de Sitter空间中具有常数量曲率的完备类空子流形的间隙现象

Gap Phenomena for Complete Space-Like Submanifolds with Constant Scalar Curvature in de Sitter Spaces

设M^n是de Sitter空间S_p^(n+p)(c)中具有常数量曲率R(≤c)的完备类空子流形.得到了M^n关于其第二基本形式模长平方‖h‖~2的间隙性定理:如果n(c-R)≤‖h‖~2≤2(n-1c)^(1/2),那么,或者‖h‖~2=n(c-R)且M^n是全脐点子流形,或者‖h‖~2=2(n-1c)^(1/2)且M^n是全脐的或是双曲柱面S^(n-1)(c-tanh^2 r)×H^1(c-coth^2 r).

Let M^n be a complete space-like submanifold with constant scalar curvature R （≤c） in a de Sitter space Sp^n＋p（c）, h be the second fundamental form of M^n in sp^n＋p（c）. This paper obtains a gap property of the squared norm ｜｜h｜｜^2： if n（c-R） ≤｜｜h｜｜^2 ≤ 2√n-1c, then either ｜｜h｜｜^2 = n（c - R） and Mn is totally umbilical, or ｜｜h｜｜^2 = 2√n- 1c and M^n is totally umbilical or a hyperbolic cylinder S^n- 1 （c - tanh^2 r） × H^1（c - coth^2 r）.