Mean curvature flow for pinched submanifolds in rank one symmetric spaces

TL;DR

This paper extends the analysis of mean curvature flow for pinched submanifolds from complex projective spaces to all rank one symmetric spaces, showing similar collapse or convergence behaviors under pinching conditions.

Contribution

It generalizes previous results to all rank one symmetric spaces, both compact and non-compact, under suitable pinching conditions for the second fundamental form.

Findings

Submanifolds in compact rank one symmetric spaces either collapse or converge smoothly.

Submanifolds in non-compact rank one symmetric spaces collapse to a round point.

Pinching conditions determine the flow's long-term behavior.

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.