>(mn-1)/(m+n)时的第1类超Cartan域给出了Einstein-Khler度量和Kobayashi度量的比较定理;对一种特殊的超Cartan域给出了其完备的Einstein-Khler度量的显表达式,这在非齐性域中还是首次得到。     

超Cartan域的Einstein-Kähler度量

给出了第1类超Cartan域的Einstein-Kähler度量生成函数的隐函数表达式; 给出了第1类超Cartan域的全纯截曲率及其估计, 并由此对K> 时的第1类超Cartan域给出了Einstein-Kähler度量和Kobayashi度量的比较定理; 对一种特殊的超Cartan域给出了其完备的Einstein-Kähler度量的显表达式, 这在非齐性域中还是首次得到.     

第三类超Cartan域的Einstein-Khler度量

设第三类超Cartan域为Y_Ⅲ,我们给出了Y_Ⅲ的Einstein-Khler度量的生成函数的隐函数表达式;给出了Y_Ⅲ的全纯截曲率及其估计,并得到Y_Ⅲ的Einstein-Khler度量和Kobayashi度量的比较定理;对Y_Ⅲ的参数K的一些特殊值,求出了其完备的Einstein-Khler度量的显表达式,此时的Y_Ⅲ一般而言是非齐性的。     

Kähler-Einstein度量和Bergman度量的等价问题

华罗庚域的特殊类型Cartan-Hartogs域YⅡ(N,p;K)当K=p/2+1/p+1时,求解了该域上的复Monge-Ampère方程的边值问题,从而得到该域的完备Kähler-Einstein度量的显表达式,并且得到此度量下的全纯截曲率的负的上下确界,最后证明了此Kähler-Einstein度量与Bergman度量等价.     

K(a)hler-Einstein度量和Bergman度量的等价问题

华罗庚域的特殊类型Cartan-Hartogs域YⅡ(N,p;K)当K=p/2+1/p+1时,求解了该域上的复Monge-Ampère方程的边值问题,从而得到该域的完备K(a)hler-Einstein度量的显表达式,并且得到此度量下的全纯截曲率的负的上下确界,最后证明了此K(a)hler-Einstein度量与Bergman度量等价.     

Khler-Einstein度量和Bergman度量的等价问题

华罗庚域的特殊类型Cartan-Hartogs域Y_Ⅱ(N,p;K)当K=p/2 1/(p 1)时,求解了该域上的复Monge-Ampère方程的边值问题,从而得到该域的完备K■hler-Einstein度量的显表达式,并且得到此度量下的全纯截曲率的负的上下确界,最后证明了此K■hler-Einstein度量与Bergman度量等价。     

Comparison theorem on Cartan-Hartogs domain of the first type

In this paper the holomorphic sectional curvature under invariant Kahler metrics on Cartan-Hartogs domain of the first type are given in explicit forms. In the meantime, we construct an invariant Kahler metric, which is not less than Bergman metric such that its holomorphic sectional curvature is bounded from above by a negative constant. Hence we obtain the comparison theorem for the Bergman metric and Kobayashi metric on Cartan-Hartogs domain of the first type.     

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.     

一类拟凸域的Bergman度量与Kobayashi度量的比较定理

本文对一类拟凸域Ｅ（ｍ，ｎ，Ｋ）给出其不变Ｋａｈｌｅｒ度量下的全纯截曲率的显表达式，并构造了Ｅ（ｍ，ｎ，Ｋ）的一个不变的完备的Ｋａｈｌｅｒ度量，使得它大于或等于Ｂｅｒｇｍａｎ度量，而且其全纯截曲率的上界是一个负常数，从而得到Ｅ（ｍ，ｎ，Ｋ）的Ｂｅｒｇｍａｎ度量和Ｋｏｂａｙａｓｈｉ度量的比较定理。     

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 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).     

Remarks on two theorems of Qi-Keng Lu

Two alternate arguments in the framework of intrinsic metrics and measures respectively of part of the proof of a famous theorem due to Qi-Keng Lu on Bergman metric with constant negative holomorphic sectional curvature are presented.A relationship between the Lu constant and the holo- morphic sectional curvature of the Bergman metric is given.Some recent progress of the Yau's porblem on the characterization of domain of holomorphy on which the Bergman metric is K(?)hler-Einstein is described.     

The equivalence on classical metrics

In this paper we study the complete invariant metrics on Cartan-Hartogs domains which are the special types of Hua domains. Firstly, we introduce a class of new complete invariant metrics on these domains, and prove that these metrics are equivalent to the Bergman metric. Secondly, the Ricci curvatures under these new metrics are bounded from above and below by the negative constants. Thirdly, we estimate the holomorphic sectional curvatures of the new metrics, and prove that the holomorphic sectional curvatures are bounded from above and below by the negative constants. Finally, by using these new metrics and Yau's Schwarz lemma we prove that the new metrics are equivalent to the Einstein-Kahler metric. That means that the Yau's conjecture is true on Cartan-Hartogs domains.     

第一类超Cartan域上的不变度量

首先证明超Cartan域Y_I(k;N;m,n)为凸域的充分必要条件是2k■m;接着讨论了在超Cartan域上四类经典的不变度量,即Bergman度量、Caratheodory度量、Kobayashi度量和Einstein-Kahler度量的等价性;最后通过计算得到了超Cartan域Y_I(1;N;2,n)和Y_I(2;N;2,n)上的Caratheodory度量(和Kobayashi度量)的显表达式.     

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.     

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.