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.     

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.     

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．     

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.     

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.     

度量凸函数的延拓

每个度量空间都能等距嵌入到实Banach空间,所以度量凸函数可视为Banach空间子集上的函数．本文举出反例说明不是所有度量凸函数都能延拓为凸函数,并给出度量凸函数能延拓为凸函数的充分条件．     

共形平坦的(α,β)-度量

本文主要研究共形平坦的(α,β)-度量.通过共形相关的Finsler度量间其测地系数间的关系,得到了(α,β)-度量是共形平坦的充分必要条件,并构造了若干共形平坦(α,β)-度量的例子.在此基础上,发现共形平坦且具有迷向S-曲率的(α,β)-度量一定是Minkowski度量或Riemann度量.     

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.     

两类Cartan张量有界的(α,β)-度量（英文）

本文研究了两类重要的(α,β)-度量-(广义)Kropina度量和Matsumoto度量的Cartan张量.证明了这两类(α,β)-度量的Cartan张量是有界的,并由此给出了这两类度量的一些刚性结果及其应用.     

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。     

A new weighted metric: the relative metric II

In the first part of this investigation we generalized a weighted distance function of R.-C. Li's and found necessary and sufficient conditions for it being a metric. In this paper some properties of this so-called M-relative metric are established. Specifically, isometries and quasiconvexity results are derived. We also illustrate connections between our approach and generalizations of the hyperbolic metric.     

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.     

Some constructions of projectively flat Finsler metrics

In this paper, we find some solutions to a system of partial differential equations that characterize the projectively flat Finsler metrics. Further, we discover that some of these metrics actually have the zero flag curvature.     

A note on L. Zhou's result on Finsler surfaces with K = 0 and J = 0

In this note, we show that the examples of non-Berwaldian Landsberg surfaces with vanishing flag curvature, obtained in [5], are in fact Berwaldian. Consequently, Bryant's claim is still unverified.     

On a class of weak Landsberg metrics

In this paper, we discuss a class of Finsler metrics defined by a Riemannian metric and a 1-form on a manifold. We characterize weak Landsberg metrics in this class and show that there exist weak Landsberg metrics which are not Landsberg metrics in dimension greater than two.     

On Generalized Douglas–Weyl(α, β)-Metrics

In this paper, we study generalized Douglas–Weyl(α, β)-metrics. Suppose that a regular(α, β)-metric F is not of Randers type. We prove that F is a generalized Douglas–Weyl metric with vanishing S-curvature if and only if it is a Berwald metric. Moreover, by ignoring the regularity, if F is not a Berwald metric, then we find a family of almost regular Finsler metrics which is not Douglas nor Weyl. As its application, we show that generalized Douglas–Weyl square metric or Matsumoto metric with isotropic mean Berwald curvature are Berwald metrics.     

反正切Finsler度量与Randers度量射影等价(英文)

本文研究了反正切Finsler度量F=α+εβ+βarctan(β/α)与Randers度量F=α+β射影等价,这里α和α表示流形上的两个黎曼度量,β和β表示流形上的两个非零的1-形式.利用射影等价具有相同的Douglas曲率的性质,获得了这两类度量射影等价的充要条件.