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

Abstract

We study mean curvature flow of $n$-dimensional submanifolds of $S_K^{n+\ell }$, the round $(n+\ell )$-sphere of sectional curvature $K>0$, under the quadratic curvature pinching condition $|A{|}^2<\frac{1}{n-2}|H{|}^2+4K$ when $n\ge 8$, $|A{|}^2<\frac{4}{3n}|H{|}^2+\frac{n}{2}K$ when $n=7$, and $|A{|}^2<\frac{3(n+1)}{2n(n+2)}|H{|}^2+\frac{2n(n-1)}{3(n+1)}K$ when $n=5$ or $6$. This condition is related to a theorem of Li and Li [Arch. Math., 58:582--594, 1992] which states that the only $n$-dimensional minimal submanifolds of $S_K^{n+\ell }$ satisfying $|A{|}^2<\frac{2n}{3}K$ are the totally geodesic $n$-spheres. We prove the existence of a suitable mean curvature flow with surgeries starting from initial data satisfying the pinching condition. As a result, we conclude that any smoothly, properly immersed submanifold of $S_K^{n+1}$ satisfying the pinching condition is diffeomorphic either to the sphere $S^n$ or to the connected sum of a finite number of handles $S^1\times S^{n-1}$. The results are sharp when $n\ge 8$ due to hypersurface counterexamples.

arXiv admin note: text overlap with arXiv:2006.08049

References in corpus (1)

• Positively curved surfaces in the three-sphere