Long time behavior II

In this section we adapt the arguments of Hamilton [H 4] to a more general setting. Hamilton considered smooth Ricci flow with bounded normalized curvature; we drop both these assumptions. In the end of [I,13.2] I claimed that the volumes of the maximal horns can be effectively bounded below, which would imply that the solution must be smooth from some time on; however, the argument I had in mind seems to be faulty. On the other hand, as we'll see below, the presence of surgeries does not lead to any substantial problems.

From now on we assume that our initial manifold does not admit a metric with nonnegative scalar curvature, and that once we get a component with nonnegative scalar curvature, it is immediately removed.

7.1 (cf. [H 4,§2,7]) Recall that for a solution to the smooth Ricci flow the scalar curvature satisfies the evolution equation

$$\frac{d}{dt}R=\triangle R+2\vert Ric\vert^2= \triangle R +2\vert Ric^{\circ}\vert^2+\frac{2}{3}R^2,$$ (7.1)

where $Ric^{\circ}$ is the trace-free part of $Ric.$ Then $R_{\mathrm{min}}(t)$ satisfies $\frac{d}{dt}R_{\mathrm{min}}\ge \frac{2}{3}R_{\mathrm{min}}^2,$ whence

$$R_{\mathrm{min}}(t)\ge -\frac{3}{2}\ \ \frac{1}{t+1/4}$$ (7.2)

for a solution with normalized initial data. The evolution equation for the volume is $\frac{d}{dt}V=-\int RdV,$ in particular

$$\frac{d}{dt}V\le -R_{\mathrm{min}}V,$$ (7.3)

whence by (7.2) the function $V(t)(t+1/4)^{-\frac{3}{2}}$ is non-increasing in $t.$ Let $\bar{V}$ denote its limit as $t\to\infty.$

Now the scale invariant quantity $\hat{R}=R_{\mathrm{min}}V^{\frac{2}{3}}$ satisfies

$$\frac{d}{dt}\hat{R}(t)\ge \frac{2}{3}\ \ \hat{R}V^{-1}\int (R_{\mathrm{min}}-R)dV ,$$ (7.4)

which is nonnegative whenever $R_{\mathrm{min}}\le 0,$ which we have assumed from the beginning of the section. Let $\bar{R}$ denote the limit of $\hat{R}(t)$ as $t\to\infty.$

Assume for a moment that $\bar{V}>0.$ Then it follows from (7.2) and (7.3) that $R_{\mathrm{min}}(t)$ is asymptotic to $-\frac{3}{2t};$ in other words, $\bar{R}\bar{V}^{-\frac{2}{3}}=-\frac{3}{2}.$ Now the inequality (7.4) implies that whenever we have a sequence of parabolic neighborhoods $P(x^{\alpha },t^{\alpha },r\sqrt{t^{\alpha }},-r^2t^{\alpha }),$ for $t^{\alpha }\to\infty$ and some fixed small $r>0,$ such that the scalings of our solution with factor $t^{\alpha }$ smoothly converge to some limit solution, defined in an abstract parabolic neighborhood $P(\bar{x},1,r,-r^2),$ then the scalar curvature of this limit solution is independent of the space variables and equals $-\frac{3}{2t}$ at time $t\in [1-r^2,1];$ moreover, the strong maximum principle for (7.1) implies that the sectional curvature of the limit at time t is constant and equals $-\frac{1}{4t}.$ This conclusion is also valid without the a priori assumption that $\bar{V}>0,$ since otherwise it is vacuous.

Clearly the inequalities and conclusions above hold for the solutions to the Ricci flow with $\delta (t)$-cutoff, defined in the previous sections. From now on we assume that we are given such a solution, so the estimates below may depend on it.

7.2 Lemma. (a) Given $w>0,r>0,\xi>0$ one can find $T=T(w,r,\xi)<\infty,$ such that if the ball $B(x_0,t_0,r\sqrt{t_0})$ at some time $t_0\ge T$ has volume at least $wr^3$ and sectional curvature at least $-r^{-2}t_0^{-1},$ then curvature at $x_0$ at time $t=t_0$ satisfies

$$\vert 2tR_{ij}+g_{ij}\vert< \xi.$$ (7.5)

(b) Given in addition $A<\infty$ and allowing $T$ to depend on $A,$ we can ensure (7.5) for all points in $B(x_0,t_0,Ar\sqrt{t_0}).$

(c) The same is true for $P(x_0,t_0,Ar\sqrt{t_0},Ar^2t_0).$

Proof. (a) If $T$ is large enough then we can apply corollary 6.8 to the ball $B(x_0,t_0,r_0)$ for $r_0=\mathrm{min}(r,\bar{r}(w))\sqrt{t_0};$ then use the conclusion of 7.1.

