Geometrization and Ricci Flow

The previous chapters described what a closed 3-manifold is made of: prime pieces, cut further along tori into Seifert fibred and atoroidal pieces, each carrying one of eight geometries. This chapter connects that picture to the Ricci flow. The flow does not know the decomposition in advance; it starts from an arbitrary metric. Yet each kind of piece has a characteristic fate: spherical pieces shrink and disappear, necks pinch, hyperbolic pieces expand into hyperbolic metrics, and the remaining pieces collapse. Watching the flow, with surgery at the necks, reveals the decomposition. That is Hamilton's program, which Perelman completed.

This is a map, not a proof. Each fate listed here is proved in Books 11A–12C, and the chapter says where. It ties together every thread of the Path: the PDE (Book 6A), the geometry (Books 9A–9B) and the topology (Books 7A, 10A).

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

• describe the fate of each model geometry under the Ricci flow, and compute it for the isotropic and product geometries;
• explain why necks force surgery, and how surgery relates to the prime decomposition;
• describe the thick–thin picture of the long-time behaviour and how it produces the JSJ decomposition;
• state what Hamilton's program had established by 1999 and what Perelman supplied;
• outline how the Poincaré conjecture and geometrization follow.

7.1Perelman's papers

7.2The fates of the geometries

For the isotropic and product geometries, the Ricci flow of the standard metric can be written down, since the Ricci tensor is a multiple of the metric on each factor (9A.4 Curvature and What It Means, 9A.5 Computing Curvature):

• $S^3$: $g(t) = (1 - 4t)g_0$ for the unit sphere. It shrinks to a point at $t = \frac14$, staying round. Hamilton's 1982 theorem (11A.6 Hamilton’s 1982 Theorem) shows that any metric with $\operatorname{Ric} > 0$ does the same after rescaling, which proves elliptization for such manifolds.
• $S^2\times\mathbb{R}$: the sphere factor shrinks, $\rho(t)^2 = \rho_0^2 - 2t$, while the line stays fixed. The cylinder pinches everywhere at $t = \frac{\rho_0^2}{2}$. This is the model neck (9A.5 Computing Curvature).
• $\mathbb{E}^3$: flat metrics do not move.
• $\mathbb{H}^3$: $g(t) = (1 + 4t)g_0$, expanding forever; $\frac{g(t)}{t} \to 4g_0$ (10A.6 Hyperbolic Three-Manifolds).
• $\mathbb{H}^2\times\mathbb{R}$: the hyperbolic factor expands, $\rho(t)^2 = \rho_0^2 + 2t$, and the line stays fixed. After rescaling by $\frac1t$, the line direction shrinks to zero: collapse.

The other three geometries, Nil, Sol and $\widetilde{SL}(2, \mathbb{R})$, are Lie groups, and their left-invariant metrics evolve by ODEs (11A.8 Homogeneous Flows). In each case some directions shrink relative to others, and the rescaled metrics collapse. Isenberg and Jackson (1992) and Knopf and McLeod (2001) worked out these homogeneous flows. All five anisotropic geometries other than $S^2\times\mathbb{R}$, and the flat one, are geometries of graph manifolds, and all collapse.

7.3Necks and surgery

A general metric does not stay homogeneous. Where the curvature becomes large, Perelman's canonical neighbourhood theorem (12B.3 The Canonical Neighbourhood Theorem) says the geometry, after rescaling, is close to one of a few models: a round neck $S^2\times(-L, L)$, a cap (a ball capping a neck), or a closed positively curved piece. When necks become too thin, the flow is stopped. Each thin neck is cut along its central sphere, the two ends are capped with standard balls, and the flow is restarted (12B.4 Surgery). Components with positive curvature that are about to disappear are recorded and discarded: they are spherical space forms or $S^2\times S^1$ (or $\mathbb{RP}^3\#\mathbb{RP}^3$).

Topologically, cutting along the sphere of a neck undoes a connected sum, or removes an $S^2\times S^1$ summand if the sphere does not separate (10A.3 The Prime Decomposition). So the flow with surgery splits off prime summands, and the discarded components are exactly the spherical and $S^2\times\mathbb{R}$ pieces of the prime decomposition. Perelman showed that surgery times do not accumulate, so on each finite time interval there are finitely many surgeries (12B.5 Ricci Flow with Surgery for All Time).

7.4The long-time picture: thick and thin

For a component that survives for all time, Perelman studied the rescaled metrics $\frac{g(t)}{t}$ as $t \to \infty$. Fix a small $w > 0$. For a point $x$, let $\rho$ be the radius at which the infimum of the sectional curvature on $B(x, \rho)$ equals $-\rho^{-2}$; call $x$ $w$-thin if $\operatorname{Vol}B(x, \rho) < w\rho^3$, and $w$-thick otherwise. Then:

• The thick part converges to a complete hyperbolic manifold of finite volume, with curvature $-\frac14$ in the normalisation $\frac{g(t)}{t}$. Its cusps are cut off by tori, which Hamilton (1999) and Perelman showed are incompressible (12C.4 Geometrization). These are the hyperbolic pieces of the JSJ decomposition, and Mostow rigidity (10A.6 Hyperbolic Three-Manifolds) says the limit is unique.
• The thin part collapses with curvature bounded below. By the collapsing theory of Shioya–Yamaguchi, Morgan–Tian, Kleiner–Lott and others, it is a graph manifold (10A.4 Seifert Spaces and the JSJ Decomposition): a union of Seifert pieces glued along tori.

Together these give the JSJ decomposition of each prime piece, with the hyperbolic pieces in the thick part and the Seifert pieces in the thin part. That is geometrization.

