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

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.