  We study global aspects of the mean curvature flow of non-separating
hypersurfaces $S$ in closed manifolds. For instance, if $S$ has non-vanishing
mean curvature, we show its level set flow converges smoothly towards an
embedded minimal hypersurface $\Gamma$. We prove a similar result for the flow
with surgery in dimension 2. As an application we show the existence of
monotone incompressible isotopies in manifolds with negative curvature.
Combining this result with min-max theory, we show that quasi-Fuchsian and
hyperbolic $3$-manifolds fibered over $\mathrm{S}^1$ admit smooth entire
foliations whose leaves are either minimal or have non-vanishing mean
curvature. We also conclude the existence of outermost minimal surfaces for
quasi-Fuchsian ends and study their continuity with respect to variations of
the quasi-Fuchsian metric.
