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 ${\int }_M|\alpha {|}^{m}dV<\infty$ and all complete 2-dimensional complete surfaces with ${\int }_M|\alpha {|}^{2}dV<\infty$ and nonpositive curvature with respect to every normal direction, since $a(M)=0$ for them.

arXiv comment8 pages