(b) The curvature control in $P(x_0,t_0,r_0/4,-\tau r_0^2),$ provided by corollary 6.8, allows us to apply proposition 6.3 (a),(b) to a controllably smaller neighborhood $P(x_0,t_0,r_0',-(r_0')^2).$ Thus by 6.3(b) we know that each point in $B(x_0,t_0,Ar\sqrt{t_0})$ with scalar curvature at least $Q=K_1'(A)r_0^{-2}$ has a canonical neighborhood. This implies that for $T$ large enough such points do not exist, since if there was a point with $R$ larger than $Q,$ there would be a point having a canonical neighborhood with $R=Q$ in the same ball, and that contradicts the already proved assertion (a). Therefore we have curvature control in the ball in question, and applying 6.3(a) we also get volume control there, so our assertion has been reduced to (a).

(c) If $\xi$ is small enough, then the solution in the ball $B(x_0,t_0,Ar\sqrt{t_0})$ would stay almost homothetic to itself on the time interval $[t_0,t_0+Ar^2t_0]$ until (7.5) is violated at some (first) time $t'$ in this interval. However, if $T$ is large enough, then this violation could not happen, because we can apply the already proved assertion (b) at time $t'$ for somewhat larger $A.$

7.3 Let $\rho(x,t)$ denote the radius $\rho$ of the ball $B(x,t,\rho)$ where $\mathrm{inf}\ Rm=-\rho^{-2}.$ It follows from corollary 6.8, proposition 6.3(c), and the pinching estimate (5.1) that for any $w>0$ we can find $\bar{\rho}=\bar{\rho}(w)>0,$ such that if $\rho(x,t)<\bar{\rho}\sqrt{t},$ then

$$Vol\ B(x,t,\rho(x,t))<w\rho^3(x,t),$$ (7.6)

provided that $t$ is large enough (depending on $w$).

Let $M^-(w,t)$ denote the thin part of $M,$ that is the set of $x\in M$ where (7.6) holds at time $t,$ and let $M^+(w,t)$ be its complement. Then for $t$ large enough (depending on $w$) every point of $M^+$ satisfies the assumptions of lemma 7.2.

Assume first that for some $w>0$ the set $M^+(w,t)$ is not empty for a sequence of $t\to\infty.$ Then the arguments of Hamilton [H 4,§ 8-12] work in our situation. In particular, if we take a sequence of points $x^{\alpha }\in M^+(w,t^{\alpha }), \ t^{\alpha }\to\infty,$ then the scalings of $g_{ij}^{\alpha }$ about $x^{\alpha }$ with factors $(t^{\alpha })^{-1}$ converge, along a subsequence of $\alpha \to\infty,$ to a complete hyperbolic manifold of finite volume. The limits may be different for different choices of $(x^{\alpha },t^{\alpha }).$ If none of the limits is closed, and $H_1$ is such a limit with the least number of cusps, then, by an argument in [H 4,§8-10], based on hyperbolic rigidity, for all sufficiently small $w',\ 0<w'<\bar{w}(H_1),$ there exists a standard truncation $H_1(w')$ of $H_1,$ such that, for $t$ large enough, $M^+(w'/2,t)$ contains an almost isometric copy of $H_1(w'),$ which in turn contains a component of $M^+(w',t);$ moreover, this embedded copy of $H_1(w')$ moves by isotopy as $t$ increases to infinity. If for some $w>0$ the complement $M^+(w,t)\setminus H_1(w)$ is not empty for a sequence of $t\to\infty,$ then we can repeat the argument and get another complete hyperbolic manifold $H_2,$ etc., until we find a finite collection of $H_j, 1\le j\le i,$ such that for each sufficiently small $w>0$ the embeddings of $H_j(w)$ cover $M^+(w,t)$ for all sufficiently large $t.$

Furthermore, the boundary tori of $H_j(w)$ are incompressible in $M.$ This is proved [H 4,§11,12] by a minimal surface argument, using a result of Meeks and Yau. This argument does not use the uniform bound on the normalized curvature, and goes through even in the presence of surgeries, because the area of the least area disk in question can only decrease when we make a surgery.

