Xu–Zhao diffeomorphic sphere conjecture for submanifolds

Let $M$ be a smooth closed $n$-dimensional submanifold, with $n\geq2$, in the space form $\mathbb{F}^{n+d}(c)$, where $c\geq0$. Let $\mathring{A}$ denote its traceless second fundamental form and $\|\mathring{A}\|_{L^n}$ its $L^n$-norm. Xu–Zhao's diffeomorphic sphere conjecture. There exists a positive constant $C_n$, depending only on $n$, such that

$$\|\mathring{A}\|_{L^n}<C_n$$

implies that $M$ is diffeomorphic to the standard sphere $\mathbb{S}^n$. The conjecture strengthens the known result that the corresponding smallness condition implies only that $M$ is homeomorphic to a sphere; the paper obtains a partial result in the unit sphere, while the full assertion remains open.