On the volume growth and the topology of complete minimal submanifolds of a Euclidean space

Abstract

Let $M$ be a $n$-dimensional complete properly immersed minimal submanifold of a Euclidean space. We show that the number of the ends of $M$ is bounded above by k=\sup{\roman{volume}(M\cap B(t)) \over ω_nt^n}, where $B(t)$ is the ball of the Euclidean space of center 0 and radius $t$, $ω_n$ is the volume of $n$-dimensional unit Euclidean ball. Moreover, we prove that the number of ends of $M$ is equal to $k$ under some curvature decay condition