Volume estimates and convergence results for solutions to Ricci flow with $L^{p}$ bounded scalar curvature

Abstract

In this paper we study $n$-dimensional Ricci flows $(M^n,g(t))_{t\in [0,T)},$ where $T< \infty$ is a potentially singular time, and for which the spatial $L^p$ norm, $p>\frac n 2$, of the scalar curvature is uniformly bounded on $[0,T).$ In the case that $M$ is closed and four dimensional, we explain why non-collapsing estimates hold and how they can be combined with integral bounds on the Ricci and full curvature tensor of the prequel paper of the authors, as well as non-inflating estimates (already known due to works of Bamler), to obtain an improved space time integral bound of the Ricci curvature. As an application of these estimates, we show that if we further restrict to $n=4$, then the solution convergences to an orbifold as $t \to T$ and that the flow can be extended using the Orbifold Ricci flow to the time interval $ [0,T+\sigma)$ for some $\sigma>0.$ We also prove local versions of many of the results mentioned above.