Ancient solutions with nonnegative curvature operator and bounded entropy

11.1   In this section we consider smooth solutions to the Ricci flow $(g_{ij})_t=-2R_{ij}, -\infty<t\le 0,$ such that for each $t$ the metric $g_{ij}(t)$ is a complete non-flat metric of bounded curvature and nonnegative curvature operator. Hamilton discovered a remarkable differential Harnack inequality for such solutions; we need only its trace version

$$R_t+2<X,\nabla R>+2 (X,X)\ge 0$$ (11.1)

and its corollary, $R_t\ge 0.$ In particular, the scalar curvature at some time $t_0\le 0$ controls the curvatures for all $t\le t_0.$

We impose one more requirement on the solutions; namely, we fix some $\kappa >0$ and require that $g_{ij}(t)$ be $\kappa$-noncollapsed on all scales (the definitions 4.2 and 8.1 are essentially equivalent in this case). It is not hard to show that this requirement is equivalent to a uniform bound on the entropy $S,$ defined as in 5.1 using an arbitrary fundamental solution to the conjugate heat equation.

11.2   Pick an arbitrary point $(p,t_0)$ and define $\tilde{V}(\tau), l(q,\tau)$ as in 7.1, for $\tau(t)=t_0-t.$ Recall that for each $\tau>0$ we can find $q=q(\tau),$ such that $l(q,\tau)\le \frac{n}{2}.$

Proposition.The scalings of $g_{ij}(t_0-\tau)$ at $q(\tau)$ with factors $\tau^{-1}$ converge along a subsequence of $\tau\to\infty$ to a non-flat gradient shrinking soliton.

Proof (sketch). It is not hard to deduce from (7.16) that for any $\epsilon >0$ one can find $\delta >0$ such that both $l(q,\tau)$ and $\tau R(q,t_0-\tau)$ do not exceed $\delta ^{-1}$ whenever $\frac{1}{2}\bar{\tau}\le \tau\le \bar{\tau}$ and dist$_{t_0-\bar{\tau}}^2(q,q(\bar{\tau}))\le \epsilon ^{-1}\bar{\tau}$ for some $\bar{\tau}>0.$ Therefore, taking into account the $\kappa$-noncollapsing assumption, we can take a blow-down limit, say $\bar{g}_{ij}(\tau),$ defined for $\tau\in(\frac{1}{2},1), (\bar{g}_{ij})_{\tau}=2\bar{R}_{ij}.$ We may assume also that functions $l$ tend to a locally Lipschitz function $\bar{l},$ satisfying (7.13),(7.14) in the sense of distributions. Now, since $\tilde{V}(\tau)$ is nonincreasing and bounded away from zero (because the scaled metrics are not collapsed near $q(\tau)$) the limit function $\bar{V}(\tau)$ must be a positive constant; this constant is strictly less than lim$_{\tau\to 0}\tilde{V}(\tau)=(4\pi)^{\frac{n}{2}},$ since $g_{ij}(t)$ is not flat. Therefore, on the one hand, (7.14) must become an equality, hence $\bar{l}$ is smooth, and on the other hand, by the description of the equality case in (7.12), $\bar{g}_{ij}(\tau)$ must be a gradient shrinking soliton with $\bar{R}_{ij}+\bar{\nabla}_i\bar{\nabla}_j \bar{l}-\frac{1}{2\tau}\bar{g}_{ij}=0.$ If this soliton is flat, then $\bar{l}$ is uniquely determined by the equality in (7.14), and it turns out that the value of $\bar{V}$ is exactly $(4\pi)^{\frac{n}{2}},$ which was ruled out.

Corollary 11.3   There is only one oriented two-dimensional solution, satisfying the assumptions stated in 11.1, - the round sphere.

Proof. Hamilton [H 10] proved that round sphere is the only non-flat oriented nonnegatively curved gradient shrinking soliton in dimension two. Thus, the scalings of our ancient solution must converge to a round sphere. However, Hamilton [H 10] has also shown that an almost round sphere is getting more round under Ricci flow, therefore our ancient solution must be round.

