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

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 (Ann Math 2(88):62–105, 1968), 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 (Invent Math 175(1):137–221, 2009) (cf. Haslhofer and Kleiner, Duke Math J 166(9):1591–1626, 2017). We conclude that any smoothly, properly immersed hypersurface \({\mathcal {M}}\) of \({\mathbb {S}}_K^{n+1}\) satisfying the pinching condition is diffeomorphic to \({\mathbb {S}}^n\) or to 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 \(n\ge 4\).