  In this paper, on 4-spheres equipped with Riemannian metrics we study some
integral conformal invariants, the sign and size of which under Ricci flow
characterize the standard 4-sphere. We obtain a conformal gap theorem, and for
Yamabe metrics of positive scalar curvature with $L^2$ norm of the Weyl tensor
of the metric suitably small, we establish the monotonic decay of the $L^p$
norm for certain $p>2$ of the reduced curvature tensor along the normalized
Ricci flow, with the metric converging exponentially to the standard 4-sphere.
