Long-time analysis of 3 dimensional Ricci flow III

TL;DR

This paper proves that correctly performed surgeries in 3D Ricci flow lead to finitely many surgeries and bounded curvature over time, confirming Perelman's conjecture and describing the long-term geometric behavior.

Contribution

It removes the topological condition previously required, generalizing results to all closed 3-manifolds using a new area evolution estimate.

Findings

Finitely many surgeries occur under correct procedures.

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

Provides a qualitative description of geometry as t approaches infinity.

Abstract

In this paper we analyze the long-time behavior of 3 dimensional Ricci flows with surgery. Our main result is that if the surgeries are performed correctly, then only finitely many surgeries occur and after some time the curvature is bounded by $C t^{-1}$. This result confirms a conjecture of Perelman. In the course of the proof, we also obtain a qualitative description of the geometry as $t \to \infty$. This paper is the third part of a series. Previously, we had to impose a certain topological condition $\mathcal{T}_2$ to establish the finiteness of the surgeries and the curvature control. The objective of this paper is to remove this condition and to generalize the result to arbitrary closed 3-manifolds. This goal is achieved by a new area evolution estimate for minimal simplicial complexes, which is of independent interest.