Ricci curvature integrals, local functionals, and the Ricci flow

Abstract

Consider a Riemannian manifold $(M^m,g)$ whose volume is the same as the standard sphere $(S^m,g_{round})$. If $p>\frac{m}{2}$ and ${\int }_M{\left\{Rc-(m-1)g\right\}}_{-}^{p}dv$ is sufficiently small, we show that the normalized Ricci flow initiated from $(M^m,g)$ will exist immortally and converge to the standard sphere. The choice of $p$ is optimal.