Asian Journal of Mathematics

Volume 15 (2011)

Number 4

A New Pinching Theorem for Closed Hypersurfaces with Constant Mean Curvature in $S^{n+1}$

Pages: 611 – 630

DOI: http://dx.doi.org/10.4310/AJM.2011.v15.n4.a6

Authors

Ling Tian

Hong-Wei Xu

Abstract

We investigate the generalized Chern conjecture, and prove that if $M$ is a closed hypersurface in $S^{n+1}$ with constant scalar curvature and constant mean curvature, then there exists an explicit positive constant $C(n)$ depending only on $n$ such that if $|H| < C(n)$ and $S > \beta (n,H)$, then $S > \beta (n,H) + \frac{3n}{7}$, where $\beta(n,H) = n + \frac{n^3 H^2}{2(n−1)} + \frac{n(n−2)}{2(n−1)} \sqrt{n^2 H^4 + 4(n − 1)H^2}$.

Keywords

Closed hypersurface; pinching phenomenon; mean curvature; scalar curvature; second fundamental form

2010 Mathematics Subject Classification

53C40, 53C42

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