复空间形式的全纯等距嵌入问题研究

In Kaehler geometry, the embedding problem, i.e. the existence of the holomorphic isometric embeddings form a Kaehler manifold into another classical Kaehler manifold, is a fundamental problem. Let the complex space forms be the classical Kaehler manifolds. In the case that the original manifold is homogeneous, there are abundant results. While in the other case, the problem becomes more difficult. That is because there is no united method to study such manifolds. In this project, we will study the embedding problem form pseudoconvex domain embedded Kaehler-Einstein or other classical Kaehler metrics into complex space forms in the function theory of several complex variables. Such domains can be considered as noncompact nonhomogeneous manifolds. To solve such problem can help us to study the classification problem of the submanifolds of complex space forms. The Kaehler potential function of Kaehler-Einstein metric on pseudoconvex domain will also be studied. This problem has many important applications in embedding problem, the comparison theorem of metrics and so on. In several complex variables and complex geometry, this research direction has attracted many mathematicians to study.

一个 Kaehler 流形能否全纯等距嵌入于一个典则的 Kaehler 流形是Kaehler几何学的一个基本问题。考虑复空间形式作为被嵌入的典则 Kaehler 流形。当嵌入流形是齐性空间时，研究结果比较丰富。当嵌入流形是非齐性空间时，问题变得更加复杂，很难用一套方法进行统一研究。本项目将在多复变函数论范畴内，利用函数分析和矩阵计算的技巧，研究赋予完备 Kaehler-Einstein 度量或其它典则 Kaehler度量的拟凸域到复空间形式的全纯等距嵌入问题。这类研究对象可看做一类非齐性非紧致的 Kaehler流形。该问题的解决将有助于研究复空间形式子流形的分类问题。 此外，拟凸域上的 Kaehler-Einstein 度量的 Kaehler 势函数也是本项目的研究内容。这个问题在全纯等距嵌入，度量比较理论，曲率计算等许多方面有重要应用，是多复变和复几何方面的专家学者一直关心的研究内容。

Kaehler流形到复空间形式的全纯等距嵌入映射的存在性问题是复几何里的基本研究内容。本项目的主要研究对象为赋予自然Kaehler度量的拟凸Hartogs域。主要结果如下：1、拟凸Hartogs域到三种复空间形式的全纯等距嵌入映射存在性的判别法则。2、拟凸Hartogs域上的全纯自同构群的结构。3、拟凸Hartogs域上自然Kaehler度量是完备Kaehler-Einstein度量的充要条件。4、正Hermite线丛上单位球丛上完备Kaehler-Einstein度量、极值度量的存在性.。研究结果不仅有助于复空间形式的嵌入子流形的分类，也有助于拟凸Hartogs域相关性质的刻画和分类。