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The stability index of hypersurfaces with constant scalar curvature in spheres

Published online by Cambridge University Press:  16 May 2014

Qing-Ming Cheng ,
Haizhong Li  and
Guoxin Wei
Qing-Ming Cheng
Affiliation:
Department of Applied Mathematics, Faculty of Sciences, Fukuoka University, Fukuoka 814-0180, Japan, ([email protected])
Haizhong Li
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People's Republic of China, ([email protected])
Guoxin Wei
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, People's Republic of China, ([email protected])
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Abstract

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.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 144 , Issue 3 , June 2014 , pp. 447 - 453
Copyright
Copyright © Royal Society of Edinburgh 2014 

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