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

We study mean curvature flow of $n$-dimensional submanifolds of $S_K^{n+\mathrm{\ell }}$, the round $(n+\mathrm{\ell })$-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 $\mathrm{\mid }A{\mathrm{\mid }}^2<\frac{3(n+1)}{2n(n+2)}\mathrm{\mid }H{\mathrm{\mid }}^2+\frac{2n(n-1)}{3(n+1)}K$ 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 $S_K^{n+\mathrm{\ell }}$ satisfying $\mathrm{\mid }A{\mathrm{\mid }}^2<\frac{2n}{3}K$ 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 $S^1\times S^{n-1}$\mathrm{.}Theresultsaresharpwhen$n\geq 8$ due to hypersurface counterexamples.