Finite time extinction

1.1 Theorem. Let $M$ be a closed oriented three-manifold, whose prime decomposition contains no aspherical factors. Then for any initial metric on $M$ the solution to the Ricci flow with surgery becomes extinct in finite time.

Proof for irreducible $M$. Let $\Lambda M$ denote the space of all contractible loops in $C^1(\mathbb{S}^1\to M).$ Given a riemannian metric $g$ on $M$ and $c\in\Lambda M,$ define $A(c,g)$ to be the infimum of the areas of all lipschitz maps from $\mathbb{D}^2$ to $M,$ whose restriction to $\partial\mathbb{D}^2=\mathbb{S}^1$ is $c.$ For a family $\Gamma\subset\Lambda M$ let $A(\Gamma ,g)$ be the supremum of $A(c,g)$ over all $c\in\Gamma.$ Finally, for a nontrivial homotopy class $\alpha\in\pi_{\ast}(\Lambda M,M)$ let $A(\alpha ,g)$ be the infimum of $A(\Gamma ,g)$ over all $\Gamma\in\alpha.$ Since $M$ is not aspherical, it follows from a classical (and elementary) result of Serre that such a nontrivial homotopy class exists.

1.2 Lemma. (cf. [H,§11]) If $g^t$ is a smooth solution to the Ricci flow, then for any $\alpha$ the rate of change of the function $A^t=A(\alpha ,g^t)$ satisfies the estimate

$$\frac{d}{dt}A^t\le-2\pi-\frac{1}{2}R^t_{\mathrm{min}}A^t$$

(in the sense of the $\mathrm{lim \ sup}$ of the forward difference quotients), where $R^t_{\mathrm{min}}$ denotes the minimum of the scalar curvature of the metric $g^t.$

A rigorous proof of this lemma will be given in §3, but the idea is simple and can be explained here. Let us assume that at time $t$ the value $A^t$ is attained by the family $\Gamma,$ such that the loops $c\in\Gamma$ where $A(c,g^t)$ is close to $A^t$ are embedded and sufficiently smooth. For each such $c$ consider the minimal disk $D_c$ with boundary $c$ and with area $A(c,g^t).$ Now let the metric evolve by the Ricci flow and let the curves $c$ evolve by the curve shortening flow (which moves every point of the curve in the direction of its curvature vector at this point) with the same time parameter. Then the rate of change of the area of $D_c$ can be computed as

$$\int_{D_c}{(-\mathrm{Tr(Ric^T)})}\ \ + \int_c{(-k_g)}$$

where $\mathrm{Ric^T}$ is the Ricci tensor of $M$ restricted to the tangent plane of $D_c,$ and $k_g$ is the geodesic curvature of $c$ with respect to $D_c$ (cf. [A-G, Lemma 3.2]). In three dimensions the first integrand equals $-\frac{1}{2}R-(K-\mathrm{det \ II}),$ where $K$ is the intrinsic curvature of $D_c$ and $\mathrm{det \ II},$ the determinant of the second fundamental form, is nonpositive, because $D_c$ is minimal. Thus, the rate of change of the area of $D_c$ can be estimated from above by

$$\int_{D_c}(-\frac{1}{2}R-K) \ + \int_c (-k_g) = \int_{D_c}(-\frac{1}{2}R) \ -2\pi$$

by the Gauss-Bonnet theorem, and the statement of the lemma follows.

The problem with this argument is that if $\Gamma$ contains curves, which are not immersed (for instance, a curve could pass an arc once in one direction and then make an about turn and pass the same arc in the opposite direction), then it is not clear how to define curve shortening flow so that it would be continuous both in the time parameter and in the family parameter. In §3 we'll explain how to circumvent this difficulty, essentially by adding one dimension to the ambient manifold. This regularization of the curve shortening flow has been worked out by Altschuler and Grayson [A-G] (who were interested in approximating the singular curve shortening flow on the plane and obtained for that case more precise results than what we need).

1.3 Now consider the solution to the Ricci flow with surgery. Since $M$ is assumed irreducible, the surgeries are topologically trivial, that is one of the components of the post-surgery manifold is diffeomorphic to the pre-surgery manifold, and all the others are spheres. Moreover, by the construction of the surgery [P,4.4], the diffeomorphism from the pre-surgery manifold to the post-surgery one can be chosen to be distance non-increasing ( more precisely, $(1+\xi)$-lipschitz, where $\xi>0$ can be made as small as we like). It follows that the conclusion of the lemma above holds for the solutions to the Ricci flow with surgery as well.

Now recall that the evolution equation for the scalar curvature

$$\frac{d}{dt}R=\triangle R + 2\vert\mathrm{Ric}\vert^2=\triangle R+\frac{2}{3}R^2+2\vert\mathrm{Ric}^{\circ}\vert^2$$

implies the estimate $R^t_{\mathrm{min}}\ge -\frac{3}{2}\frac{1}{t+\mathrm{const}}.$ It follows that $\hat{A}^t=\frac{A^t}{t+\mathrm{const}}$ satisfies $\frac{d}{dt}\hat{A}^t\le -\frac{2\pi}{t+\mathrm{const}},$ which implies finite extinction time since the right hand side is non-integrable at infinity whereas $\hat{A}^t$ can not become negative.

1.4 Remark. The finite time extinction result for irreducible non-aspherical manifolds already implies (in conjuction with the work in [P,§1-5] and the Kneser finiteness theorem) the so called "elliptization conjecture", claiming that a closed manifold with finite fundamental group is diffeomorphic to a spherical space form. The analysis of the long time behavior in [P,§6-8] is not needed in this case; moreover the argument in [P,§5] can be slightly simplified, replacing the sequences $r_j, \kappa_j, \bar{\delta}_j$ by single values $r, \kappa, \bar{\delta},$ since we already have an upper bound on the extinction time in terms of the initial metric.

In fact, we can even avoid the use of the Kneser theorem. Indeed, if we start from an initial metric on a homotopy sphere (not assumed irreducible), then at each surgery time we have (almost) distance non-increasing homotopy equivalences from the pre-surgery manifold to each of the post-surgery components, and this is enough to keep track of the nontrivial relative homotopy class of the loop space.

1.5 Proof of theorem 1.1 for general $M$. The Kneser theorem implies that our solution undergoes only finitely many topologically nontrivial surgeries, so from some time $T$ on all the surgeries are trivial. Moreover, by the Milnor uniqueness theorem, each component at time $T$ satisfies the assumption of the theorem. Since we already know from 1.4 that there can not be any simply connected prime factors, it follows that every such component is either irreducible, or has nontrivial $\pi_2;$ in either case the proof in 1.1-1.3 works.