  Some new differentiable sphere theorems are obtained via the Ricci flow and
stable currents. We prove that if $M^n$ is a compact manifold whose normalized
scalar curvature and sectional curvature satisfy the pointwise pinching
condition $R_0>\sigma_{n}K_{\max}$, where $\sigma_n\in (\frac{1}{4},1)$ is an
explicit positive constant, then $M$ is diffeomorphic to a spherical space
form. This gives a partial answer to Yau's conjecture on pinching theorem.
Moreover, we prove that if $M^n(n\geq3)$ is a compact manifold whose $(n-2)$-th
Ricci curvature and normalized scalar curvature satisfy the pointwise condition
$Ric^{(n-2)}_{\min}>\tau_n(n-2)R_0,$ where $\tau_n\in (\frac{1}{4},1)$ is an
explicit positive constant, then $M$ is diffeomorphic to a spherical space
form. We then extend the sphere theorems above to submanifolds in a Riemannian
manifold. Finally we give a classification of submanifolds with weakly pinched
curvatures, which improves the differentiable pinching theorems due to Andrews,
Baker and the authors.
