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THE RIGIDITY OF MINIMAL SUBMANIFOLDS IN A LOCALLY SYMMETRIC SPACE

  • Cao, Shunjuan
  • Received : 2011.06.03
  • Published : 2013.01.31

Abstract

In the present paper, we discuss the rigidity phenomenon of closed minimal submanifolds in a locally symmetric Riemannian manifold with pinched sectional curvature. We show that if the sectional curvature of the submanifold is no less than an explicitly given constant, then either the submanifold is totally geodesic, or the ambient space is a sphere and the submanifold is isometric to a product of two spheres or the Veronese surface in $S^4$.

Keywords

minimal submanifold;rigidity;sectional curvature;locally symmetric space

References

  1. S. S. Chern, M. do Carmo, and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, In: Functional analysis and relathed fields, pp. 59-75. Berlin, Heidelberg, New York, Springer, 1970.
  2. Q. Ding and Y. L. Xin, On Chern's problem for rigidity of minimal hypersurfaces in the spheres, Adv. Math. 227 (2011), no. 1, 131-145. https://doi.org/10.1016/j.aim.2011.01.018
  3. J. Q. Ge and Z. Z. Tang, A proof of the DDVV conjecture and its equality case, Pacific J. Math. 237 (2008), no. 1, 87-95. https://doi.org/10.2140/pjm.2008.237.87
  4. S. I. Goldberg, Curvature and Homology, Academic Press, London, 1998.
  5. J. R. Gu and H. W. Xu, On Yau rigidity theorem for minimal submanifolds in spheres, preprint, arxiv:1102.5732v1.
  6. T. Itoh, On Veronese manifolds, J. Math. Soc. Japan 27 (1975), no. 3, 497-506. https://doi.org/10.2969/jmsj/02730497
  7. T. Itoh, Addendum to my paper "On Veronese manifolds", J. Math. Soc. Japan 30 (1978), no. 1, 73-74. https://doi.org/10.2969/jmsj/03010073
  8. B. Lawson, Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2) 89 (1969), 179-185.
  9. A. M. Li and J. M. Li, An intrinsic rigidity theorem for minimal submanifolds in a sphere, Arch. Math. (Basel) 58 (1992), no. 6, 582-594. https://doi.org/10.1007/BF01193528
  10. Z. Lu, Proof of the normal scalar curvature conjecture, arXiv:0711.3510v1.
  11. J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. (2) 88 (1968), 62-105. https://doi.org/10.2307/1970556
  12. W. D. Song, On minimal submanifolds in a locally symmetric space, Chinese Ann. Math. Ser. A 19 (1998), 693-698.
  13. S. M. Wei and H. W. Xu, Scalar curvature of minimal hypersurfaces in a sphere, Math. Res. Lett. 14 (2007), no. 3, 423-432. https://doi.org/10.4310/MRL.2007.v14.n3.a7
  14. H. W. Xu, On closed minimal submanifolds in pinched Riemannian manifolds, Trans. Amer. Math. Soc. 347 (1995), no. 5, 1743-1751. https://doi.org/10.1090/S0002-9947-1995-1243175-X
  15. H. C. Yang and Q. M. Cheng, Chern's conjecture on minimal hypersurfaces, Math. Z. 227 (1998), no. 3, 377-390. https://doi.org/10.1007/PL00004382
  16. S. T. Yau, Submanifolds with constant mean curvature I, Amer. J. Math. 96, (1974), 346-366 https://doi.org/10.2307/2373638
  17. S. T. Yau, Submanifolds with constant mean curvature II, Amer. J. Math. 97 (1975), 76-100. https://doi.org/10.2307/2373661