On closed minimal submanifolds in pinched Riemannian manifolds
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- by Hong Wei Xu
- Trans. Amer. Math. Soc. 347 (1995), 1743-1751
- DOI: https://doi.org/10.1090/S0002-9947-1995-1243175-X
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Abstract:
In this paper, we first prove a generalized Simons integral inequality for closed minimal submanifolds in a Riemannian manifold. Second, we prove a pinching theorem for closed minimal submanifolds in a complete simply connected pinched Riemannian manifold, which generalizes the results obtained by S. S. Chern, M. do Carmo, and S. Kobayashi and A. M. Li and J. M. Li respectively. Finally, we obtain a distribution theorem for the square norm of the second fundamental form of $M$ under the assumption that $M$ is a minimal submanifold with parallel second fundamental form in a Riemannian manifold.References
- Hong Cang Yang and Qing Ming Cheng, A note on the pinching constant of minimal hypersurfaces with constant scalar curvature in the unit sphere, Chinese Sci. Bull. 36 (1991), no. 1, 1–6. MR 1137751
- S. S. Chern, M. do Carmo, and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968) Springer, New York, 1970, pp. 59–75. MR 0273546
- Hillel Gauchman, Minimal submanifolds of a sphere with bounded second fundamental form, Trans. Amer. Math. Soc. 298 (1986), no. 2, 779–791. MR 860393, DOI 10.1090/S0002-9947-1986-0860393-5
- Samuel I. Goldberg, Curvature and homology, Pure and Applied Mathematics, Vol. XI, Academic Press, New York-London, 1962. MR 0139098
- H. Blaine Lawson Jr., Rigidity theorems in rank-$1$ symmetric spaces, J. Differential Geometry 4 (1970), 349–357. MR 267492
- Li An-Min and Li Jimin, An intrinsic rigidity theorem for minimal submanifolds in a sphere, Arch. Math. (Basel) 58 (1992), no. 6, 582–594. MR 1161925, DOI 10.1007/BF01193528
- Masafumi Okumura, Hypersurfaces and a pinching problem on the second fundamental tensor, Amer. J. Math. 96 (1974), 207–213. MR 353216, DOI 10.2307/2373587
- Chia-Kuei Peng and Chuu-Lian Terng, Minimal hypersurfaces of spheres with constant scalar curvature, Seminar on minimal submanifolds, Ann. of Math. Stud., vol. 103, Princeton Univ. Press, Princeton, NJ, 1983, pp. 177–198. MR 795235
- Yi Bing Shen, On intrinsic rigidity for minimal submanifolds in a sphere, Sci. China Ser. A 32 (1989), no. 7, 769–781. MR 1057998
- James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. MR 233295, DOI 10.2307/1970556 H. W. Xu, Some results on geometry of Riemannian submanifolds, Ph.D. dissertation, Fudan Univ., 1990.
- Shing Tung Yau, Submanifolds with constant mean curvature. I, II, Amer. J. Math. 96 (1974), 346–366; ibid. 97 (1975), 76–100. MR 370443, DOI 10.2307/2373638
Bibliographic Information
- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 347 (1995), 1743-1751
- MSC: Primary 53C42; Secondary 53C20, 53C40
- DOI: https://doi.org/10.1090/S0002-9947-1995-1243175-X
- MathSciNet review: 1243175