Small curvature laminations in hyperbolic $3$–manifolds

Abstract

We show that if $\mathcal{ℒ}$ is a codimension-one lamination in a finite volume hyperbolic $3$–manifold such that the principal curvatures of each leaf of $\mathcal{ℒ}$ are all in the interval $\left(-\delta ,\delta \right)$ for a fixed $\delta$ with $0\le \delta <1$ and no complementary region of $\mathcal{ℒ}$ is an interval bundle over a surface, then each boundary leaf of $\mathcal{ℒ}$ has a nontrivial fundamental group. We also prove existence of a fixed constant ${\delta }_0>0$ such that if $\mathcal{ℒ}$ is a codimension-one lamination in a finite volume hyperbolic $3$–manifold such that the principal curvatures of each leaf of $\mathcal{ℒ}$ are all in the interval $\left(-{\delta }_0,{\delta }_0\right)$ and no complementary region of $\mathcal{ℒ}$ is an interval bundle over a surface, then each boundary leaf of $\mathcal{ℒ}$ has a noncyclic fundamental group.