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

arXiv comment38 pages, 5 figures

References in corpus (1)

• The Dynamics Theorem for properly embedded minimal surfaces