  We consider the problem of deforming a one-parameter family of hypersurfaces
immersed into closed Riemannian manifolds with positive curvature operator. The
hypersurface in this family satisfies mean curvature flow while the ambient
metric satisfying the normalized Ricci flow. We prove that if the initial
metric of the background manifold is sufficiently pinched and the initial
hypersurface also satisfies a suitable pinching condition, then either the
hypersurfaces shrink to a round point in finite time or converge to a totally
geodesic sphere as the time tends to infinity.
