The stability index of hypersurfaces with constant scalar curvature in spheres
The stability index of hypersurfaces with constant scalar curvature in spheres
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DOI：
10.1017/s030821051200056x
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发表时间：
2014-05
期刊：
影响因子：
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通讯作者：
Q. Cheng;Haizhong Li;G. Wei
中科院分区：
文献类型：
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作者：
Q. Cheng;Haizhong Li;G. Wei
(n + 1) 维球体中的全脐超曲面和非全测地线超曲面被描述为唯一具有弱稳定性指数 0 的超曲面。在我们 2010 年的论文中，我们证明了在 (n + 1) 维球体 Sn + 1(1) 中，具有恒定标量曲率 n(n − 1)r，r> 1 的致密超曲面 M 的弱稳定性指数，该球体不是全脐球如果平均曲率 H 和 H3 恒定，则超曲面大于或等于 n + 2。在本文中，我们在不假设 H3 恒定的情况下证明了相同的结果。事实上，我们表明，如果平均曲率 H 恒定，则在 Sn + 1(1) 中，具有恒定标量曲率 n(n − 1)r (r> 1) 的紧致超曲面 M 的弱稳定性指数大于或等于 n + 2，该超曲面不是完全脐状超曲面。
The totally umbilical and non-totally geodesic hypersurfaces in the (n + 1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. In our 2010 paper we proved that the weak stability index of a compact hypersurface M with constant scalar curvature n(n − 1)r, r> 1, in an (n + 1)-dimensional sphere Sn + 1(1), which is not a totally umbilical hypersurface, is greater than or equal to n + 2 if the mean curvature H and H3 are constant. In this paper, we prove the same results, without the assumption that H3 is constant. In fact, we show that the weak stability index of a compact hypersurface M with constant scalar curvature n(n − 1)r, r> 1, in Sn + 1(1), which is not a totally umbilical hypersurface, is greater than or equal to n + 2 if the mean curvature H is constant.