球面上k-极值子流形的Pinching定理

A Pinching theorem for k-extremal submanifolds in a sphere

令M^n是n维单位球空间S^(n+p)(n≥3)中的紧致k-极值子流形(1≤k<n/2),证明当(∫_(M^n)ρ~ndv)2/n<C时,|A|~2=nH^2且M^n全脐,其中C依赖于n,p,M^n.记ρ~2=|A|~2-nH^2,H和|A|~2分别表示Mn的平均曲率和第2基本型模长平方.

Let Mn be a n -dimensional compact k -extremal submanifolds（ 1 ≤k 〈n/2） in a unit sphere Sn＋P（ n ≥ 3）, it is proved that if （ ∫Mn ρdv） 1/n〈 C, then ｜A｜2 =nH2 and are a totally umbilical, where only depends on n,ρ,Mn. Set ρ2 = ｜ A ｜2 - nH2, H and ｜ A ｜ 2 and respectively denote the mean curvature and the squared length of the sec-ond fundamental form of Mn.