Rigidity of submanifolds with parallel mean curvature in a hyperbolic space

doi:10.3969/j.issn.1000-5641.201911009

Abstract

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 sup_{x∈_M}|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

CLC Number:

O186.1

ZHOU Jundong. Rigidity of submanifolds with parallel mean curvature in a hyperbolic space[J]. Journal of East China Normal University(Natural Science), 2020, 2020(2): 8-14.

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