  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 $$\frac{\partial F}{\partial t}=-(H-c)\vec{\nu},$$
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.
