Volume estimates and convergence results for solutions to Ricci flow with $L^{p}$ bounded scalar curvature
Jiawei Liu, Miles Simon
TL;DR
The paper analyzes $n$-dimensional Ricci flows on closed manifolds with finite-time singularities under a uniform $L^p$ bound on the scalar curvature for $p> frac{n}{2}$. It develops non-collapsing and non-inflating estimates, combines them with prior integral-curvature bounds to obtain strengthened spacetime control of the Ricci curvature, and proves local versions of these results. In dimension four, these tools yield convergence to a $C^0$ Riemannian orbifold as $t o T$, with the flow extendable past $T$ via the Orbifold Ricci flow for a short time; the paper also provides local heat-flow estimates in a $c/t$ regime and supplies harmonic-coordinate and neck-lemma apparatus in the appendices to support the orbifold convergence analysis. Collectively, the results illuminate the structure of singular regions under $L^{p}$ curvature bounds and enable a controlled extension of Ricci flow beyond potential singular times in the 4D closed setting.
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+σ)$ for some $σ>0.$ We also prove local versions of many of the results mentioned above.