Title: Mean Curvature Flows of Closed Hypersurfaces in Warped Product Manifolds,
Abstract: We investigate the mean curvature flows in a class of warped product
manifolds with closed hypersurfaces fibering over $\mathbb{R}$. In particular,
we prove that under natural conditions on the warping function and Ricci
curvature bound for the ambient space, there exists a large class of closed
initial hypersurfaces, as geodesic graphs over the totally geodesic
hypersurface $\Sigma$, such that the mean curvature flow starting from $S_0$
exists for all time and converges to $\Sigma$.