Mean Curvature Flows of Closed Hypersurfaces in Warped Product Manifolds

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 $\mathrm{\Sigma }$, such that the mean curvature flow starting from $S_0$ exists for all time and converges to $\mathrm{\Sigma }$.