Some sharp differential sphere theorems for nonnegative scalar curvature manifolds

In this paper, we obtain several new intrinsic and extrinsic differential sphere theorems via Ricci flow. For intrinsic case, we show that a closed simply connected $n(\ge 4)$-dimensional Riemannian manifold $M$ is diffeomorphic to $S^n$ if one of the following conditions holds pointwisely: $$(i)\ R_0>\left(1-\frac{24(\sqrt{10}-3)}{n(n-1)}\right)K_{max};\quad \ (ii)\ \frac{Ric^{[4]}}{4(n-1)}>\left(1-\frac{6(\sqrt{10}-3)}{n-1}\right)K_{max}.$$ Here $K_{max}$, $Ric^{[k]}$ and $R_0$ stand for the maximal sectional curvature, the $k$-th weak Ricci curvature and the normalized scalar curvature. For extrinsic case, i.e., when $M$ is a closed simply connected $n(\ge 4)$-dimensional submanifold immersed in $\bar{M}$. We prove that $M$ is diffeomorphic to $S^n$ if it satisfies some pinching curvature conditions. The only involved extrinsic quantities in our pinching conditions are the maximal sectional curvature $\bar K_{max}$ and the squared norm of mean curvature vector $\vert H\vert^2$. More precisely, we show that $M$ is diffeomorphic to $S^n$ if one of the following conditions holds: \begin{itemize} \item[(1)] $R_0\ge \left(1-\frac{2}{n(n-1)}\right)\bar{K}_{max} +\frac{n(n-2)}{(n-1)^2}\vert H\vert^2$, and strict inequality is achieved at some point; \item[(2)] $\dfrac{Ric^{[2]}}{2}\ge (n-2)\bar K_{max}+\frac{n^2}{8}\vert H\vert^2,$ and strict inequality is achieved at some point; \item[(3)] $\dfrac{Ric^{[2]}}{2} \ge\frac{n(n-3)}{n-2}\left(\bar K_{max}+\vert H\vert^2\right),$ and strict inequality is achieved at some point. \end{itemize} It is worth pointing out that, in the proof of extrinsic case, we apply suitable complex orthonormal frame and simplify the calculations considerably. We also emphasize that both of the pinching constants in (2) and (3) are optimal for $n=4$.