Two-Spheres Disappear

12C.1 Reading Off the Topology showed that the topology of $M$ can be read off from a Ricci flow with surgery that becomes extinct: $M$ is then a connected sum of spherical space forms and $S^2$-bundles over $S^1$. What remains is to prove extinction. Perelman's third preprint and Colding and Minicozzi's paper prove it for manifolds whose prime decomposition has no aspherical factors, which includes every simply connected one, by finding a geometric quantity, defined by the topology, that the Ricci flow forces to decrease at a definite rate until it would become negative. Morgan and Tian split the argument into two stages, according to the homotopy group that supplies the quantity.

1. $\pi_2$. The minimal area of a homotopically nontrivial $2$-sphere decreases at rate at least $4\pi$ (up to a curvature correction). So after a finite time, no component has nontrivial $\pi_2$. This is this chapter.
2. $\pi_3$. Once $\pi_2 = 0$, a nontrivial element of $\pi_3$ is a sphere's worth of loops, and the maximal area of their minimal spanning disks decreases at rate at least $2\pi$. So the components must disappear: 12C.3 Finite Extinction gives the overview and Colding and Minicozzi's version, and 12C.4 Loops, Disks and Curve Shrinking the rigorous proof with the curve-shrinking flow.

The method is Hamilton's: in his 1999 paper on nonsingular solutions he used minimal disks to show that the boundary tori of almost hyperbolic pieces become incompressible, by proving that the area of a compressing disk would have to become negative in finite time. The same idea, with spheres instead of disks, kills $\pi_2$.

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

• state Morgan and Tian's finite-extinction theorem and derive the Poincaré conjecture and the elliptization conjecture from it;
• prove that a compact three-manifold has a bound on the number of disjoint, essential, pairwise non-homotopic $2$-spheres, and deduce that only finitely many surgeries are along essential spheres;
• prove that the minimal area $W_2(t)$ of an essential sphere satisfies $\frac{d}{dt}W_2 \leq -4\pi - \frac12R_{\min}W_2$, is continuous between surgeries and does not jump up at surgeries;
• integrate the differential inequality and conclude that every component eventually has $\pi_2 = 0$;
• deduce the sphere theorem for closed three-manifolds.

2.1The theorem

The theorem gives the Poincaré conjecture at once: by Morgan and Tian's Corollary 15.4 (12B.10 Surgery on a δ-Neck), if some time slice is empty, $M$ is a connected sum of spherical space forms and $S^2$-bundles over $S^1$; if $M$ is simply connected, every summand is simply connected, so every summand is $S^3$ and $M \cong S^3$. If $\pi_1(M)$ is finite, $M$ is a connected sum of one spherical space form and copies of $S^3$, so $M$ is a spherical space form: the elliptization conjecture.

Perelman's version is phrased through prime decompositions ("whose prime decomposition contains no aspherical factors"); Morgan and Tian's group-theoretic hypothesis is equivalent by standard three-manifold topology, and is what the proof uses. The first claim is a short topological argument: an $\mathbb{RP}^2$ with trivial normal bundle does not separate (it has Euler characteristic $1$, so it bounds no compact three-manifold), can be moved off the spheres of a connected sum decomposition by innermost-disk surgery, and then, in the prime factor containing it, forces $H_1$ to surject onto $\mathbb{Z}$ while $\pi_1$ contains an element of order $2$: neither finite nor infinite cyclic. So Assumption (1) of 12B.10 Surgery on a δ-Neck holds, and the flow with surgery exists.

Paths of components (Definition 18.2). Since surgery can split a component into several, the objects that persist through time are paths of components: a connected open subset $\mathcal X$ of space-time over an interval $I$ whose slice at each time is a single component. Between surgery times it is the Ricci flow of one component; at a surgery time, if the component is cut along spheres, the path continues into one of the pieces, and if the component is removed, the path ends.

