球面中常平均曲率超曲面的整体Pinching定理

Global Pinching Theorem for Hypersurfaces with Constant Mean Curvature in Sphere

假设M是标准球面Sn+1中的紧致嵌入超曲面。本文利用P.Li的Sobolev不等式,对一个联系到平均曲率H和第二基本形式的张量φ的模长作Lp估计,建立了球面中常平均曲牢超曲面的整体Pinching定理。即证明了:如果M具有常平均曲率且Ricci曲率有正的下界(n-1)k,于是必存在一个仅依赖n,H和k的常数C,当σ的Ln/2模小于C时,M为球面的全脐点超曲面,其中σ表示M的第二基本形式长度的平方。

Let M be a compact embedded hypersurface with constant mean curvature H and positive Ricci curvature in the unit sphere Sn+1. By using the Sobolev inequalties of P.Li to the norm of a tensor φ, related to the second fundamental form, we set up a pinching theorem. Denote by ||φ||p the Lv norm of the square length of the second fundamental form. It is shown that there is a constant C depending only on n, H and k where (n - 1)k is the lower bound of Ricci curvature such that if ||σ||n/2 < C, then M is totally umbilic.