Mean curvature flow for pinched submanifolds in rank one symmetric spaces

Abstract

G. Pipoli and C. Sinestrari considered the mean curvature flow starting from a closed submanifold in the complex projective space. They proved that if the submanifold is of small codimension and satisfies a suitable pinching condition for the second fundamental form, then the flow has two possible behaviors: either the submanifold collapses to a round point in finite time, or it converges smoothly to a totally geodesic submanifold in infinite time. In this paper, we prove the similar results for the mean curvature flow starting from pinched closed submanifolds in (general) rank one symmetric spaces of compact type. Also, we prove that closed submanifolds in (general) rank one symmetric spaces of non-compact type collapse to a round point along the mean curvature flow under certain strict pinching condition for the norm of the second fundamental form.