The behaviour of the mean curvature flow for pinched submanifolds in rank one symmetric spaces

Abstract

In this paper, we consider the mean curvature flow starting from closed submanifolds in rank one symmetric spaces satisfying some pinching condition for the norm of the second fundamental form. We prove that, under some additional condition, the closed submanifold satisfying the pinching condition collapses to a round point in finite time or converges to a totally geodesic submanifold in infinite time along the mean curvature flow.