Ricci flow with $L^p$ bounded scalar curvature

In this talk, we show that localised, weighted curvature integral estimates for solutions to the Ricci flow in the setting of a smooth four dimensional Ricci flow or a closed $n$-dimensional Kähler Ricci flow always hold. These integral estimates improve and extend the integral curvature estimates shown in an earlier paper by the speaker. If $M^4$ is closed and four dimensional, and the spatial $L^p$ norm of the scalar curvature is uniformly bounded for some $p>2$, for $t\in [0,T),$ $T< \infty$, then we show:
a) a uniform bound on the spatial $L^2$ norm of the Riemannian curvature tensor for $t\in [0,T)$,
b) uniform non-expanding and non-inflating estimates for $t\in [0,T)$,
c) convergence to an orbifold as $t \to T$,
d) existence of an extension of the flow to times $t\in [0,T+\sigma)$ for some $\sigma>0$ using the orbifold Ricci flow.

This is joint work with Jiawei Liu.