Asian Journal of Mathematics

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

Hong-Wei Xu and Ling Tian

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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 $\beta(n,H) = n + \frac{n^3 H^2}{2(n−1)} + \frac{n(n−2)}{2(n−1)} \sqrt{n^2 H^4 + 4(n − 1)H^2}$.

Article information

Source
Asian J. Math., Volume 15, Number 4 (2011), 611-630.

Dates
First available in Project Euclid: 12 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.ajm/1331583350

Mathematical Reviews number (MathSciNet)
MR2853651

Zentralblatt MATH identifier
1243.53104

Subjects
Primary: 53C40: Global submanifolds [See also 53B25] 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

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

Citation

Xu, Hong-Wei; Tian, Ling. A New Pinching Theorem for Closed Hypersurfaces with Constant Mean Curvature in $S^{n+1}$. Asian J. Math. 15 (2011), no. 4, 611--630. https://projecteuclid.org/euclid.ajm/1331583350


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