The Classification of CMC foliations of {\Bbb R}^3 and {\Bbb S}^3 with countably many singularities

In this paper we generalize our previous Local Removable Singularity Theorem for minimal laminations to the case of weak $H$-laminations (with $H\in{\Bbb R}$ constant) in a punctured ball of a Riemannian three-manifold. We also obtain a curvature estimate for any weak CMC foliation (with possibly varying constant mean curvature from leaf to leaf) of a compact Riemannian three-manifold $N$ with boundary solely in terms of a bound of the absolute sectional curvature of $N$ and of the distance to the boundary of $N$. We then apply these results to classify weak CMC foliations of ${\Bbb R}^3$ and ${\Bbb S}^3$ with a closed countable set of singularities.