Complete submanifolds of Rn with finite topology

Abstract

We show that a complete m-dimensional immersed submanifold M of Rn with a(M) < 1 is properly immersed and have finite topology, where a(M) ⊂ [0,∞] is a scaling invariant number that gives the rate that the norm of the second fundamental form decays to zero at infinity. The class of submanifolds M ⊂ Rn with a(M) < 1 contains all complete minimal surfaces with finite total curvature, all m-dimensional minimal submanifolds with finite total scalar curvature ∫M|α|m dV < ∞ and all complete 2-dimensional surfaces with ∫M|α|2 dV < ∞ and non-positive curvature with respect to every normal direction.