Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

The pinching constant of minimal hypersurfaces in the unit spheres

Author(s): Qin Zhang
Journal: Proc. Amer. Math. Soc. 138 (2010), 1833-1841.
MSC (2000): Primary 53C40
Posted: December 31, 2009
MathSciNet review: 2587468
Retrieve article in: PDF

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Abstract: In this paper, we prove that if ( $n\leq8$ ) is a closed minimal hypersurface in a unit sphere $S^{n+1}(1)$ , then there exists a positive constant $\alpha (n)$ depending only on such that if $n\leq S \leq n+\alpha(n)$ , then is isometric to a Clifford torus, where is the squared norm of the second fundamental form of .

References:

1.: J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. (2), 88 (1968), 62-105. MR 0233295 (38:1617)
2.: H. B. Lawson Jr., Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2), 89 (1969), 187-197. MR 0238229 (38:6505)
3.: S. S. Chern, M. do Carmo, S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, Functional analysis and related fields, Springer, New York, 1970, pp. 59-75. MR 0273546 (42:8424)
4.: C. K. Peng, C. L. Terng, Minimal hypersurfaces of spheres with constant scalar curvature, Ann. of Math. Stud., 103, Princeton University Press, 1983, 179-198. MR 795235 (87k:53143)
5.: C. K. Peng, C. L. Terng, The scalar curvature of minimal hypersurfaces in spheres, Math. Ann., 266 (1983), 105-113. MR 722930 (85c:53099)
6.: Q. M. Cheng, S. Ishikawa, A characterization of the Clifford torus, Proc. Amer. Math. Soc., 127 (1999), 819-828. MR 1636934 (99g:53064)
7.: Q. M. Cheng, The classification of complete hypersurfaces with nonzero constant mean curvature of space form of dimension , Mem. Fac. Sci. Kyushu Univ. Ser. A, 47 (1993), 79-102. MR 1222356 (94h:53067)
8.: S. Y. Cheng, S. T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann., 225 (1977), no. 3, 195-204. MR 0431043 (55:4045)
9.: A. M. Li, J. M. Li, An intrinsic rigidity theorem for minimal submanifolds in a sphere, Arch. Math. (Basel), 58 (1992), no. 6, 582-594. MR 1161925 (93b:53050)
10.: H. W. Xu, A rigidity theorem for submanifolds with parallel mean curvature in a sphere, Arch. Math. (Basel), 61 (1993), no. 5, 489-496. MR 1241055 (94m:53084)
11.: H. W. Xu, On closed minimal submanifolds in pinched Riemannian manifolds, Trans. Amer. Math. Soc., 347 (1995), no. 5, 1743-1751. MR 1243175 (95h:53088)
12.: H. W. Xu, W. Fang, F. Xiang, A generalization of Gauchman's rigidity theorem, Pacific J. Math., 228 (2006), no. 1, 185-199. MR 2263029 (2007j:53048)
13.: S. T. Yau, Submanifolds with constant mean curvature. I, II, Amer. J. Math., 96 (1974), 346-366; ibid., 97 (1975), 76-100. MR 0370443 (51:6670)
14.: Q. M. Cheng, H. C. Yang, Chern's conjecture on minimal hypersurfaces, Math. Z., 227 (1998), no. 3, 377-390. MR 1612653 (99c:53070)
15.: Q. M. Cheng, The rigidity of Clifford torus $S^1(\sqrt{\frac{1}{n}}) \times S^{n - 1}(\sqrt{\frac{n - 1}{n}})$ , Comment. Math. Helvetici, 71 (1996), 60-69. MR 1371678 (97a:53094)
16.: Y. T. Zhang, S. L. Xu, Rigidity of the Clifford torus $S^m(\sqrt{m/n})$ $\times$ $S^{n-m}(\sqrt{(n-m)/n})$ , Acta Mathematica Scientia Ser. A Chin. Ed., 28 (2008), 128-132. MR 2392041 (2008m:53145)

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