Some new differentiable sphere theorems are obtained via the
Ricci flow and stable currents. We prove that if Mn is a compact
manifold whose normalized scalar curvature and sectional curvature
satisfy the pointwise pinching condition R0 > nKmax, where
n 2 ( 1
4 , 1) is an explicit positive constant, then M is diffeomorphic
to a spherical space form. We also provide a partial answer
to Yau’s conjecture on the pinching theorem. Moreover, we prove
that if Mn(n 3) is a compact manifold whose (n . 2)-th Ricci
curvature and normalized scalar curvature satisfy the pointwise
condition Ric(n.2)
min > n(n .2)R0, where n 2 ( 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.