Some aspects of Ricci flow on the 4-sphere

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 standard 4-sphere. We obtain a gap theorem, for Yamabe positive scalar curvature $L^2$ norm Weyl tensor metric suitably small, establish monotonic decay $L^p$ certain $p>2$ reduced along normalized flow, converging exponentially to