2.2Essential spheres are finite in number

A $2$-sphere embedded in $X$ is homotopically essential if it is not homotopic to a point. A family of disjoint embedded spheres is homotopically essential if each is, and no two are homotopic to each other (Definition 18.4).

A surgery at time $t$ along a sphere $S_0$ is essential if $S_0$, flowed back to slightly earlier times, is homotopically essential (Definition 18.6).

So after some time $T_0$ every sphere surgery is along an inessential sphere. From then on, a path of components with nontrivial $\pi_2$ is determined by its starting component: at each surgery its component is either untouched, or split into one piece homotopy equivalent to it and some homotopy spheres, or removed. So the number of components with nontrivial $\pi_2$ does not increase after $T_0$, and it suffices to show that each such path ends.

2.3The minimal essential sphere

The proof is the Sacks–Uhlenbeck existence theory for harmonic spheres (6C.7 Harmonic Maps and Their Heat Flow): minimize the perturbed energy $E_\alpha(s) = \int_{S^2}(1 + |ds|^2)^\alpha$ for $\alpha > 1$, which satisfies the Palais–Smale condition, among nontrivial maps, and let $\alpha \to 1$. The minimizers converge away from finitely many points; if the convergence is strong, the limit is the answer, and if energy concentrates, rescaling at a concentration point produces a bubble, a nonconstant harmonic sphere. Morgan and Tian show that the bubble is itself homotopically nontrivial, since otherwise one could cut it off and replace it by a small disk, lowering the energy below the infimum by a definite amount; so the bubble realises the minimum. A minimizing harmonic sphere is conformal (there are no nonzero holomorphic quadratic differentials on $S^2$), hence a branched minimal immersion.

For a path of components $\mathcal X$ with $\pi_2 \neq 0$ define

The $4\pi$ is topology: it is $2\pi\chi(S^2)$. The minimal sphere's area decreases because the Ricci flow shrinks the metric in the directions tangent to the sphere at a rate given by curvature, and Gauss–Bonnet turns the total intrinsic curvature of any sphere into the constant $4\pi$.

2.4Every component loses its $\pi_2$

Notice that only finitely many surgeries are along essential spheres, but the proof never needs to know when they happen: the area argument works on each path of components, and the bound on $T_1$ comes from the finitely many components at time $T_0$. Notice too that nothing here uses the fundamental group hypothesis of Theorem 2.1: every Ricci flow with surgery on a closed three-manifold eventually has only components with $\pi_2 = 0$.

2.5The sphere theorem

The classical sphere theorem (Papakyriakopoulos, 1957) is proved by topology; here it falls out of geometry, an example of the Ricci flow proving topological theorems that were already known, as a test of its strength. Remark 18.15 also records what the flow does to a manifold without $\mathbb{RP}^3$ summands and without non-separating spheres: by time $T_0$ it has performed a connected sum decomposition into components with $\pi_2 = 0$, that is, into prime pieces of the two kinds in the proposition.

2.6History

Hamilton used minimal disks and the Gauss–Bonnet estimate in "Non-singular solutions of the Ricci flow on three-manifolds" (1999, §11) to show that compressing disks for the boundary tori of hyperbolic pieces disappear in finite time. Perelman's third preprint (2003) applied the method to disks spanning loops, and Colding and Minicozzi (2005) to spheres in sweepouts. Morgan and Tian (2007, §18.2) added the separate treatment of $\pi_2$ with minimal spheres, using the Sacks–Uhlenbeck theory (1981), and derived the sphere theorem as a corollary.

