  In this paper we prove some general results on constant mean curvature
lamination limits of certain sequences of compact surfaces $M_n$ embedded in
$\mathbb R^3$ with constant mean curvature $H_n$ and fixed finite genus, when
the boundaries of these surfaces tend to infinity. Two of these theorems
generalize to the non-zero constant mean curvature case, similar structure
theorems by Colding and Minicozzi in~[6,8] for limits of sequences of minimal
surfaces of fixed finite genus.
