共形紧致流形与分裂定理

Conformally Compact Manifold and Splitting Type Theorem

通过对给定共形紧致流形上的L2调和1形式空间的研究,确定了共形紧致流形的结构.利用Wang的方法以及流形的曲率和第一特征值条件可知,流形上不存在非平凡的L2调和1形式,或者流形上成立一些微分方程.通过解这些微分方程可以证明给定的流形分裂成一个欧氏空间和一个曲率有下界全测地子流形的乘积,并且流形上的度量能够被显式表达.对于一般的完备流形,如果对其上的L2调和1形式的增长做一定限制,类似的结果也成立.

The main purpose of our paper is to understand the structure of conformally compact manifolds by studying the space of L2 harmonic 1-forms on it. First following the Wang’s method and using the condition for curvature and the first eigenvalue, we know that either there does not exist any nontrivial L2 harmonic 1-form or some differential equations hold on these conformally compact manifolds. By solving these equations, we can prove that the given manifolds can be written as the product of a Euclidean space and a totally geodesic submanifolds whose curvature has lower bound, and the metrics of the manifolds can be expressed explicitly. For the generalized complete manifolds, if we restrict the growth of the L2 harmonic 1-forms on them, the similar theorem holds, too.