Mean curvature flow of pinched submanifolds of $\mathbb{CP}^n$

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