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

Abstract

We study mean curvature flow of n-dimensional submanifolds of S n+ℓ K , the round (n + ℓ)-sphere of sectional curvature K > 0, under the quadratic curvature pinching condition3(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 n+ℓ K satisfying |A| 2 < 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 n+1 K 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 × S n−1 . The results are sharp when n ≥ 8 due to hypersurface counterexamples.