  In a previous paper, we proved a number of optimal rigidity results for
Riemannian manifolds of dimension greater than four whose curvature satisfy an
integral pinching. In this article, we use the same integral Bochner technique
to extend the results in dimension three. Then, by using the classification of
closed three-manifolds with nonnegative scalar curvature and a few topological
considerations, we deduce optimal sphere theorems for three-dimensional
manifolds with integral pinched curvature.
