Finite Extinction

By 12C.1 Reading Off the Topology, the Poincaré conjecture now rests on one statement: on a closed simply connected 3-manifold, the Ricci flow with surgery becomes extinct in finite time. The idea of the proof fits in a line. Find a geometric quantity that is positive as long as anything is left, and show that it decreases at a definite rate. Then it would become negative, which is impossible, so the flow must end first. The quantity is an area: of discs spanning loops in Perelman's version, of 2-spheres sweeping out the manifold in Colding and Minicozzi's. Its rate of decrease comes from the Gauss–Bonnet theorem, so it is set by topology, not by the metric.

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

• explain the one-dimensional model: under curve shortening, enclosed area drops at rate $2\pi$;
• state Perelman's finite extinction theorem and the topological input that makes the relevant quantity nontrivial;
• derive the area inequalities, $\frac{dA}{dt} \leq -2\pi - \frac12R_{\min}A$ for Perelman's discs and $\frac{dW}{dt} \leq -4\pi + \frac{3}{4(t + C)}W$ for the width;
• deduce finite extinction from either inequality, with an explicit bound on the extinction time;
• explain how surgery is handled and what the two versions of the argument have in common.

2.1Shrinking at a rate set by topology

2.2The one-dimensional model

Under curve shortening flow, a closed embedded plane curve moves with normal velocity equal to its curvature $\kappa$, and the area it encloses changes at the rate

because the total curvature of a simple closed curve is $2\pi$ (6A.8 Curve Shortening and the First Geometric Flows). So the curve cannot survive past time $\frac{A(0)}{2\pi}$, whatever its shape. The Ricci flow arguments below follow this pattern, with two complications. The relevant area is not the area of a fixed surface but a min–max quantity, which must be shown to be nontrivial using topology. And the metric's own curvature enters, through the scalar curvature, with a sign that can work against the argument; the bound $R_{\min} \geq -\frac{3}{2(t + C)}$ (11A.4 Maximum Principles under Ricci Flow) keeps it under control.

2.3Perelman's theorem

A closed simply connected $M$ satisfies the hypothesis: its prime factors are simply connected, and an aspherical closed 3-manifold has infinite fundamental group. So, with 12C.1 Reading Off the Topology, this theorem proves the Poincaré conjecture. Perelman's own summary is that the proof needs no substantially new ideas: a least-area disc argument from Hamilton's work on nonsingular solutions, and a regularisation of the curve shortening flow due to Steven Altschuler and Matthew Grayson.

2.3.1The quantity: least-area discs spanning a family of loops

Let $\Lambda M$ be the space of contractible loops in $M$. For a metric $g$ and a loop $c$, let $A(c, g)$ be the least area of a disc spanning $c$. For a family $\Gamma$ of loops, let $A(\Gamma, g)$ be the largest of these areas; for a nontrivial homotopy class $\alpha$ of families, relative to the constant loops, let

This is a min–max area, of the same kind as the width of 10A.8 Min–Max and Width. That a nontrivial class $\alpha$ exists when $M$ is not aspherical follows, Perelman notes, from a classical and elementary result of Serre. Some higher homotopy group of $M$ is nonzero, and the loop space detects it.

The difficulty is that a family may contain loops that are not immersed, such as a loop that runs along an arc and back, and curve shortening is then not continuous in the family. Perelman's §3 avoids this by adding one dimension to the ambient manifold, following Altschuler and Grayson, who had regularised the singular curve shortening flow in the plane.

2.3.2From the inequality to extinction

In dimension three, $\partial_tR = \Delta R + \frac23R^2 + 2|\operatorname{Ric}^\circ|^2$, which gives $R_{\min}(t) \geq -\frac{3}{2(t + C)}$. With Lemma 2.2,

(Exercise 2.4). The right-hand side is not integrable at infinity, while $\frac{A}{t + C}$ cannot become negative, so the flow must become extinct by a time that depends only on $A(0)$ and $C$.

2.3.3Surgery

For irreducible $M$, surgeries are topologically trivial: one component after the surgery is diffeomorphic to the manifold before, and the others are spheres. Perelman notes that the identification can be chosen $(1 + \xi)$-Lipschitz with $\xi$ as small as one likes, so the min–max area does not increase at a surgery, and the lemma holds for the flow with surgery. For general $M$, the Kneser finiteness theorem (10A.3 The Prime Decomposition) shows that only finitely many surgeries are topologically nontrivial, so from some time on all surgeries are trivial, and by Milnor's uniqueness of prime decomposition each component then satisfies the hypothesis and the irreducible argument applies. For a homotopy sphere, Perelman adds, even Kneser can be avoided, by following homotopy equivalences through each surgery.

2.4Colding and Minicozzi's width

On 25 April 2003, at a dinner in New York, Perelman asked Tobias Colding what happens to the Ricci flow on $S^3$ from an arbitrary metric, and whether it becomes extinct in finite time; he added that he knew no good way of constructing minimal surfaces for a general metric. Colding and William Minicozzi answered with min–max: the smallest of the largest slices of sweepouts is a minimal surface. Perelman posted his own proof soon after, and their paper, which avoids the curve shortening flow, appeared in 2005.

