Global pinching theorems of submanifolds in spheres
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 Lp estimate for the square length σ of the second fundamental form and the norm of a tensor Φ, related to...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2002-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171202106247 |
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| Summary: | 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 Lp 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 Lp 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. |
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| ISSN: | 0161-1712 1687-0425 |