Cartan-Hartogs域上Kaihler-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Kahler-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．     

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.     

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。     

底空间为对称域乘积的Hartogs域的度量等价

研究一类非齐性Hartogs域上经典度量的等价问题.首先证明了Bergman-度量和Einstein-Khler度量在这类域上等价;其次,当域的参数满足mσ+nΤ<1时,此类域上Bergman度量,Carathéodary度量,Kobalyashi度量和Einstein-Khler度量是等价的.     

华罗庚域研究的综述

华罗庚域的创建,统一了多复变中的对称典型域和蛋型域的研究,给多复变函数论提供了一个新的研究领域．对华罗庚域的研究,至今已经取得了一系列重要成果．本文简单介绍了华罗庚域创建的历史并着重介绍了华罗庚域上的Bergman核函数和Einstein-Khler度量的显表达式的计算,以及4个经典度量(Bergman度量,Carathéodory度量,Einstein-Kahler度量, Kobayashi度量)之间的等价关系,包括这些度量与Kobayashi度量的比较定理,阐述了Bergman度量等价于Einstein-Khler度量的这一丘成桐猜想在华罗庚域的特例Cartan-Hartogs域上也成立．着重指出了获得这些结果的新的思想和方法并提出了一些尚未解决的问题,以期更多的学者对华罗庚域感到兴趣并进行更深入的研究．     

Cartan-Hartogs域的K(a)hler-Einstein度量

研究以不可约有界对称域Ω为底空间的一类Hartogs域Ω上的K(a)ler-Einstein度量,这种域称之为Cartan-Hartogs域,是华罗庚域的一种,其中K(a)ler-Einstein度量的生成函数满足一带有边界条件的复Monge-Ampère方程.一般地,域Ω是非齐性域,其上有一全纯自同构子群以及群不变轨道X∈[0,1],因此可以把复Monge-Ampère方程化为常微分方程,并且此方程在临界值μ0=μ时能够显式解出.临界值μ0对于研究其他不变度量如Bergman度量也是非常有意义的.文中还给出一个猜想,并且证明了该猜想对于两类两类例外域是成立的.     

Cartan-Hartogs域的Kähler-Einstein度量

研究以不可约有界对称域Ω为底空间的一类Hartogs域Ω上的K(a)ler-Einstein度量,这种域称之为Cartan-Hartogs域,是华罗庚域的一种,其中K(a)ler-Einstein度量的生成函数满足一带有边界条件的复Monge-Ampère方程.一般地,域Ω是非齐性域,其上有一全纯自同构子群以及群不变轨道X∈[0,1],因此可以把复Monge-Ampère方程化为常微分方程,并且此方程在临界值μ0=μ时能够显式解出.临界值μ0对于研究其他不变度量如Bergman度量也是非常有意义的.文中还给出一个猜想,并且证明了该猜想对于两类两类例外域是成立的.     

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.     

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.     

Geometric aspects of the moduli space of Riemann surfaces

We describe some recent progress in the study of moduli space of Riemann surfaces in this survey paper. New complete Kahler metrics were introduced on the moduli space and Teichmuller space. Their curvature properties and asymptotic behavior were studied in details. These natural metrics served as bridges to connect all the known canonical metrics, especially the Kahler-Einstein metric. We showed that all the known complete metrics on the moduli space are equivalent and have Poincare type growth. Furthermore, the Kahler-Einstein metric has strongly bounded geometry. This also implied that the logarithm cotangent bundle of the moduli space is stable in the sense of Mumford.     

第三类超Cartan域上的比较定理

本文给出了第三类超Cartan域上不变Kalher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Kalher度量,证明了在此Kalher度量下的全纯截曲率有一个负上界,从而证明了第三类超Cartan域的Bergman度量与Kobayashi度量的比较定理.     

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.     

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.     

Canonical Metrics on Generalized Cartan-Hartogs Domains

In this paper, the author considers a class of bounded pseudoconvex domains,i.e., the generalized Cartan-Hartogs domains Ω(μ, m). The first result is that the natural Khler metric g~(Ω(μ,m)) of Ω(μ, m) is extremal if and only if its scalar curvature is a constant. The second result is that the Bergman metric, the Ka¨hler-Einstein metric, the Carathéodary metric, and the Koboyashi metric are equivalent for Ω(μ, m).     

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。     

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.     

On the Szegő Metric

We introduce a new biholomorphically invariant metric based on Fefferman’s invariant Szeg? kernel and investigate the relation of the new metric to the Bergman and Carathéodory metrics. A key tool is a new absolutely invariant function assembled from the Szeg? and Bergman kernels.     

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.