High Codimension Mean Curvature Flow with Surgery

Abstract

We construct a mean curvature flow with surgery for submanifolds of arbitrary codimension. The theory applies to closed submanifolds satisfying a natural quadratic pinching condition, which serves as the high-codimension analogue of 2-convexity and is preserved under the flow in dimensions $n\ge 8$. Our results therefore are in line with the current state-of-the-art in codimension one (where at present 2-convexity is required for surgery). Central to our analysis is a collection of new a priori estimates for the second fundamental form, uniform across surgeries, which yield a precise description of high-curvature regions and permit controlled surgeries. This provides the first notion of mean curvature flow through singularities with topological control in higher codimensions. As a consequence we obtain a sharp classification: Every closed quadratically 2-convexity submanifold is diffeomorphic either to ${\mathbb{S}}^n$ or to a finite connected sum of ${\mathbb{S}}^{n-1}$-bundles over ${\mathbb{S}}^1$.

This replaces an earlier version by the second author and also subsumes the paper arXiv:1805.11808 [math.DG]. It contains a number of new estimates and results