On complete submanifolds with bounded mean curvature

Abstract

We consider M n , n ≥ 3 , an n -dimensional complete and connected submanifold of a space form M n + p ( c ) , whose mean curvature H does not vanish and is bounded with a parallel normalized mean curvature vector. We prove that if S ≤ n 2 H 2 n − 1 + 2 c , where S denotes the squared norm of the second fundamental form of M n , 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 M n is a compact submanifold of hyperbolic space form with constant mean curvature H ≠ 0 , then M n is a geodesic sphere. When M n is a submanifold of the unit sphere with constant mean curvature H ≠ 0 , then either M n is a great or small sphere in S n + 1 ( 1 ) or M n is a product of spheres S l ( r ) × S m ( s ) .