Long-time analysis of 3 dimensional Ricci flow II

TL;DR

This paper extends the analysis of 3D Ricci flows with surgery, showing finite surgeries and curvature bounds for broader initial topologies, partially answering an open question in geometric analysis.

Contribution

It generalizes previous methods to handle more complex initial topologies like 3-torus and surface products, establishing long-time behavior results.

Findings

Finite number of surgeries occur under the new topological conditions.

Curvature is bounded by Ct^{-1} after some time.

Provides a description of large-time geometry even when initial conditions are violated.

Abstract

This is the second part of a series of papers analyzing the long-time behaviour of 3 dimensional Ricci flows with surgery. We generalize the methods developed in the first part and use them to treat cases in which the initial manifold satisfies a certain purely topological condition which is far more general than the one that we previously had to impose. Amongst others, we are able to treat initial topologies such as the 3-torus or $\Sigma \times S^1$ where $\Sigma$ is any surface of genus $\geq 1$. We prove that under this condition, only finitely many surgeries occur and that after some time the curvature is bounded by $C t^{-1}$. This partially answers an open question in Perelman's work, which was made more precise by Lott and Tian. In the process of the proof, we also find an interesting description of the geometry at large times, which even holds when the condition on the initial topology is violated. The methods presented in this paper will be refined to treat a more general case in a subsequent paper.