Beyond almost Fuchsian space
Zheng Huang, Ben Lowe
TL;DR
We study complete hyperbolic 3-manifolds diffeomorphic to $S\times \mathbb{R}$ that admit minimal surfaces with small principal curvatures, focusing on almost Fuchsian and weakly almost Fuchsian spaces. The authors prove that every weakly almost Fuchsian manifold is geometrically finite and develop a Canary–Storm type compactification of the weakly AF space using data from the unique minimal surface; from this, they derive uniform bounds on the convex core volume and on the Hausdorff dimension of the limit set, and establish a gap theorem for principal curvatures in certain fibered manifolds, while also producing quasi-Fuchsian manifolds with a unique stable minimal surface that are not weakly AF. They further construct a robust framework for extending these results beyond the weakly AF setting, including analysis of algebraic vs geometric limits in sequences with curvature bounds near or above 1 and the existence of unique-minimal-surface quasi-Fuchsian examples. Altogether, the work connects minimal-surface theory with global Kleinian geometry, providing compactifications and quantitative controls that illuminate the boundary between AF, weakly AF, and broader Kleinian spaces.
Abstract
An almost Fuchsian manifold is a hyperbolic 3-manifold of the type $S\times \mathbb{R}$ which admits a closed minimal surface (homeomorphic to $S$) with the maximum principal curvature $λ_0 <1$, while a weakly almost Fuchsian manifold is of the same type but it admits a closed minimal surface with $λ_0 <= 1$. We first prove that any weakly almost Fuchsian manifold is geometrically finite, and we construct a Canary-Storm type compactification for the weakly almost Fuchsian space. We use this to prove uniform upper bounds on the volume of the convex core and Hausdorff dimension for the limit set of weakly almost Fuchsian manifolds, and to prove a gap theorem for the principal curvatures of minimal surfaces in hyperbolic 3-manifolds that fiber over the circle. We also give examples of quasi-Fuchsian manifolds which admit unique stable minimal surfaces without being weakly almost Fuchsian.