Extending four-dimensional Ricci flows with bounded scalar curvature

We consider solutions $(M, g(t)), 0 \leq t \lt T$, to Ricci flow on compact, connected four dimensional manifolds without boundary. We assume that the scalar curvature is bounded uniformly, and that $T \lt \infty$. In this case, we show that the metric space $(M, d(t))$ associated to $(M, g(t))$ converges uniformly in the $C^0$ sense to $(X, d)$, as $t \nearrow T$, where $(X, d)$ is a $C^0$ Riemannian orbifold with at most finitely many orbifold points. Estimates on the rate of convergence near and away from the orbifold points are given. We also show that it is possible to continue the flow past $(X, d)$ using the orbifold Ricci flow.

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Received 18 July 2016

Accepted 4 February 2018

Published 7 December 2020