High Codimension Mean Curvature Flow with Surgery

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 $\mathbb{S}^1$.

💡 Research Summary

This paper presents the groundbreaking construction of a mean curvature flow with surgery for submanifolds of arbitrary codimension, providing the first method to flow through singularities with topological control beyond the hypersurface case.

The theory applies to closed, smoothly immersed submanifolds satisfying a quadratic 2-convexity pinching condition: |A|² < 1/(n-2) |H|². This condition serves as the high-codimension analogue of 2-convexity for hypersurfaces and is preserved under the mean curvature flow for dimensions n ≥ 8, as shown by Andrews and Baker. The main theorem establishes the existence of a mean curvature flow with surgery starting from any such submanifold, which terminates after finitely many steps.

The core challenge in high codimension is the complexity of the second fundamental form, including a non-flat normal bundle, components orthogonal to the mean curvature direction (denoted A⁻), and intricate reaction terms in its evolution equation. To overcome this, the authors develop a suite of new, uniform a priori estimates that form the analytical backbone of the surgery construction:

1. A scale-invariant pointwise gradient estimate for the second fundamental form, enabling controlled comparison of curvature even near surgery regions.
2. A planarity estimate for flows with surgery, proving that at points of high curvature, the flow becomes asymptotically codimension one (i.e., A⁻ is of lower order). This is complemented by a planarity improvement theorem, which ensures that just before each surgery, the submanifold is close to a hypersurface with much stronger quantitative control, guaranteeing the estimate remains uniform globally in time.
3. Cylindrical estimates quantifying how close high-curvature regions are to a cylinder Sⁿ⁻¹ × ℝ after rescaling.
4. A neck detection lemma, which uses the cylindrical estimates to prove that near a potential singular time, either the entire submanifold is a positively curved sphere, or it contains large, standard neck regions where surgery can be performed.
5. A neck continuation theorem, characterizing the geometry when moving out of a neck: the curvature either drops by a fixed factor or the neck eventually closes into a positively curved cap.

Using these tools, the authors detail a surgery algorithm. When the maximum curvature reaches a threshold, the flow is stopped. The neck detection lemma identifies suitable neck regions. A standard surgery procedure then removes the central part of each neck and replaces it with two convex caps. The flow is restarted on the remaining components. This process is repeated and is shown to terminate finitely, with all remaining components being diffeomorphic to either spheres or standard necks (cylinders with caps).

A major topological consequence is a complete classification (Corollary 1.2): Every closed n-manifold (n≥8) admitting a smooth immersion into some Euclidean space satisfying the quadratic 2-convexity condition is diffeomorphic either to the sphere Sⁿ or to a finite connected sum of Sⁿ⁻¹-bundles over S¹. This includes both the trivial product Sⁿ⁻¹ × S¹ and a single non-trivial twisted bundle, revealing a richer topology than in the codimension-one case. This result is sharp, as the pinching constant 1/(n-2) is the threshold that allows neck-pinch singularities but rules out more complex bubble-sheet singularities.

The paper discusses the necessity of the dimension restriction (n≥8), obstacles in lower dimensions, and outlines future directions for studying flows under other quadratic pinching conditions (e.g., quadratic k-convexity).