Global pinching theorems of submanifolds in spheres

Kairen Cai

International Journal of Mathematics and Mathematical Sciences, 2002, vol. 31, 1-9

Abstract:

Let M be a compact embedded submanifold with parallel mean curvature vector and positive Ricci curvature in the unit sphere S   n + p ( n ≥ 2   , p ≥ 1 ) . By using the Sobolev inequalities of P. Li (1980) to L p estimate for the square length σ of the second fundamental form and the norm of a tensor Φ , related to the second fundamental form, we set up some rigidity theorems. Denote by ‖ σ ‖ p the L p norm of σ and H the constant mean curvature of M . It is shown that there is a constant C depending only on n , H , and k where ( n − 1 )   k is the lower bound of Ricci curvature such that if ‖ σ ‖   n / 2 < C , then M is a totally umbilic hypersurface in the sphere S   n + 1 .

Date: 2002
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:476906

DOI: 10.1155/S0161171202106247

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