Function Theory of pseudoconvex subdomains and Geometry of boundaries in Kahler manifolds

利用微分几何方法，研究了Kahler流形与伪凸子域及其边界的关系。主要工具是边界的Levi形式，并对其进行了明确的研究。结果如下。(1)在M = C^n的情况下，利用S的定义函数显式地写出C^n中到复子流形S的距离函数的Levi形式，并应用于复切向条件下Levi形式退化的条件和子流形S可展开的条件之间的关系。(2)在M = C^2的情况下，利用S的定义函数明确地写出C^2中到实子流形S的距离函数的列维形式。作为它的应用，讨论了Grauert给出的复环面中伪凸域的例子。(3)在M = C^n的情况下，利用S的定义函数，给出了C^n中到实子流形S的距离函数的列维形式，并给出了列维形式退化的充分必要条件。

The purpose of this research is to show the relation with pseudoconvex subdomains and its boundary in Kahler manifolds by using differential-geometric methods. The main tool is the Levi form to the boundary and we studied it explicitly.The results are the following.(1) In the case M = C^n , we write explicitly the Levi form of the distance function to a complex submanifold S in C^n by using the defining function of S. As its application, we found the relation with the conditoin for the Levi form to degenerate in complex tangential condition and the condition for the submanifold S to be developable.(2) In the case M = C^2, we write explicitly the Levi form of the distance function to a real submanifold S in C^2 by using the defining function of S. As its application, we discussed the example of pseudoconvex domains in complex tori showed by Grauert.(3) In the case M = C^n , we write explicitly the Levi form of the distance function to a real submanifold S in C^n by using the defining function of S, and give a necessary and sufficient condition for the Levi form to degenrrate.