Lei–Xu's optimal pinching conjecture for mean curvature flow in spheres

Let $M_0$ be an $n$-dimensional complete submanifold of the sphere $\mathbb{S}^{n+p}(1/\sqrt{\bar K})$, where $\bar K>0$ is the sectional curvature of the ambient sphere. Write $h$ for the second fundamental form, $H$ for the mean curvature vector, and let $\alpha(|H|^2)$ denote the pinching function introduced in the paper. The mean curvature flow starts from $M_0$.

Lei–Xu's conjecture. If

$$\sup_{M_0}\bigl(|h|^2-\alpha(|H|^2)\bigr)<0,$$

then the mean curvature flow with initial value $M_0$ converges to a round point in finite time, or converges to a totally geodesic sphere as $t\to\infty$. In particular, if

$$|A|^2<2\sqrt{n-1}\,\bar K,$$

then $M_0$ is diffeomorphic to $\mathbb{S}^n$.

This conjecture proposes that $\alpha(|H|^2)$ is the optimal pinching condition for mean curvature flow in spheres, extending rigidity and topological sphere theorems. The supplied text does not establish the conjecture or provide evidence that it has been resolved.