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In this paper we generalize the Local Removable Singularity Theorem in [16]for minimal laminations to the case of weak $H$-laminations (with $H\in\mathbb{R}$ constant) in a punctured ball of a Riemannian three-manifold. Wealso obtain a curvature estimate for any weak CMC foliation (with possiblyvarying constant mean curvature from leaf to leaf) of a compact Riemannianthree-manifold $N$ with boundary solely in terms of a bound of the absolutesectional curvature of $N$ and of the distance to the boundary of $N$. We thenapply these results to classify weak CMC foliations of $\mathbb{R}^3$ and$\mathbb{S}^3$ with a closed countable set of singularities.

The classification of CMC foliations of $\mathbb{R}^3$ and $\mathbb{S}^3$ with countably many singularities