Let $M$ be prime and not aspherical. If $M = S^2\times S^1$, then $\pi_3(M) = \mathbb{Z}$; otherwise $M$ is irreducible, $\pi_2(M) = 0$ by the sphere theorem, and the Hurewicz theorem gives $\pi_3(M) \neq 0$ (10A.8 Min–Max and Width). Either way, the space of maps $S^2 \to M$ is not simply connected, and a nontrivial class $[\beta]$ of sweepouts by 2-spheres exists: one-parameter families $\beta(s)$, $s \in [0, 1]$, of maps $S^2 \to M$, constant at both ends (Figure 2.1). The width is

with $E$ the energy of a map; using area instead gives the same value.

The $4\pi$ comes from Gauss–Bonnet for a sphere, and the $\frac34$ from the lower bound on $R_{\min}$; both matter, while $C$ depends on the initial metric. For a single minimal sphere $\Sigma$ the computation is the one of 10A.8 Min–Max and Width: $\frac{d}{dt}\operatorname{Area}(\Sigma) = -\frac12\int_\Sigma(R + R_\Sigma + |A|^2) \leq -4\pi + \frac{3}{4(t + C)}\operatorname{Area}(\Sigma)$, using $\int_\Sigma R_\Sigma = 8\pi$. The work in their paper is to pass from minimal spheres to the width: they show that a nearly optimal sweepout can be chosen so that every slice whose energy is close to the maximum is close to a collection of harmonic spheres, to which the computation applies. Multiplying by $(t + C)^{-3/4}$ and integrating gives

whose right-hand side becomes negative for large $T$ (Figure 2.2). For the flow with surgery and general $M$ without aspherical summands, they argue as in Perelman's §1.5.

2.5Two versions, one argument

	curve shortening (6A.8 Curve Shortening and the First Geometric Flows)	Perelman III	Colding–Minicozzi
quantity	enclosed area	min–max area of spanning discs	width: min–max area of 2-spheres
topological input	the curve is closed and embedded	a nontrivial class of loop families (Serre)	a nontrivial class in $\pi_3$ (Hurewicz, sphere theorem)
Gauss–Bonnet	turning number: $2\pi$	disc: $2\pi$	sphere: $4\pi$
curvature term	none	$-\frac12R_{\min}A$	$-\frac12R_{\min}W$
technical core	—	regularised curve shortening (Altschuler–Grayson)	min–max with energy and bubbling
conclusion	extinct by $\frac{A(0)}{2\pi}$	extinct in finite time	extinct in finite time

2.6History

Hamilton had used least-area discs in his 1999 paper on nonsingular solutions to show that the boundary tori of hyperbolic pieces are incompressible, and Perelman adapted that argument. Perelman's third preprint was posted on 17 July 2003. Colding and Minicozzi's paper was submitted in August 2003 and published in 2005; their expository paper with full proofs appeared in 2008. Perelman also observed that finite extinction gives the elliptization conjecture without the long-time analysis of his second preprint (12C.1 Reading Off the Topology).

2.7Exercises

Solution

(a) $\hat A' = \frac{A'}{t + C} - \frac{A}{(t + C)^2} \leq -\frac{2\pi}{t + C} + \frac{3A}{4(t + C)^2} - \frac{A}{(t + C)^2} \leq -\frac{2\pi}{t + C}$, since $A \geq 0$. (b) Integrating, $\hat A(t) \leq \frac{A(0)}{C} - 2\pi\log\frac{t + C}{C}$. The left side is nonnegative while the flow exists, so $\log\frac{t + C}{C} \leq \frac{A(0)}{2\pi C}$, that is $t \leq C\big(e^{A(0)/(2\pi C)} - 1\big)$.

Solution

Rate $= \int_D(-\frac12R - K + \det\mathrm{II}) - \oint k_g \leq \int_D(-\frac12R - K) - \oint k_g$, using $\det\mathrm{II} \leq 0$. Minimality: the principal curvatures are $\pm\lambda$, so $\det\mathrm{II} = -\lambda^2 \leq 0$. Gauss–Bonnet: $\int_DK + \oint k_g = 2\pi$. So the rate is at most $-\frac12\int_DR - 2\pi \leq -\frac12R_{\min}\operatorname{Area}(D) - 2\pi$. If $\det\mathrm{II}$ could be positive, the estimate would fail.

Solution

Covering maps induce isomorphisms on $\pi_k$ for $k \geq 2$, and $S^2\times\mathbb{R} \simeq S^2$, so $\pi_3(S^2\times S^1) \cong \pi_3(S^2) \cong \mathbb{Z}$. The argument needs only a nontrivial class of sweepouts by 2-spheres, which $\pi_3 \neq 0$ provides; simple connectivity is not used.

Solution

(a) The universal cover $\mathbb{R}^3$ is contractible. A trivial class contains sweepouts by arbitrarily small spheres, so $W = 0$. (b) $\operatorname{Ric} = 0$, so $g(t) = g(0)$. (c) The theorem assumes $M$ is not aspherical; $T^3$ is aspherical, there is no nontrivial class, and the inequality says nothing.

Solution

(a) $W = 4\pi(r_0^2 - 4t)$, so $W' = -16\pi$. (b) $R_{\min} = \frac{6}{r^2}$, so $-4\pi - \frac12\cdot\frac{6}{r^2}\cdot4\pi r^2 = -4\pi - 12\pi = -16\pi$: equality, because a great sphere is totally geodesic ($|A|^2 = 0$) and $R$ is constant. (c) True extinction: $t = \frac{r_0^2}{4}$. From $W' \leq -4\pi$: $W(0)/4\pi = r_0^2$, four times too late. The curvature term does real work when $R > 0$; when $R$ may be negative, it is the term that must be controlled.