Minimal hypersurfaces of unit sphere
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 isoparametric hypersurfaces with \(3\) principal curvatures. In fact, \textit{S. Chang} [Pac. J. Math. 165, 67-76 (1994; 846.53040)] proves that a closed hypersurface in \(S^{n+1}\) with constant mean and scalar curvatures that has 3 distinct principal curvatures at each point is isoparametric. This complements a result of R. Miyaoka proving the same conclusion under an assumption of three non-simple principal curvatures. Other generalizations are studied by various authors: assuming a hypersurface to be non-compact, or considering minimal submanifolds of higher codimension in \(S^{n+1}.\)
scientific article; zbMATH DE number 1140479
scientific article