On complete submanifolds with parallel mean curvature in negative pinched manifolds

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/ ) or the Veronese surface in S ⁴(1/ ). Research supported by the NSFC (10231010); Trans-Century Training Programme Foundation for Talents by the Ministry of Education of China; Natural Science Foundation of Zhejiang Province (101037).