On closed minimal submanifolds in pinched Riemannian manifolds

Xu, Hong Wei

doi:10.1090/S0002-9947-1995-1243175-X

On closed minimal submanifolds in pinched Riemannian manifolds
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Trans. Amer. Math. Soc. 347 (1995), 1743-1751 Request permission

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

Some results on geometry of Riemannian submanifolds

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