  Let $M$ be a quasi-Fuchsian three-manifold that contains a closed
incompressible surface with principal curvatures within the range of the unit
interval, for a prescribed function $H$ (with mild conditions) on $M$, we
construct a closed incompressible surface with mean curvature $H$ . A direct
application is the existence of embedded surfaces of prescribed constant mean
curvatures with constants in $(-2,2)$.
