Quasi-Fuchsian manifolds close to the Fuchsian locus are foliated by constant mean curvature surfaces

TL;DR

This paper proves that quasi-Fuchsian manifolds near the Fuchsian locus admit a unique monotone foliation by constant mean curvature surfaces, confirming Thurston's conjecture for almost-Fuchsian manifolds.

Contribution

It establishes the existence and uniqueness of monotone CMC foliations for quasi-Fuchsian manifolds close to the Fuchsian locus, advancing understanding of their geometric structure.

Findings

Existence of a unique monotone CMC foliation near the Fuchsian locus

Confirmation of Thurston's conjecture for almost-Fuchsian manifolds

Foliation persists in a small neighborhood of the Fuchsian locus

Abstract

Even though it is known that there exist quasi-Fuchsian hyperbolic three-manifolds that do not admit any monotone foliation by constant mean curvature (CMC) surfaces, a conjecture due to Thurston asserts the existence of CMC foliations for all almost-Fuchsian manifolds, namely those quasi-Fuchsian manifolds that contain a closed minimal surface with principal curvatures in (-1,1). In this paper we prove that there exists a (unique) monotone CMC foliation for all quasi-Fuchsian manifolds that lie in a sufficiently small neighborhood of the Fuchsian locus.