11.4   Recall that for any non-compact complete riemannian manifold $M$ of nonnegative Ricci curvature and a point $p\in M,$ the function $VolB(p,r)r^{-n}$ is nonincreasing in $r>0;$ therefore, one can define an asymptotic volume ratio $\mathcal{V}$ as the limit of this function as $r\to\infty.$

Proposition. Under assumptions of 11.1, $\mathcal{V}=0$ for each $t.$

Proof. Induction on dimension. In dimension two the statement is vacuous, as we have just shown. Now let $n\ge 3,$ suppose that $\mathcal{V}>0$ for some $t=t_0,$ and consider the asymptotic scalar curvature ratio $\mathcal{R}=$lim sup$R(x,t_0)d^2(x)$ as $d(x)\to\infty.$ ($d(x)$ denotes the distance, at time $t_0,$ from $x$ to some fixed point $x_0$) If $\mathcal{R}=\infty,$ then we can find a sequence of points $x_k$ and radii $r_k>0,$ such that $r_k/d(x_k)\to 0, R(x_k)r_k^2\to\infty ,$ and $R(x)\le 2R(x_k)$ whenever $x\in B(x_k,r_k).$ Taking blow-up limit of $g_{ij}(t)$ at $(x_k,t_0)$ with factors $R(x_k),$ we get a smooth non-flat ancient solution, satisfying the assumptions of 11.1, which splits off a line (this follows from a standard argument based on the Aleksandrov-Toponogov concavity). Thus, we can do dimension reduction in this case (cf. [H 4,$\S 22$]).

If $0<\mathcal{R}<\infty ,$ then a similar argument gives a blow-up limit in a ball of finite radius; this limit has the structure of a non-flat metric cone. This is ruled out by Hamilton's strong maximum principle for nonnegative curvature operator.

Finally, if $\mathcal{R}=0,$ then (in dimensions three and up) it is easy to see that the metric is flat.

Corollary 11.5   For every $\epsilon >0$ there exists $A< \infty$ with the following property. Suppose we have a sequence of ( not necessarily complete) solutions $(g_k)_{ij}(t)$ with nonnegative curvature operator, defined on $M_k\times[t_k,0],$ such that for each $k$ the ball $B(x_k,r_k)$ at time $t=0$ is compactly contained in $M_k,$ $\frac{1}{2}R(x,t)\le R(x_k,0)=Q_k$ for all $(x,t), t_kQ_k\to -\infty , r_k^2Q_k\to\infty$ as $k\to\infty.$ Then $VolB(x_k,A/\sqrt{Q_k})\le\epsilon (A/\sqrt{Q_k})^n$ at $t=0$ if $k$ is large enough.

Proof. Assuming the contrary, we may take a blow-up limit (at $(x_k,0)$ with factors $Q_k$) and get a non-flat ancient solution with positive asymptotic volume ratio at $t=0,$ satisfying the assumptions in 11.1, except, may be, the $\kappa$-noncollapsing assumption. But if that assumption is violated for each $\kappa>0,$ then $\mathcal{V}(t)$ is not bounded away from zero as $t\to -\infty.$ However, this is impossible, because it is easy to see that $\mathcal{V}(t)$ is nonincreasing in $t.$ (Indeed, Ricci flow decreases the volume and does not decrease the distances faster than $C\sqrt{R}$ per time unit, by lemma 8.3(b)) Thus, $\kappa$-noncollapsing holds for some $\kappa>0,$ and we can apply the previous proposition to obtain a contradiction.

Corollary 11.6   For every $w>0$ there exist $B=B(w)<\infty , C=C(w)<\infty , \tau_0=\tau_0(w)>0,$ with the following properties.

