Deforming hypersurfaces of the sphere by their mean curvature

is satisfied. In [6] we studied hypersurfaces moving along their mean curvature vector in a general Riemannian manifold N" + 1. It was shown that all hypersurfaces Mo satisfying a suitable convexity condition will contract to a single point in finite time during this evolution. Here we want to show that in a spherical spaceform some convergence results can be obtained without assuming convexity for the initial hypersurface Mo. In particular, we will see that some hypersurfaces do not contract during this flow, but straighten out and become totally geodesic, i.e. in case N" + 1 = S" + 1 they converge to a "big S" ". To be precise, let g = {gij} and A = {hi j} be the induced metric and the second fundamental form on M and denote by H = giih~j, [A 12= h~Jh~j the mean curvature and the squared norm of the second fundamental form respectively.