For the Poincaré conjecture the long-time picture is not needed. If $M$ is simply connected, the flow with surgery becomes extinct in finite time: every component disappears (12C.2 Finite Extinction). Perelman and, independently, Colding and Minicozzi proved this with a min–max quantity, the width (10A.8 Min–Max and Width), which decreases at a definite rate. Then $M$ was a connected sum of the discarded pieces, spherical space forms and copies of $S^2\times S^1$. All are simply connected only if they are copies of $S^3$, so $M = S^3$ (12C.3 The Poincaré Conjecture, Assembled).

7.5Hamilton's program and Perelman's contribution

Richard Hamilton introduced the Ricci flow in 1982 and proved that closed 3-manifolds with $\operatorname{Ric} > 0$ are spherical space forms. Over the next 17 years he developed the program: maximum principles for tensors and the Hamilton–Ivey pinching estimate (11A.4 Maximum Principles under Ricci Flow, 11A.5 Hamilton–Ivey Pinching), the Harnack inequality (11B.2 Ancient Solutions and the Harnack Inequality), the compactness theorem (11B.3 Compactness of Ricci Flows), the classification of singularity models in good cases (11B.4 Singularities), surgery in dimension four, and in 1999 the long-time thick–thin picture for flows without singularities and with bounded normalised curvature. By 2002 the missing steps were (11B.5 Hamilton’s Program in 2002):

1. Noncollapsing. Blow-up limits could collapse, like the cigar (9B.3 Collapsing and Noncollapsing), and nothing ruled this out. Perelman's entropy gave the $\kappa$-noncollapsing theorem (12A.3 The 𝓦-Entropy, 12A.4 κ-Noncollapsing).
2. Canonical neighbourhoods. A description of all high-curvature regions, needed to know where to cut. Perelman classified $\kappa$-solutions (12B.1 κ-Solutions, 12B.2 The Structure of κ-Solutions) and proved the canonical neighbourhood theorem (12B.3 The Canonical Neighbourhood Theorem).
3. Surgery that can be continued. The constants must be chosen so that the estimates survive each surgery and surgeries do not accumulate (12B.4 Surgery, 12B.5 Ricci Flow with Surgery for All Time).
4. The end of the flow. Finite extinction for simply connected manifolds (12C.2 Finite Extinction), and for geometrization, the thick–thin analysis without Hamilton's curvature assumption (12C.4 Geometrization).

7.6History

Thurston stated the geometrization conjecture in 1982, the year Hamilton introduced the Ricci flow. Hamilton's survey "The formation of singularities in the Ricci flow" (1995) laid out the program, and his 1999 paper on nonsingular solutions gave the long-time picture under extra assumptions. Perelman's three preprints appeared between November 2002 and July 2003, and the detailed expositions between 2006 and 2014. Morgan's Bulletin survey appeared in 2005.

7.7Exercises

Solution

$\operatorname{Ric}(g_1\oplus g_2) = \operatorname{Ric}(g_1)\oplus\operatorname{Ric}(g_2)$, so $\partial_t(g_1\oplus g_2) = -2\operatorname{Ric}(g_1)\oplus-2\operatorname{Ric}(g_2)$ is solved by flowing each factor. On a surface of curvature $\kappa_0$, $\operatorname{Ric} = \kappa_0g_0$ and the flow is $(1 - 2\kappa_0t)g_0$: $1 - 2t$ for $S^2$ and $1 + 2t$ for $\mathbb{H}^2$. The line has $\operatorname{Ric} = 0$.

Solution

The surface factor evolves by $(1 + 2t)g_\Sigma$ (curvature $-1$ at $t = 0$), so its area is $(1 + 2t)\cdot2\pi|\chi|$ by Gauss–Bonnet, growing like $4\pi|\chi|t$. The circle has $\operatorname{Ric} = 0$ and keeps its length. Dividing the metric by $t$, the circle's length is $\frac{\ell}{\sqrt t}$, and $\frac{1 + 2t}{t}g_\Sigma \to 2g_\Sigma$, of curvature $-\frac12$. The volume of unit balls of $\frac{g(t)}{t}$ goes to zero: the whole manifold collapses.

Solution

Sphere: $r(t)^2 = r^2 - 4t$, $T = \frac{r^2}{4}$, and $R = \frac{6}{r(t)^2} = \frac{6}{4(T - t)} = \frac{3}{2(T - t)}$. Cylinder: $\rho(t)^2 = \rho^2 - 2t$, $T = \frac{\rho^2}{2}$, $R = \frac{2}{\rho(t)^2} = \frac{1}{T - t}$.

Solution

Each surgery replaces the manifold by a disjoint union whose connected sum (with copies of $S^2\times S^1$ added for non-separating spheres) is the manifold before surgery. Going backwards from extinction, $M$ is a connected sum of the extinct components and copies of $S^2\times S^1$. Since $\pi_1(M) = 1$, every summand has trivial $\pi_1$ (10A.3 The Prime Decomposition); spherical space forms with trivial $\pi_1$ are $S^3$, and $S^2\times S^1$ and $\mathbb{RP}^3\#\mathbb{RP}^3$ have infinite $\pi_1$, so they cannot occur. So $M$ is a connected sum of copies of $S^3$, which is $S^3$.

Solution

$S^3$ and $L(5, 2)$ (quotients of the round sphere): shrink round, extinct at $t = \frac14$ for curvature $1$. $S^2\times S^1$: the $S^2$ pinches at $t = \frac12$ while the circle stays fixed, a neck; with surgery it is discarded. $T^3$: static; $\frac{g(t)}{t} \to 0$, entirely thin. Hyperbolic: expands as $1 + 4t$, thick, $\frac{g(t)}{t} \to 4g_0$. $\Sigma_2\times S^1$: the surface expands, the circle stays fixed, thin.