On complete submanifolds with parallel mean curvature in negative pinched manifolds

  • Leng Yan 
  • Xu Hongwei 
Article
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Abstract

A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian (n + p)-dimensional manifold N n+p with negative sectional curvature is proved. For given positive integers n(≥ 2), p and for a constant H satisfying H > 1 there exists a negative number τ(n, p, H) ∈ (−1,0) with the property that if the sectional curvature of N is pinched in [−1, τ(n, p, H)], and if the squared length of the second fundamental form is in a certain interval, then N n+p is isometric to the hyperbolic space H n+p(−1). As a consequence, this submanifold M is congruent to S n (1/\(\sqrt {H^2 - 1} \)) or the Veronese surface in S 4(1/\(\sqrt {H^2 - 1} \)).

Keywords

complete submanifold rigidity theorem mean curvature second fundamental form pinched Riemannian manifold 

MR Subject Classification

53C40 53C20 
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Copyright information

© Editorial Committee of Applied Mathematics 2007

Authors and Affiliations

  • Leng Yan 
    • 1
  • Xu Hongwei 
    • 1
  1. 1.Center of Mathematical SciencesZhejiang UniversityHangzhouChina

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