On complete submanifolds with bounded mean curvature

摘要

We consider Mn, n≥3, an n-dimensional complete and connected submanifold of a space form Mn+p(c), whose mean curvature H does not vanish and is bounded with a parallel normalized mean curvature vector. We prove that if S≤n2H2n-1+2c, where S denotes the squared norm of the second fundamental form of Mn, then the codimension of isometric immersion reduces to 1. This result generalizes the case where the mean curvature vector is parallel or mean curvature is constant. If Mn is a compact submanifold of hyperbolic space form with constant mean curvature H≠0, then Mn is a geodesic sphere. When Mn is a submanifold of the unit sphere with constant mean curvature H≠0, then either Mn is a great or small sphere in Sn+1(1) or Mn is a product of spheres Sl(r)χSm(s).