Mean curvature flow of arbitrary codimension in complex projective spaces

Abstract

In this paper, we investigate the mean curvature flow of submanifolds of arbitrary codimension in $\mathbb{C}{\mathbb{P}}^m$. We prove that if the initial submanifold satisfies a pinching condition, then the mean curvature flow converges to a round point in finite time, or converges to a totally geodesic submanifold as $t\to \infty$. Consequently, we obtain a new differentiable sphere theorem for submanifolds in $\mathbb{C}{\mathbb{P}}^m$. Our work improves the convergence theorem for mean curvature flow due to Pipoli and Sinestrari {\cite{PiSi2015}}.

31 pages