Minimal hypersurfaces of unit sphere
type: Item
instance of: scholarly article
title: Minimal hypersurfaces of unit sphere
zbMATH Open document ID: 0938.53029
author: Koichi Ogiue
author: Huafei Sun
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2000-06-14T00:00:00Z
review text: If \(S\) denotes the square of the length of the second fundamental form of a closed minimal hypersurface \(M^n\) in the unit sphere \(S^{n+1},\) the celebrated Chern's conjecture states that the set of possible values of \(S\) (or, equivalently, set of values of the scalar curvature) for minimal hypersurfaces with constant scalar curvature is discrete. Starting with works of J. Simons, S. S. Chern et al. and H. B. Lawson, it has been known for a long time that there are no such hypersurfaces with \(S\) strictly between \(0\) and \(n\), and that \(S = n\) is realized only for the Clifford torus. So, the natural problem was to find the next value after \(n\) that \(S\) can take. \textit{C. K. Peng} and \textit{C. L. Terng} [Ann. Math. Stud. 103, 177-198 (1983; Zbl 0534.53048)] proved the existence of a constant \(\varepsilon (n) = 1/(12n)\) such that if \(n \leq S \leq n + \varepsilon (n), \) then \(S = n\), giving the Clifford torus again. Moreover, in a related work [Math. Ann. 266, 105-113 (1983; Zbl 0515.53043)], they were able to dispose of the assumption on constancy of the scalar curvature in low dimensions (\(n \leq 5\)), this time using a different value for \(\varepsilon (n).\) In the paper under review, the authors improve the result of Peng and Terng removing the dimension restriction \(n \leq 5\) altogether, and producing a better \(\varepsilon (n).\) Namely, they prove that if \(M^n\) is a closed minimally immersed hypersurface of \(S^{n+1}\) then there exists a constant \(\varepsilon (n) = 2n^2(n+4)/3(n+2)^2\) such that, if \(n \leq S \leq n + \varepsilon (n)\), then \(S = n,\) and \(M\) is again a Clifford torus. This is also a stronger result than that of \textit{H. Yang} and \textit{Q. M. Cheng} [Math. Z. 227, 377-390 (1998; Zbl 0893.53025)] who give the value \(\varepsilon (n) = (1/3)n.\) Together with other results, this supports the conjecture that the next value for \(S\) after \(0\) and \(n\) is \(2n\) which is realized for Cartan's isoparametr…
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 1140479
zbMATH Keywords: minimal hypersurface of a sphere
zbMATH Keywords: length of the second fundamental form
zbMATH Keywords: Clifford torus
zbMATH Keywords: constant scalar curvature
zbMATH Keywords: isoparametric hypersurfaces
reviewed by: Ivko Dimitrić
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178225113
OpenAlex ID: W2090636363
cites work: Minimal varieties in Riemannian manifolds
cites work: Minimal Submanifolds of a Sphere with Second Fundamental Form of Constant Length
cites work: Local rigidity theorems for minimal hypersurfaces
cites work: The scalar curvature of minimal hypersurfaces in spheres
cites work: Q3317736
cites work: The rigidity of Clifford torus \(S^ 1(\sqrt{\frac1n})\times S^{n-1}(\sqrt{\frac{n-1}n})\)
DOI: 10.2748/TMJ/1178225113
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description: scientific article; zbMATH DE number 1140479