Global pinching theorems of submanifolds in spheres
2002 ◽
Vol 31
(3)
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pp. 183-191
Keyword(s):
LetMbe a compact embedded submanifold with parallel mean curvature vector and positive Ricci curvature in the unit sphereS n+p(n≥2 ,p≥1). By using the Sobolev inequalities of P. Li (1980) toLpestimate for the square lengthσof the second fundamental form and the norm of a tensorΦ, related to the second fundamental form, we set up some rigidity theorems. Denote by‖σ‖ptheLpnorm ofσandHthe constant mean curvature ofM. It is shown that there is a constantCdepending only onn,H, andkwhere(n−1) kis the lower bound of Ricci curvature such that if‖σ‖ n/2<C, thenMis a totally umbilic hypersurface in the sphereS n+1.
2000 ◽
Vol 69
(1)
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pp. 1-7
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2011 ◽
Vol 54
(1)
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pp. 67-75
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2011 ◽
Vol 22
(01)
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pp. 131-143
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2016 ◽
Vol 18
(06)
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pp. 1550073
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2004 ◽
Vol 76
(3)
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pp. 489-497
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2009 ◽
Vol 51
(2)
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pp. 413-423
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2013 ◽
Vol 20
(4)
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pp. 715-733
1972 ◽
Vol 78
(2)
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pp. 247-251
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