Ricci curvature integrals, local functionals, and the Ricci flow

TL;DR

This paper proves that if a Riemannian manifold's Ricci curvature is close to that of a sphere in an integral sense, then the normalized Ricci flow will exist forever and converge to the standard sphere, with optimal conditions on the integrability parameter.

Contribution

It establishes optimal integral conditions on Ricci curvature that guarantee convergence of the Ricci flow to a sphere, extending previous pointwise results.

Findings

Normalized Ricci flow exists eternally under small integral Ricci curvature deviations.

Flow converges to the standard sphere when initial conditions are sufficiently close.

Optimality of the integrability parameter p is demonstrated.

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.