Complete submanifolds of $R^n$ with finite topology

Abstract

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