The global picture of the Ricci flow in dimension three

13.1 Let $g_{ij}(t)$ be a smooth solution to the Ricci flow on $M\times [1,\infty),$ where $M$ is a closed oriented three-manifold. Then, according to [H 6, theorem 4.1], the normalized curvatures $\tilde{Rm}(x,t)=tRm(x,t)$ satisfy an estimate of the form $\tilde{Rm}(x,t)\ge -\phi(\tilde{R}(x,t))\tilde{R}(x,t),$ where $\phi$ behaves at infinity as $\frac{1}{\mbox{log}}.$ This estimate allows us to apply the results 12.3,12.4, and obtain the following

Theorem. For any $w>0$ there exist $K=K(w)<\infty , \rho=\rho(w)>0,$ such that for sufficiently large times $t$ the manifold $M$ admits a thick-thin decomposition $M=M_{thick}\bigcup M_{thin}$ with the following properties. (a) For every $x\in M_{thick}$ we have an estimate $\vert\tilde{Rm}\vert\le K$ in the ball $B(x,\rho(w)\sqrt{t}).$ and the volume of this ball is at least $\frac{1}{10}w(\rho(w)\sqrt{t})^n.$ (b) For every $y\in M_{thin}$ there exists $r=r(y), 0<r<\rho(w)\sqrt{t},$ such that for all points in the ball $B(y,r)$ we have $Rm\ge -r^{-2},$ and the volume of this ball is $<wr^n.$

Now the arguments in [H 6] show that either $M_{thick}$ is empty for large $t,$ or , for an appropriate sequence of $t\to 0$ and $w\to 0,$ it converges to a (possibly, disconnected) complete hyperbolic manifold of finite volume, whose cusps (if there are any) are incompressible in $M.$ On the other hand, collapsing with lower curvature bound in dimension three is understood well enough to claim that, for sufficiently small $w>0,$ $\ M_{thin}$ is homeomorphic to a graph manifold.

The natural questions that remain open are whether the normalized curvatures must stay bounded as $t\to\infty,$ and whether reducible manifolds and manifolds with finite fundamental group can have metrics which evolve smoothly by the Ricci flow on the infinite time interval.

13.2 Now suppose that $g_{ij}(t)$ is defined on $M\times [1,T), T<\infty ,$ and goes singular as $t\to T.$ Then using 12.1 we see that, as $t\to T,$ either the curvature goes to infinity everywhere, and then $M$ is a quotient of either $\mathbb{S}^3$ or $\mathbb{S}^2\times \mathbb{R},$ or the region of high curvature in $g_{ij}(t)$ is the union of several necks and capped necks, which in the limit turn into horns (the horns most likely have finite diameter, but at the moment I don't have a proof of that). Then at the time $T$ we can replace the tips of the horns by smooth caps and continue running the Ricci flow until the solution goes singular for the next time, e.t.c. It turns out that those tips can be chosen in such a way that the need for the surgery will arise only finite number of times on every finite time interval. The proof of this is in the same spirit, as our proof of 12.1; it is technically quite complicated, but requires no essentially new ideas. It is likely that by passing to the limit in this construction one would get a canonically defined Ricci flow through singularities, but at the moment I don't have a proof of that. (The positive answer to the conjecture in 11.9 on the uniqueness of ancient solutions would help here)

Moreover, it can be shown, using an argument based on 12.2, that every maximal horn at any time $T,$ when the solution goes singular, has volume at least $cT^n;$ this easily implies that the solution is smooth (if nonempty) from some finite time on. Thus the topology of the original manifold can be reconstructed as a connected sum of manifolds, admitting a thick-thin decomposition as in 13.1, and quotients of $\mathbb{S}^3$ and $\mathbb{S}^2\times\mathbb{R}.$

13.3* Another differential-geometric approach to the geometrization conjecture is being developed by Anderson [A]; he studies the elliptic equations, arising as Euler-Lagrange equations for certain functionals of the riemannian metric, perturbing the total scalar curvature functional, and one can observe certain parallelism between his work and that of Hamilton, especially taking into account that, as we have shown in 1.1, Ricci flow is the gradient flow for a functional, that closely resembles the total scalar curvature.