(a) Suppose we have a (not necessarily complete) solution $g_{ij}(t)$ to the Ricci flow, defined on $M\times [t_0,0],$ so that at time $t=0$ the metric ball $B(x_0,r_0)$ is compactly contained in $M.$ Suppose that at each time $t, t_0\le t\le 0,$ the metric $g_{ij}(t)$ has nonnegative curvature operator, and $VolB(x_0,r_0)\ge wr_0^n.$ Then we have an estimate $R(x,t)\le Cr_0^{-2}+B(t-t_0)^{-1}$ whenever dist$_t(x,x_0)\le \frac{1}{4}r_0.$

(b) If, rather than assuming a lower bound on volume for all $t,$ we assume it only for $t=0,$ then the same conclusion holds with $-\tau_0r_0^2$ in place of $t_0,$ provided that $-t_0\ge \tau_0r_0^2.$

Proof. By scaling assume $r_0=1.$ (a) Arguing by contradiction, consider a sequence of $B,C\to \infty,$ of solutions $g_{ij}(t)$ and points $(x,t),$ such that dist$_t(x,x_0)\le \frac{1}{4}$ and $R(x,t)> C+ B(t-t_0)^{-1}.$ Then, arguing as in the proof of claims 1,2 in 10.1, we can find a point $(\bar{x},\bar{t}),$ satisfying dist$_{\bar{t}}(\bar{x},x_0)<\frac{1}{3}, Q=R(\bar{x},\bar{t})>C+B(\bar{t}-t_0)^{-1},$ and such that $R(x',t')\le 2Q$ whenever $\bar{t}-AQ^{-1}\le t'\le \bar{t},$   dist$_{\bar{t}}(x',\bar{x})<AQ^{-\frac{1}{2}},$ where $A$ tends to infinity with $B,C.$ Applying the previous corollary at $(\bar{x},\bar{t})$ and using the relative volume comparison, we get a contradiction with the assumption involving $w.$

(b) Let $B(w),C(w)$ be good for (a). We claim that $B=B(5^{-n}w),C=C(5^{-n}w)$ are good for (b) , for an appropriate $\tau_0(w)>0.$ Indeed, let $g_{ij}(t)$ be a solution with nonnegative curvature operator, such that $VolB(x_0,1)\ge w$ at $t=0,$ and let $[-\tau ,0]$ be the maximal time interval, where the assumption of (a) still holds, with $5^{-n}w$ in place of $w$ and with $-\tau$ in place of $t_0.$ Then at time $t=-\tau$ we must have $VolB(x_0,1)\le 5^{-n}w.$ On the other hand, from lemma 8.3 (b) we see that the ball $B(x_0,\frac{1}{4})$ at time $t=-\tau$ contains the ball $B(x_0,\frac{1}{4}-10(n-1)(\tau\sqrt{C}+2\sqrt{B\tau}))$ at time $t=0,$ and the volume of the former is at least as large as the volume of the latter. Thus, it is enough to choose $\tau_0=\tau_0(w)$ in such a way that the radius of the latter ball is $>\frac{1}{5}.$

Clearly, the proof also works if instead of assuming that curvature operator is nonnegative, we assumed that it is bounded below by $-r_0^{-2}$ in the (time-dependent) metric ball of radius $r_0,$ centered at $x_0.$

11.7   From now on we restrict our attention to oriented manifolds of dimension three. Under the assumptions in 11.1, the solutions on closed manifolds must be quotients of the round $\mathbb{S}^3$ or $\mathbb{S}^2\times\mathbb{R}$ - this is proved in the same way as in two dimensions, since the gradient shrinking solitons are known from the work of Hamilton [H 1,10]. The noncompact solutions are described below.

Theorem.The set of non-compact ancient solutions , satisfying the assumptions of 11.1, is compact modulo scaling. That is , from any sequence of such solutions and points $(x_k,0)$ with $R(x_k,0)=1,$ we can extract a smoothly converging subsequence, and the limit satisfies the same conditions.

