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There has been a conjecture, often attributed to Thurston, which asserts thatevery almost Fuchsian manifold is foliated by closed incompressible constantmean curvature (CMC) surfaces. In this paper, for a certain class of almostFuchsian manifolds, we prove the long-time existence and convergence of themodified 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 asubclass of almost Fuchsian manifolds.

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