High Codimension Mean Curvature Flow in the Complex Projective Space

We study the mean curvature flow of smooth, compact submanifolds of high codimension with quadratic pinching in CPᵏ C P k . We establish a codimension estimate, where in regions of high curvature, the submanifold becomes approximately codimension one, in a quantifiable way. Under a cylindrical type pinching, this limiting flow is weakly convex and moves by translation or is a self shrinker. These estimates allow us to analyse the behaviour of the flow near singularities and establish the existence of the limiting flow. With a decay estimate, the rescaling converges smoothly to a totally geodesic limit in infinite time, without using Stampacchia iteration or integral analysis.