空间形式中具有三个互异主曲率的超曲面

Hypersurfaces in the Space Form with Three Principal Curvatures

设M^(n+1)(c)是常曲率c的n+1维空间形式,M^n是M^(n+1)(c)中超曲面.在不同条件下,对M进行分类.Miyaoka,R.给出了具有三个互异主曲率的极小超曲面M^n(n≥4)在满足一定条件下一个完全分类.本文把Miyaoka,R.的结果推广到具有三个互异主曲率的常平均曲率超曲面的情况,得到一个类似的完全分类.

Let Mn+1(c) be an (n + 1)-dimensional space form of curvature c and Mn a hypersurface in Mn+1(c), In 1980, under a suitable condition related to the curvature distributions, R. Miyaoka Classified minimal hypersurfaces with three principal curvatures in the space form. In this paper, we obtain the following main vesults:1 . Under a suitable condition related to the curvature distributions, we classify Mn with three principal curvatures and constant mean curvature.2 . Let Mn be a complete hypersurface of Mn+1(c) with constant scalar curvature where c>0. If Mn has theee principal curvatures such that one is identically mean curvature of Mn, and each of other two is non-simple, then M is isoparametric.