球面上Willmore子流形的Pinching定理

A Pinching Theorem for Willmore Submanifolds in a Sphere

设Mn是单位球Sn+p中的一个n维Willmore子流形,H和S分别表示M的平均曲率和第二基本形式模长的平方,记ρ2=S-nH2。证明了当‖ρ2‖n2<C时,S=nH2且M是全脐的球面。其中C只依赖于n,ρ和M。

Let Mn be an n-dimetional Willmore submanifold in a unit sphere Sn ＋ p.Denote by H and S the mean curvature and the squared length of the second fundamental form of M,respectively.Set ρ2 = S-nH2.By using the Sobolev inequality,we prove that if ‖ ρ2‖n 2 C,then S = nH2 and M is a totally umbilical sphere,where C only depends on n,ρand M.