The stability index of hypersurfaces with constant scalar curvature in spheres. Issue 3 (16th May 2014)

• Record Type: Journal Article
• Authors: Cheng, Qing-Ming; Li, Haizhong; Wei, Guoxin
• Is Part Of: Proceedings. Volume 144:Issue 3(2014)
• Page Start: 447
• Page End: 453
• Publication Date: 2014-05-16
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1017/S030821051200056X
• Languages: English
• ISSNs: 0308-2105

Abstract

<abstract abstract-type="normal"> <title> <x content-type="archive" xml:space="preserve">Abstract</x> </title> <p>The totally umbilical and non-totally geodesic hypersurfaces in the (<italic>n</italic> + 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 <italic>M</italic> with constant scalar curvature <italic>n</italic>(<italic>n</italic> − 1)<italic>r</italic>, <italic>r</italic>&gt; 1, in an (<italic>n</italic> + 1)-dimensional sphere <italic>S</italic>^{<italic>n</italic> + 1}(1), which is not a totally umbilical hypersurface, is greater than or equal to <italic>n</italic> + 2 if the mean curvature <italic>H</italic> and <italic>H</italic>₃ are constant. In this paper, we prove the same results, without the assumption that <italic>H</italic>₃ is constant. In fact, we show that the weak stability index of a compact hypersurface <italic>M</italic> with constant scalar curvature <italic>n</italic>(<italic>n</italic> − 1)<italic>r</italic>, <italic>r</italic>&gt; 1, in <italic>S</italic>^{<italic>n</italic> + 1}(1), which is not a totally umbilical hypersurface, is greater than or equal to <italic>n</italic> + 2 if the mean curvature <italic>H</italic> is constant.</p> </abstract>