A sharp convergence theorem for the mean curvature flow in the sphere

In this paper, we prove a sharp convergence theorem for the mean curvature flow of arbitrary codimension in the sphere Sn+p(1/root K). Note that Baker proved a convergence theorem for the mean curvature flow in the sphere under the pinching condition vertical bar A vertical bar(2) <= 1/n-1 vertical bar H vertical bar(2) + 2 (K) over bar, where A is the second fundamental form and H is themean curvature vector. Baker's theorem is already optimal in the sense that it is the best linear relationship between vertical bar A vertical bar(2), vertical bar H vertical bar(2) and (K) over bar. We show that if the initial submanifold satisfies some optimal nonlinear relationships between vertical bar A vertical bar(2), vertical bar H vertical bar(2) and (K) over bar, then the mean curvature flow either converges to a round point in finite time, or converges to a totally geodesic sphere of Sn+ p(1/root(K) over bar) as t -> infinity. This extends and improves the results due to Baker and Baker-Nguyen. In particular, we also obtain a new differentiable sphere theorem for submanifolds in the sphere.