欧氏空间中常高阶平均曲率紧致凸超曲面与高斯映像

On the Gauss image of compact and convex hypersurfaces with a constant higher order mean curvature in Euclidean space

针对(n+1)维欧氏空间Rn+1中紧致无边凸超曲面M,利用一个已知的积分公式,并提出一种新的技巧,证明了:如果存在一个整数r(1≤r≤n-1)使得M的第r阶高阶平均曲率Hr是常数,并且M的高斯映照是到标准单位球面Sn的拓扑同胚,则M全脐.

For an oriented,compact and convex hypersurface M without boundary in the（ n ＋ 1）-dimensional Euclidean space Rn＋1,we apply a known integral formula and put out a new skill to prove that if there exits an integer r（ 1≤r≤n-1） such that the r-mean curvature Hris a constant and if the Gauss map of M is a topological homeomorphism onto the standard unit sphere Sn,then M is totally umbilical.