Hamilton’s Program in 2002

By the end of the 1990s Hamilton had built, piece by piece, almost everything needed to prove the geometrization conjecture with the Ricci flow. He had the flow and its evolution equations, the maximum principles and pinching, the Harnack inequality, compactness, and the theory of singularities. He had carried out surgery in a four-dimensional setting. And he had shown how the long-time behaviour of the flow would produce the thick–thin decomposition, under an extra assumption. What was missing were a few specific statements, each of which looked out of reach. This chapter sets out the program as it stood in 2002 and lists the gaps precisely, each paired with the chapter of Book 12 where Perelman fills it.

By the end of this chapter you will be able to:

• describe Hamilton's surgery for four-manifolds with positive isotropic curvature as the template for Perelman's;
• state the conclusion and hypothesis of Hamilton's theorem on nonsingular solutions;
• list the four gaps in the program in 2002, and say which part of Perelman's work fills each;
• place the main events from 1982 to 2010 on a timeline.

5.1The record

5.2Surgery in dimension four

A four-manifold has positive isotropic curvature (PIC) if a certain combination of curvatures is positive on every totally isotropic complex 2-plane in the complexified tangent space; for example $S^4$, $S^3\times\mathbb{R}$ and their quotients have it. Hamilton showed in 1997 that the Ricci flow preserves PIC, that high-curvature regions of such a flow are necks $S^3\times\mathbb{R}$ (or caps), and that one can perform surgery: stop the flow, cut along the central $S^3$ of each sufficiently thin neck, glue in standard caps, and continue. The pinching of the curvature towards a cylindrical shape in necks was enforced by the curvature conditions themselves. The intended conclusion, for compact four-manifolds with PIC and no essential incompressible space forms, was a classification: they are $S^4$, $\mathbb{RP}^4$, $S^3\times S^1$, the twisted product $S^3\tilde\times S^1$, or connected sums of these. A gap in the surgery argument was filled by Bing-Long Chen and Xi-Ping Zhu (2006), and Chen, Siu-Hung Tang and Zhu completed the classification in 2012. Perelman's surgery (12B.4 Surgery) follows this template, with the canonical neighbourhood theorem in place of PIC.

5.3Nonsingular solutions

Hamilton (1999) studied a closed 3-manifold with a Ricci flow that exists for all time and satisfies $|\operatorname{Rm}| \leq \frac Ct$ for large $t$: a nonsingular (Type III) solution. Under this assumption, he proved that the rescaled metrics $\frac{g(t)}{t}$ split the manifold, for large $t$, into a thick part converging to finite-volume hyperbolic metrics, with incompressible boundary tori, and a thin part that collapses with bounded curvature: exactly the decomposition predicted by geometrization (10A.7 Geometrization and Ricci Flow). The homogeneous examples of 11A.8 Homogeneous Flows and the hyperbolic metrics of 10A.6 Hyperbolic Three-Manifolds satisfy the hypothesis. The assumption was the problem: there was no reason a general flow, even after surgery, should satisfy $|\operatorname{Rm}| \leq \frac Ct$. Perelman's long-time analysis (12C.4 Geometrization) does without it.

5.4The gaps in 2002

gap	why it mattered	filled in
1. Noncollapsing. No uniform lower bound on volume ratios at small scales.	Without it, blow-up limits need not exist (11B.3 Compactness of Ricci Flows), and the cigar could appear as a singularity model (11B.4 Singularities).	Perelman's entropy and the $\kappa$-noncollapsing theorem, 12A.3 The 𝓦-Entropy, 12A.4 κ-Noncollapsing
2. Classification of singularity models. No list of the possible blow-up limits in 3D.	Surgery needs to know what high-curvature regions look like.	$\kappa$-solutions and their structure, 12B.1 κ-Solutions, 12B.2 The Structure of κ-Solutions
3. Control of surgery. No canonical neighbourhoods at all high-curvature points, no proof that surgeries can be done with estimates that survive, and no proof that surgery times do not accumulate.	The flow with surgery might not exist for all time.	Canonical neighbourhood theorem and Ricci flow with surgery, 12B.3 The Canonical Neighbourhood Theorem–12B.5 Ricci Flow with Surgery for All Time
4. The end of the flow. No route from the flow with surgery to topology: for simply connected manifolds, no proof that the flow becomes extinct; in general, no long-time analysis without the nonsingular hypothesis.	The topology must be read off from the flow.	Finite extinction, 12C.2 Finite Extinction; long-time analysis, 12C.4 Geometrization

Two tools in the first column, Perelman's entropy and reduced volume, are new monotone quantities (12A.3 The 𝓦-Entropy, 12A.5 Reduced Distance and Reduced Volume). They do for the Ricci flow in every dimension what Hamilton's entropy did for surfaces (11A.7 Ricci Flow on Surfaces) and what Bishop–Gromov does for Riemannian manifolds (9B.2 Volume Comparison). Perelman's own summary, at the start of his first preprint, is that the flow can be viewed as a gradient flow (12A.2 Ricci Flow as a Gradient Flow).

5.5History

This chapter is a history. Beyond the dates in the box above: Grisha Perelman had worked on Alexandrov spaces and the soul conjecture (9B.5 Splitting and Soul Theorems) in the early 1990s, before turning to the Ricci flow; Hamilton's 1995 survey had set out the program and its obstacles in detail, and Perelman's first preprint answered several of them directly.

5.6Exercises

Solution

Round $S^3$: no, it becomes extinct. Hyperbolic: yes, $|\operatorname{Rm}| \sim \frac{c}{1 + 4t}$. Flat torus: yes, $\operatorname{Rm} = 0$. Nil: yes, curvature $\sim\frac1t$. $S^2\times S^1$: no, the sphere factor pinches at a finite time.

Solution

For a simply connected manifold the flow with surgery becomes extinct in finite time (12C.2 Finite Extinction), so there is no long-time behaviour to analyse; the topology is read from the extinct pieces and the surgeries (10A.7 Geometrization and Ricci Flow). Gaps 1–3 are all needed, because the flow must be continued through surgeries up to extinction, which requires noncollapsing, the classification of singularity models, and control of surgery.

Solution

$\operatorname{Ric}(S^3(\rho)) = \frac{2}{\rho^2}g$, so $(\rho^2)' = -4$. Pinching at $T = \frac{\rho_0^2}{4}$; $R = \frac{6}{\rho^2} = \frac{6}{4(T - t)} = \frac{3}{2(T - t)}$.

Solution

(a) $\frac{d}{dt}\operatorname{Vol} = -\int R\,dV \leq \frac{3}{2(t + C)}\operatorname{Vol}$, so $\operatorname{Vol}(t) \leq \operatorname{Vol}(0)\big(\frac{t + C}{C}\big)^{3/2}$ between surgeries, and surgeries only decrease volume. (b) On $[0, T]$ the volume is at most $V_T = \operatorname{Vol}(0)\big(\frac{T + C}{C}\big)^{3/2}$, and each surgery removes at least $\delta$, so there are at most $\frac{V_T}{\delta}$ of them.