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 $t \rightarrow \infty$. 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}}.
Categories: math.DG