复空间形式中常数量曲率的完备全实伪脐子流形

Complete totally real pseudo-umbilical submanifolds with constant scalar curvature in a complex space form

设CNcn是具有常全纯截面曲率c(≤0)的复n维的复空间形式,Mn是CNcn中常数量曲率的完备全实伪脐子流形,R,‖h‖2分别表示Mn的标准数量曲率和第二基本形式模长的平方.假设R≥ c/4.利用丘成桐的广义极大值原理和自伴随算子研究了关于‖h‖2的pinching问题,得到了两个Mn成为全测地或全脐的刚性定理.

Let CNcn be a complex space form,of complex dimension n,with constant holomorphic sectional curvature c（≤ 0）.Mn is a complete totally real pseudo-umbilical submanifold with constant scalar curvaturein CNcn.Denoted by R and h 2,the normalized scalar curvature and the square of the length of the second fundamental form of Mn,respectively.Suppose R ≥ c/4.We studied the pinching problem on h 2 by making use of the generalized maximal principle and self-adjoint differential operator of Yau,and obtained two rigidity theorems for Mn to be a totally geodesic or totally umblilic submanifold.