Mean Curvature Flow of High Codimension in Complex Projective Space

Abstract

We study the mean curvature flow of smooth $m$-dimensional compact submanifolds with quadratic pinching in the Riemannian manifold $\mathbb{C}P^n$. Our main focus is on the case of high codimension, $k\geq 2$. We establish a codimension estimate that shows in regions of high curvature, the submanifold becomes approximately codimension one in a quantifiable way. This estimate enables us to prove at a singular time of the flow, there exists a rescaling that converges to a smooth codimension-one limiting flow in Euclidean space. Under a cylindrical type pinching, we show that this limiting flow is weakly convex and moves by translation. These estimates allow us to analyse the behaviour of the flow near singularities and establish the existence of the limiting flow. Lastly, we prove a decay estimate that shows that the rescaling converges smoothly to a totally geodesic limit in infinite time. This behaviour is only possible if the dimension of the submanifold is even. Our approach relies on the preservation of the quadratic pinching condition along the flow and a gradient estimate that controls the mean curvature in regions of high curvature. This result generalises the work of Pipoli and Sinestrari on the mean curvature flow of submanifolds of the complex projective space.