Quadratically pinched hypersurfaces of the sphere via mean curvature flow with surgery

TL;DR

This paper investigates mean curvature flow with surgery for hypersurfaces in spheres under a quadratic curvature pinching condition, leading to classification results and demonstrating the flow's convergence to either convex or cylindrical shapes.

Contribution

It introduces a new quadratic curvature pinching condition for hypersurfaces in spheres, proves its preservation under mean curvature flow, and applies surgery techniques to classify the hypersurfaces.

Findings

Flow becomes either convex or cylindrical in high curvature regions.

Hypersurfaces satisfying the pinching condition are topologically spheres or connected sums of spheres and tori.

Results are sharp for dimensions n ≥ 4.

Abstract

We study mean curvature flow in $\mathbb S_K^{n+1}$, the round sphere of sectional curvature $K>0$, under the quadratic curvature pinching condition $|A|^{2} < \frac{1}{n-2} H^{2} + 4 K$ when $n\ge 4$ and $|A|^{2} < \frac{3}{5}H^{2}+\frac{8}{3}K$ when $n=3$. This condition is related to a famous theorem of Simons, which states that the only minimal hypersurfaces satisfying $\vert A\vert^2<nK$ are the totally geodesic hyperspheres. It is related to but distinct from two-convexity. Notably, in contrast to two-convexity, it allows the mean curvature to change sign. We show that the pinching condition is preserved by mean curvature flow, and obtain a cylindrical estimate and corresponding pointwise derivative estimates for the curvature. As a result, we find that the flow becomes either uniformly convex or quantitatively cylindrical in regions of high curvature. This allows us to apply the surgery apparatus developed by Huisken and Sinestrari. We conclude that any smoothly, properly, isometrically immersed hypersurface $\mathcal{M}$ of $\mathbb S_K^{n+1}$ satisfying the pinching condition is diffeomorphic to $\mathbb S^n$ or the connected sum of a finite number of copies of $\mathbb S^1\times \mathbb S^{n-1}$. If $\mathcal M$ is embedded, then it bounds a 1-handlebody. The results are sharp when $n\ge 4$.