2.7Exercises

Solution

(a) $R = 2\sum_{i<j}K(e_i, e_j)$ over an orthonormal frame $e_1, e_2, e_3 = \nu$, and $\operatorname{Ric}(\nu, \nu) = K(e_1, \nu) + K(e_2, \nu)$; so $R = 2K(e_1, e_2) + 2\operatorname{Ric}(\nu, \nu)$. (b) With principal curvatures $\kappa_1 + \kappa_2 = 0$: $\det\mathrm{II} = \kappa_1\kappa_2 = -\kappa_1^2 = -\frac12(\kappa_1^2 + \kappa_2^2) = -\frac12|\mathrm{II}|^2$. (c) $-\int(R - \operatorname{Ric}(\nu, \nu)) = -\int(\frac12R + K_\Sigma + \frac12|\mathrm{II}|^2)$.

Solution

With $u = (1 + 4t)^{-3/4}w$: $u' = (1 + 4t)^{-3/4}(w' - \frac{3w}{1 + 4t}) = -4\pi(1 + 4t)^{-3/4}$, so $u(t) = u(T_0) - 4\pi\int_{T_0}^t(1 + 4s)^{-3/4}ds = u(T_0) - 4\pi\big((1 + 4t)^{1/4} - (1 + 4T_0)^{1/4}\big)$. Multiplying back gives the formula in the text. With $T_0 = 0$: $w(t) = (A + 4\pi)(1 + 4t)^{3/4} - 4\pi(1 + 4t)$, negative once $(1 + 4t)^{1/4} > 1 + \frac{A}{4\pi}$, that is $t > \frac14\big((1 + \frac{A}{4\pi})^4 - 1\big)$. With $R_{\min} \geq 0$, $w = A - 4\pi t$, negative after $\frac{A}{4\pi}$: a much shorter time. The negative curvature allowed by pinching delays extinction polynomially but cannot prevent it.

Solution

(a) The essential spheres include the fibres $S^2\times\{\theta\}$, of area $4\pi a^2$, which is the minimum (an essential sphere maps with nonzero degree onto the $S^2$ factor, and projection to $S^2$ does not increase area). (b) $W_2 = 4\pi(a^2 - 2t)$, so $\frac{d}{dt}W_2 = -8\pi$. The lemma gives $-4\pi - \frac12R_{\min}W_2 = -4\pi - \frac12\cdot\frac{2}{a^2 - 2t}\cdot4\pi(a^2 - 2t) = -8\pi$: equality, because the fibre is totally geodesic ($\mathrm{II} = 0$) and $R$ is constant. (c) The whole manifold becomes singular and is removed by surgery (it is a component covered by necks), consistent with $W_2 \to 0$.

Solution

(a) By Grushko's theorem the minimal number of generators is additive under free products. A nontrivial group needs at least one generator, so a free product of $m$ nontrivial groups needs at least $m$ generators; $F_r$ is generated by $r$ elements, so $m \leq r$, with equality for $\mathbb{Z}*\cdots*\mathbb{Z}$. (b) $\mathbb{Z}/2*\mathbb{Z}/3$ is generated by $2$ elements, so it has at most $2$ nontrivial free factors. (c) The spheres $S^2\times\{\theta_i\}$, one in each summand: each is non-separating, hence essential, and no two are homotopic (they are dual to different generators of $H_1$). For $r \geq 2$ one can add separating spheres between summands (each complementary piece has $\pi_1 \neq 1$, so they are essential and not homotopic to the fibres), so a maximal family is larger than $r$, but still bounded by Lemma 2.2. For $r = 1$ the fibre is all there is: any two disjoint fibres are homotopic.

Solution

(a) Gauss–Bonnet for a disk: $\int_DK + \int_{\partial D}k_g = 2\pi\chi(D) = 2\pi$. (b) The first variation of area has a boundary term from the motion of the boundary, and the Gauss–Bonnet formula has the geodesic curvature of the boundary. If the boundary moves within a nearly totally geodesic, nearly flat torus that the flow keeps nearly stationary up to scaling, these terms are small compared with $2\pi$. Perelman's disks have boundaries on loops that move by curve shortening, which makes the geodesic curvature term exactly cancel, as 12C.4 Loops, Disks and Curve Shrinking shows.