  We prove that if the initial hypersurface of the mean curvature flow in
spheres satisfies a sharp pinching condition, then the solution of the flow
converges to a round point or a totally geodesic sphere. Our result improves
the famous convergence theorem due to Huisken [9]. Moreover, we prove a
convergence theorem under the weakly pinching condition. In particular, we
obtain a classification theorem for weakly pinched hypersurfaces. It should be
emphasized that our pinching condition implies that the Ricci curvature of the
initial hypersurface is positive, but does not imply positivity of the
sectional curvature.
