Beyond almost Fuchsian space

TL;DR

This paper investigates the geometric properties of weakly almost Fuchsian hyperbolic 3-manifolds, establishing finiteness, compactification, and bounds on volume and dimension, and exploring minimal surface stability and uniqueness.

Contribution

It proves that weakly almost Fuchsian manifolds are geometrically finite, constructs a compactification, and derives bounds on volume and Hausdorff dimension, also analyzing minimal surface stability.

Findings

Weakly almost Fuchsian manifolds are geometrically finite.

Established a Canary-Storm type compactification for these manifolds.

Derived uniform bounds on convex core volume and limit set Hausdorff dimension.

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 $\lambda_0 <1$, while a weakly almost Fuchsian manifold is of the same type but it admits a closed minimal surface with $\lambda_0 <= 1$. 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.