CONSTANT MEAN CURVATURE HYPERSURFACES IN SPHERES

In this paper, we first summarise the progress for the famous Chern conjecture, and then we consider n-dimensional closed hypersurfaces with constant mean curvature H in the unit sphere n+1 with n ≤ 8 and generalise the result of Cheng et al. (Q. M. Cheng, Y. J. He and H. Z. Li, Scalar curvature of...

Full description

Saved in:
Bibliographic Details
Published in:Glasgow mathematical journal 2012-01, Vol.54 (1), p.77-86
Main Authors: DENG, QIN-TAO, GU, HUI-LING, SU, YAN-HUI
Format: Article
Language:English
Subjects:
Curvature
Geometry
Mathematical analysis
Mathematics
Scalars
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:In this paper, we first summarise the progress for the famous Chern conjecture, and then we consider n-dimensional closed hypersurfaces with constant mean curvature H in the unit sphere n+1 with n ≤ 8 and generalise the result of Cheng et al. (Q. M. Cheng, Y. J. He and H. Z. Li, Scalar curvature of hypersurfaces with constant mean curvature in a sphere, Glasg. Math. J. 51(2) (2009), 413–423). In order to be precise, we prove that if |H| ≤ ϵ(n), then there exists a constant δ(n, H) > 0, which depends only on n and H, such that if S0 ≤ S ≤ S0 + δ(n, H), then S = S0 and M is isometric to the Clifford hypersurface, where ϵ(n) is a sufficiently small constant depending on n.
ISSN:0017-0895
1469-509X
DOI:10.1017/S001708951100036X