  Mean curvature flow for isoparametric submanifolds in Euclidean spaces and
spheres was studied by the authors in [LT]. In this paper, we will show that
all these solutions are ancient solutions. We also discuss rigidity of ancient
mean curvature flows for hypersurfaces in spheres and its relation to the
Chern's conjecture on the norm of the second fundamental forms of minimal
hypersurfaces in spheres.