7.4 Let us redefine the thin part in case the thick one isn't empty, $\ {M}^-(w,t)=M\setminus(H_1(w)\cup...\cup H_i(w)).$ Then, for sufficiently small $w>0$ and sufficiently large $t,$ $M^-(w,t)$ is diffeomorphic to a graph manifold, as implied by the following general result on collapsing with local lower curvature bound, applied to the metrics $t^{-1}g_{ij}(t).$

Theorem. Suppose $(M^{\alpha },g_{ij}^{\alpha })$ is a sequence of compact oriented riemannian 3-manifolds, closed or with convex boundary, and $w^{\alpha }\to 0.$ Assume that

(1) for each point $x\in M^{\alpha }$ there exists a radius $\rho=\rho^{\alpha }(x), 0<\rho<1,$ not exceeding the diameter of the manifold, such that the ball $B(x,\rho)$ in the metric $g_{ij}^{\alpha }$ has volume at most $w^{\alpha }\rho^3$ and sectional curvatures at least $-\rho^{-2};$

(2) each component of the boundary of $M^{\alpha }$ has diameter at most $w^{\alpha },$ and has a (topologically trivial) collar of length one, where the sectional curvatures are between $-1/4-\epsilon$ and $-1/4+\epsilon ;$

(3) For every $w'>0$ there exist $\bar{r}=\bar{r}(w')>0$ and $K_m=K_m(w')<\infty ,\ m=0,1,2...,$ such that if $\alpha$ is large enough, $0<r\le\bar{r},$ and the ball $B(x,r)$ in $g_{ij}^{\alpha }$ has volume at least $w'r^3$ and sectional curvatures at least $-r^2,$ then the curvature and its $m$-th order covariant derivatives at $x, \ m=1,2...,$ are bounded by $K_0r^{-2}$ and $K_mr^{-m-2}$ respectively.

Then $M^{\alpha }$ for sufficiently large ${\alpha }$ are diffeomorphic to graph manifolds.

Indeed, there is only one exceptional case, not covered by the theorem above, namely, when $M=M^-(w,t),$ and $\rho(x,t),$ for some $x\in M,$ is much larger than the diameter $d(t)$ of the manifold, whereas the ratio $V(t)/d^3(t)$ is bounded away from zero. In this case, since by the observation after formula (7.3) the volume $V(t)$ can not grow faster than $\mathrm{const}\cdot t^{\frac{3}{2}},$ the diameter does not grow faster than $\mathrm{const} \cdot \sqrt{t},$ hence if we scale our metrics $g_{ij}(t)$ to keep the diameter equal to one, the scaled metrics would satisfy the assumption (3) of the theorem above and have the minimum of sectional curvatures tending to zero. Thus we can take a limit and get a smooth solution to the Ricci flow with nonnegative sectional curvature, but not strictly positive scalar curvature. Therefore, in this exceptional case $M$ is diffeomorphic to a flat manifold.

The proof of the theorem above will be given in a separate paper; it has nothing to do with the Ricci flow; its main tool is the critical point theory for distance functions and maps, see [P,§2] and references therein. The assumption (3) is in fact redundant; however, it allows to simplify the proof quite a bit, by avoiding 3-dimensional Aleksandrov spaces, and in particular, the non-elementary Stability Theorem.

Summarizing, we have shown that for large $t$ every component of the solution is either diffeomorphic to a graph manifold, or to a closed hyperbolic manifold, or can be split by a finite collection of disjoint incompressible tori into parts, each being diffeomorphic to either a graph manifold or to a complete noncompact hyperbolic manifold of finite volume. The topology of graph manifolds is well understood [W]; in particular, every graph manifold can be decomposed in a connected sum of irreducible graph manifolds, and each irreducible one can in turn be split by a finite collection of disjoint incompressible tori into Seifert fibered manifolds.