Ricci flow with surgery on three-manifolds

TL;DR

Perelman's work develops Ricci flow with surgery on 3-manifolds, introducing a $\delta$-cutoff procedure to perform controlled surgeries and preserve pinching while maintaining canonical neighborhoods. It combines analyses of ancient $\kappa$-solutions, the standard solution, and first singular-time structure to build a framework for continuing the flow with surgery and for long-time curvature control. The long-time behavior is elaborated through a thick-thin decomposition, yielding a geometrization-like picture with hyperbolic pieces and graph manifolds, supported by first-eigenvalue considerations of $-4\Delta + R$. The paper also corrects earlier statements and provides a pathway toward a canonical Ricci flow on a maximal space-time subset, bridging topology and geometric analysis.

Abstract

This is a technical paper, which is a continuation of math.DG/0211159. Here we construct Ricci flow with surgeries and verify most of the assertions, made in section 13 of that e-print; the exceptions are (1) the statement that manifolds that can collapse with local lower bound on sectional curvature are graph manifolds - this is deferred to a separate paper, since the proof has nothing to do with the Ricci flow, and (2) the claim on the lower bound for the volume of maximal horns and the smoothness of solutions from some time on, which turned out to be unjustified and, on the other hand, irrelevant for the other conclusions.