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

TL;DR

This paper extends the classification of constant mean curvature (CMC) foliations in three-dimensional spaces to include those with countably many singularities, providing new curvature estimates and a generalized singularity theorem.

Contribution

It generalizes the Local Removable Singularity Theorem for minimal laminations to weak H-laminations and classifies weak CMC foliations with countably many singularities in b3 and b3 spaces.

Findings

Generalized singularity removal theorem for weak H-laminations.

Curvature estimates for weak CMC foliations.

Classification of weak CMC foliations with countable singularities.

Abstract

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. 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 $\mathbb{R}^3$ and $\mathbb{S}^3$ with a closed countable set of singularities.