The minimal lamination closure theorem

Abstract

We prove that the closure of a complete embedded minimal surface $M$ in a Riemannian three-manifold $N$ has the structure of a minimal lamination when $M$ has positive injectivity radius. When $N$ is ${\mathbb{R}}^3$, we prove that such a surface $M$ is properly embedded. Since a complete embedded minimal surface of finite topology in ${\mathbb{R}}^3$ has positive injectivity radius, the previous theorem implies a recent theorem of Colding and Minicozzi in [5, Corollary 0.7]; a complete embedded minimal surface of finite topology in ${\mathbb{R}}^3$ is proper. More generally, we prove that if $M$ is a complete embedded minimal surface of finite topology and $N$ has nonpositive sectional curvature (or is the Riemannian product of a Riemannian surface with $\mathbb{R}$), then the closure of $M$ has the structure of a minimal lamination