Mean curvature flow of pinched submanifolds of CPⁿ

We consider the evolution by mean curvature flow of a closed submanifold of complex projective space. We show that, if the submanifold has small and satisfies a suitable pinching condition on the second form, then the evolution has two possible behaviors: either the shrinks to a round point in finite time, or it converges smoothly a totally geodesic limit in infinite time. The latter behavior is only if the dimension is even. These results generalize previous works by and Baker on the mean curvature flow of submanifolds of the sphere.