On Almost-Fuchsian Manifolds

TL;DR

This paper studies almost-Fuchsian manifolds, a special class of hyperbolic 3-manifolds with unique minimal surfaces, providing geometric estimates and constructing examples with multiple minimal surfaces.

Contribution

It offers new bounds on volume and limit set dimension, and constructs examples with multiple minimal surfaces, advancing understanding of almost-Fuchsian manifolds.

Findings

Upper bound for hyperbolic volume of the convex core

Upper bound on Hausdorff dimension of the limit set

Existence of quasi-Fuchsian manifolds with multiple minimal surfaces

Abstract

Almost-Fuchsian manifold is a class of complete hyperbolic three manifolds. Such a three-manifold is a quasi-Fuchsian manifold which contains a closed incompressible minimal surface with principal curvatures everywhere in the range of (-1, 1). In such a manifold, the minimal surface is unique and embedded, hence one can parametrize these hyperbolic three-manifolds by their minimal surfaces. In this paper we obtain estimates on several geometric and analytical quantities of an almost-Fuchsian manifold M in terms of the data on the minimal surface. In particular, we obtain an upper bound for the hyperbolic volume of the convex core of M, and an upper bound on the Hausdor? dimension of the limit set associated to M. We also constructed a quasi-Fuchsian manifold which admits more than one minimal surface, and it does not admit a foliation of closed surfaces of constant mean curvature.