Complete subamanifolds of $\mathbb{R}^{n}$ with finite topology

Abstract

We show that a complete $m$-dimensional immersed submanifold $M$ of $\mathbb{R}^{n}$ with $a(M)<1$ is properly immersed and have finite topology, where $a(M)\in [0,\infty]$ 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 submanifolds $M$ with $a(M)<1$ contains all complete minimal surfaces in $\mathbb{R}^{n}$ with finite total curvature, all $m$-dimensional minimal submanifolds $M $ of $ \mathbb{R}^{n}$ with finite total scalar curvature $\smallint_{M}| \alpha |^{m} dV<\infty $ and all complete 2-dimensional complete surfaces with $\smallint_{M}| \alpha |^{2} dV<\infty $ and nonpositive curvature with respect to every normal direction, since $a(M)=0$ for them.