Modified mean curvature flow and CMC foliation conjecture in almost Fuchsian manifolds

There has been a conjecture, often attributed to Thurston, which asserts that every almost Fuchsian manifold is foliated by closed incompressible constant mean curvature (CMC) surfaces. In this paper, for a certain class of almost Fuchsian manifolds, we prove the long-time existence and convergence of the modified mean curvature flow $$\frac{\mathrm{\partial }F}{\mathrm{\partial }t}=-(H-c)\vec{\nu },$$ which was first introduced by Xiao and the second named author in \cite{LX12}. As an application, we confirm Thurston’s CMC foliation conjecture for such a subclass of almost Fuchsian manifolds.