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 $n\geq 8$, $|A|^{ 2} < \frac{ 4}{ 3n}|H|^{ 2}+\frac{ n}{ 2}K$ 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 $n\geq 8$ due to hypersurface counterexamples.
Categories: math.DG