Journal of East China Normal University(Natural Science) ›› 2020, Vol. 2020 ›› Issue (2): 8-14.doi: 10.3969/j.issn.1000-5641.201911009

• Mathematics • Previous Articles     Next Articles

Rigidity of submanifolds with parallel mean curvature in a hyperbolic space

ZHOU Jundong   

  1. 1. School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China;
    2. School of Mathematics and Statistics, Fuyang Normal University, Fuyang Anhui 236037, China
  • Received:2019-02-22 Published:2020-03-16

Abstract: Let M be a complete submanifold with parallel mean curvature in a hyperbolic space and Φ be the traceless second fundamental form of M. In this paper, it is shown that the submanifold is totally umbilical if the L2 norm of |Φ| has less than quadratic growth on any geodesic ball of M and either supxM|Φ|2(x) is less than some constant or Ln norm of |Φ| is less than some constant. This is a generalization of the results on complete minimal submanifolds.

Key words: hyperbolic space, traceless second fundamental form, the first eigenvalue

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