High Codimension Mean Curvature Flow with Surgery

TL;DR

This paper develops a mean curvature flow with surgery for high-codimension submanifolds satisfying a quadratic pinching condition, enabling topological classification and singularity analysis in higher dimensions.

Contribution

It introduces a new flow with surgery for high-codimension submanifolds, extending techniques from codimension one and providing topological classification results.

Findings

Flow applies to closed submanifolds with quadratic pinching

Provides a priori estimates for second fundamental form during surgeries

Classifies submanifolds as spheres or connected sums of sphere bundles

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 $n \geq 8$. 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 $\mathbb{S}^1$.