Proof. To ensure a converging subsequence it is enough to show that whenever $R(y_k,0)\to\infty,$ the distances at $t=0$ between $x_k$ and $y_k$ go to infinity as well. Assume the contrary. Define a sequence $z_k$ by the requirement that $z_k$ be the closest point to $x_k$ (at $t=0$), satisfying $R(z_k,0)$dist$_0^2(x_k,z_k)=~1.$ We claim that $R(z,0)/R(z_k,0)$ is uniformly bounded for $z\in B(z_k,2R(z_k,0)^{-\frac{1}{2}}).$ Indeed, otherwise we could show, using 11.5 and relative volume comparison in nonnegative curvature, that the balls $B(z_k,R(z_k,0)^{-\frac{1}{2}})$ are collapsing on the scale of their radii. Therefore, using the local derivative estimate, due to W.-X.Shi (see [H 4,$\S 13$]), we get a bound on $R_t(z_k,t)$ of the order of $R^2(z_k,0).$ Then we can compare $1=R(x_k,0)\ge cR(z_k,-cR^{-1}(z_k,0))\ge cR(z_k,0)$ for some small $c>0,$ where the first inequality comes from the Harnack inequality, obtained by integrating (11.1). Thus, $R(z_k,0)$ are bounded. But now the existence of the sequence $y_k$ at bounded distance from $x_k$ implies, via 11.5 and relative volume comparison, that balls $B(x_k,c)$ are collapsing - a contradiction.

It remains to show that the limit has bounded curvature at $t=0.$ If this was not the case, then we could find a sequence $y_i$ going to infinity, such that $R(y_i,0)\to\infty$ and $R(y,0)\le 2R(y_i,0)$ for $y\in B(y_i,A_iR(y_i,0)^{-\frac{1}{2}}), A_i\to\infty .$ Then the limit of scalings at $(y_i,0)$ with factors $R(y_i,0)$ satisfies the assumptions in 11.1 and splits off a line. Thus by 11.3 it must be a round infinite cylinder. It follows that for large $i$ each $y_i$ is contained in a round cylindrical "neck" of radius $(\frac{1}{2}R(y_i,0))^{-\frac{1}{2}}\to 0,$ - something that can not happen in an open manifold of nonnegative curvature.

11.8   Fix $\epsilon >0.$ Let $g_{ij}(t)$ be an ancient solution on a noncompact oriented three-manifold $M,$ satisfying the assumptions in 11.1. We say that a point $x_0\in M$ is the center of an $\epsilon$-neck, if the solution $g_{ij}(t)$ in the set $\{(x,t): -(\epsilon Q)^{-1}<t\le 0,$   dist$_0^2(x,x_0)<(\epsilon Q)^{-1}\},$ where $Q=R(x_0,0),$ is, after scaling with factor $Q,$ $\epsilon$-close (in some fixed smooth topology) to the corresponding subset of the evolving round cylinder, having scalar curvature one at $t=0.$

Corollary (from theorem 11.7 and its proof) For any $\epsilon >0$ there exists $C=C(\epsilon ,\kappa)>0,$ such that if $g_{ij}(t)$ satisfies the assumptions in 11.1, and $M_\epsilon$ denotes the set of points in $M,$ which are not centers of $\epsilon$-necks, then $M_{\epsilon }$ is compact and moreover, diam$M_{\epsilon } \le CQ^{-\frac{1}{2}},$ and $C^{-1}Q\le R(x,0)\le CQ$ whenever $x\in M_{\epsilon },$ where $Q=R(x_0,0)$ for some $x_0\in \partial M_{\epsilon }.$

11.9 Remark. It can be shown that there exists $\kappa_0>0,$ such that if an ancient solution on a noncompact three-manifold satisfies the assumptions in 11.1 with some $\kappa>0,$ then it would satisfy these assumptions with $\kappa=\kappa_0.$ This follows from the arguments in 7.3, 11.2, and the statement (which is not hard to prove) that there are no noncompact three-dimensional gradient shrinking solitons, satisfying 11.1, other than the round cylinder and its $\mathbb{Z}_2$-quotients.

Furthermore, I believe that there is only one (up to scaling) noncompact three-dimensional $\kappa$-noncollapsed ancient solution with bounded positive curvature - the rotationally symmetric gradient steady soliton, studied by R.Bryant. In this direction, I have a plausible, but not quite rigorous argument, showing that any such ancient solution can be made eternal, that is, can be extended for $t\in (-\infty ,+\infty);$ also I can prove uniqueness in the class of gradient steady solitons.

11.10* The earlier work on ancient solutions and all that can be found in [H 4, $\S 16-22,25,26$].