具有平行平均曲率的类空子流形的Pinching定理

Pinching theorems of space-like submnifolds with parallel mean curvature vector

证明若 Mn 是 de Sitter空间 Sn+ PP (1 ) (P >1 )中具有单位平行平均曲率向量的紧致类空子流形 ,若关于平均曲率向量的第二基本形式长度的平方σξ <2 n,则 Mn 是全脐点的 .在相同条件下还证明了一个整体Pinching定理 :若σ为第二基本形式长度的平方 ,c和 Vol M分别为 M的等周常数和体积 ,则存在仅与 n,c,Vol M有关的常数 A,当满足 ∫σn2 d V2n <A时 ,Mn

The author has proved the following: Let \$M\+n\$ be a compact space\|like submanifold with unit parallel mean curvature vector in an de sitter space \$S\+\{n+P\}\-P(1)(P>1)\$ and \$σ\-ξ\$ the square of the length of the second fundamental form of \%M\+n\% with respect to mean curvature vector. If \$σ\-ξ<2n\$, then \$M\+n\$ is a totally umbilic submanifold. A global pinching theorem is also obtained. Denoted by \$‖σ‖\-P,\$ and Vol \%M\% are the \$L\-P\$ norm of the square of the length of the second fundamental form of \$M\+n\$, isoperimetric constant and volume of \$M\+n\$ respectively. Then there is a constant A, depending only on \$n,\$ and Vol \%M\%, such that if \$‖σ‖\-\{n2\}<A\$, then \$M\+n\$ is a totally umbilic submanifold.\;