# Long-time analysis of 3 dimensional Ricci flow

# Long-time analysis of 3 dimensional Ricci flow

It is still an open problem how Perelman's Ricci flow with surgeries behaves for large times. For example, it is unknown whether surgeries eventually stop to occur and whether the full geometric decomposition of the underlying manifold is exhibited by the flow as $t \to \infty$. In this talk, I will present new tools to treat this question after providing a quick review of Perelman's results. In particular, I will show that in the case in which the initial manifold satisfies a certain purely topological condition, surgeries do in fact stop to occur after some time and the curvature is globally bounded by $C t^{-1}$. For example, this condition is satisfied by manifolds of the form $\Sigma \times S^1$ where $\Sigma$ is a surface of genus $\geq 1$.