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A MAXIMUM PRINCIPLE FOR COMPLETE HYPERSURFACES IN LOCALLY SYMMETRIC RIEMANNIAN MANIFOLD

  • Zhang, Shicheng
  • Received : 2013.05.09
  • Published : 2014.01.31

Abstract

In this article, we apply the weak maximum principle in order to obtain a suitable characterization of the complete linearWeingarten hypersurfaces immersed in locally symmetric Riemannian manifold $N^{n+1}$. Under the assumption that the mean curvature attains its maximum and supposing an appropriated restriction on the norm of the traceless part of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or hypersurface is an isoparametric hypersurface with two distinct principal curvatures one of which is simple.

Keywords

locally symmetric;linear Weingarten hypersurfaces;totally umbilical

References

  1. L. J. Alias, S. C. Garcia-Martinez, and M. Rigoli, A maximum principle for hypersurfaces with constant scalar curvature and applications, Ann. Global Anal. Geom. 41 (2012), no. 3, 307-320. https://doi.org/10.1007/s10455-011-9284-y
  2. A. Brasil Jr., A. G. Colares, and O. Palmas, Complete hypersurfaces with constant scalar curvature in spheres, Monatsh. Math. 161 (2010), no. 4, 369-380. https://doi.org/10.1007/s00605-009-0128-9
  3. A. Brasil Jr., A. G. Colares, and O. Palmas, Complete hypersurfaces with constant scalar curvature in spheres, Monatsh. Math. 161 (2010), no. 4, 369-380. https://doi.org/10.1007/s00605-009-0128-9
  4. E. Cartan, Familles de surfaces isoparametriques dans les espaces a courbure constante, Ann. Mat. Pura Appl. 17 (1938), no. 1, 177-191. https://doi.org/10.1007/BF02410700
  5. Q. M. Cheng, Hypersurfaces in a unit sphere $S^{n+1}$ with constant scalar curvature, J. London Math. Soc. (2) 64 (2001), no. 3, 755-768. https://doi.org/10.1112/S0024610701002587
  6. Q. M. Cheng and H. Nakagawa, Totally umbilic hypersurfaces, Hiroshima Math. J. 20 (1990), no. 1, 1-10.
  7. Q. M. Cheng and Susumu, Characterization of the clifford torus, Proc. Amer. Math. Soc. 127 (1999), no. 3, 819-831. https://doi.org/10.1090/S0002-9939-99-05088-1
  8. S. S. Cheng, M. do Carmo, and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, in Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968), 59-75, Springer, New York, 1970.
  9. S. Y. Cheng and S. T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), no. 3, 195-204. https://doi.org/10.1007/BF01425237
  10. S. H. Ding and J. F. Zhang, Hypersurfaces in a locally symmetric manifold with constant mean curvature, Pure Appl. Math. 22 (2006), no. 1, 94-99.
  11. Z. H. Hou, Hypersurfaces in a sphere with constant mean curvature, Proc. Amer. Math. Soc. 125 (1997), no. 4, 1193-1196. https://doi.org/10.1090/S0002-9939-97-03668-X
  12. H. B. Lawson, Jr., Local rigidity theorems for minimal hypersurfaces, Ann. of Math. 89 (1969), no. 2, 187-197. https://doi.org/10.2307/1970816
  13. H. Li, Global rigidity theorems of hypersurfaces, Ark. Mat. 35 (1997), no. 2, 327-351. https://doi.org/10.1007/BF02559973
  14. H. Li, Y. Suh, and G. Wei, Linear Weingarten hypersurfaces in a unit sphere, Bull. Korean Math. Soc. 46 (2009), no. 2, 321-329. https://doi.org/10.4134/BKMS.2009.46.2.321
  15. X. X. Liu and H. Li, Complete hypersurfaces with constant scalar curvature in a sphere, Comment. Math. Univ. Carolin. 46 (2005), no. 3, 567-575.
  16. M. Okumura, Hypersurfaces and a pinching problem on the second fundamental tensor, Amer. J. Math. 96 (1974), no. 4, 207-213. https://doi.org/10.2307/2373587
  17. S. Pigola, M. Rigoli, and A. G. Setti, Maximum principles on Riemannian manifolds and applications, Mem. Amer. Math. Soc. 174 (2005), no. 822, x+99 pp.
  18. S. Pigola, A Liouville-type result for quasi-linear elliptic equations on complete Riemannian manifolds, J. Funct. Anal. 219 (2005), no. 2, 400-432. https://doi.org/10.1016/j.jfa.2004.05.009
  19. B. Segre, Famiglie di ipersuperficie isoparametriche negli spazi euclidei ad un qualunque numero di dimensioni, Atti Accad. Naz. Lincei, Rend., VI. Ser. 27 (1938), 203-207.
  20. J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. 88 (1968), no. 2, 62-105. https://doi.org/10.2307/1970556
  21. S. C. Shu and S. Y. Liu, Complete hypersurfaces with constant mean curvature in locally symmetric manifold, Adv. Math. (China) 33 (2004), no. 5, 563-569.
  22. H.W. Xu, Pinching theorems, global pinching theorems and eigenvalues for Riemannian submanifolds, Ph. D. dissertation, Fudan University, 1990.
  23. 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
  24. H. W. Xu and X. Ren, Closed hypersurfaces with constant mean curvature in a symmetric manifold, Osaka J. Math. 45 (2008), no. 3, 747-756.
  25. S. Zhang and B. Wu, Rigidity theorems for complete spacelike hypersurfaces with constant scalar curvature in locally symmetric Lorentz spaces, J. Geom. Phys. 60 (2010), no. 2, 333-340. https://doi.org/10.1016/j.geomphys.2009.10.005
  26. S. Zhang and B. Wu, Complete hypersurfaces with constant mean curvature in a locally symmetric Riemannian manifold, Acta Math. Sci. Ser. A Chin. Ed. 30 (2010), no. 4, 1000-1005.