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

TL;DR

This paper studies the mean curvature flow of closed submanifolds in complex projective space, showing that under certain conditions, they either shrink to a point or converge to a totally geodesic submanifold, depending on dimension and pinching conditions.

Contribution

It extends previous results on mean curvature flow from spheres to complex projective spaces, identifying conditions for finite-time shrinking or infinite-time convergence.

Findings

Submanifolds with small codimension and pinched second fundamental form either shrink to a point or converge to a geodesic submanifold.

The convergence to a totally geodesic submanifold occurs only in even dimensions.

Results generalize earlier work by Huisken and Baker to complex projective spaces.

Abstract

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