Ancient solutions with bounded entropy

1.1 In this section we review some of the results, proved or quoted in [I,§11], correcting a few inaccuracies. We consider smooth solutions $g_{ij}(t)$ to the Ricci flow on oriented 3-manifold $M$, defined for $-\infty<t\le 0$, such that for each $t$ the metric $g_{ij}(t)$ is a complete non-flat metric of bounded nonnegative sectional curvature, $\kappa$-noncollapsed on all scales for some fixed $\kappa>0;$ such solutions will be called ancient $\kappa$-solutions for short. By Theorem I.11.7, the set of all such solutions with fixed $\kappa$ is compact modulo scaling, that is from any sequence of such solutions $(M^{\alpha },g_{ij}^{\alpha }(t))$ and points $(x^{\alpha },0)$ with $R(x^{\alpha },0)=1$, we can extract a smoothly (pointed) convergent subsequence, and the limit $(M,g_{ij}(t))$ belongs to the same class of solutions. (The assumption in I.11.7. that $M^{\alpha }$ be noncompact was clearly redundant, as it was not used in the proof. Note also that $M$ need not have the same topology as $M^{\alpha }.)$ Moreover, according to Proposition I.11.2, the scalings of any ancient $\kappa$-solution $g_{ij}(t)$ with factors $(-t)^{-1}$ about appropriate points converge along a subsequence of $t\to -\infty$ to a non-flat gradient shrinking soliton, which will be called an asymptotic soliton of the ancient solution. If the sectional curvature of this asymptotic soliton is not strictly positive, then by Hamilton's strong maximum principle it admits local metric splitting, and it is easy to see that in this case the soliton is either the round infinite cylinder, or its $\mathbb{Z}_2$ quotient, containing one-sided projective plane. If the curvature is strictly positive and the soliton is compact, then it has to be a metric quotient of the round 3-sphere, by [H 1]. The noncompact case is ruled out below.

1.2 Lemma. There is no (complete oriented 3-dimensional) noncompact $\kappa$-noncollapsed gradient shrinking soliton with bounded positive sectional curvature.

Proof. A gradient shrinking soliton $g_{ij}(t), -\infty<t<0,$ satisfies the equation

$$\nabla_i\nabla_j f+R_{ij}+\frac{1}{2t}g_{ij}=0$$ (1.1)

Differentiating and switching the order of differentiation, we get

$$\nabla_i R=2R_{ij}\nabla_j f$$ (1.2)

Fix some $t<0,$ say $t=-1,$ and consider a long shortest geodesic $\gamma(s), 0\le s\le \bar{s};$ let $x=\gamma(0), \bar{x}=\gamma(\bar{s}), X(s)=\dot{\gamma}(s).$ Since the curvature is bounded and positive, it is clear from the second variation formula that $\int_0^{\bar{s}}{\mathrm{Ric}(X,X)ds}\le \mathrm{const}.$ Therefore, $\int_0^{\bar{s}}{\vert\mathrm{Ric}(X,\cdot)\vert^2ds}\le \mathrm{const},$ and $\int_0^{\bar{s}}{\vert\mathrm{Ric}(X,Y)\vert ds}\le\mathrm{const}(\sqrt{\bar{s}}+1)$ for any unit vector field $Y$ along $\gamma,$ orthogonal to $X.$ Thus by integrating (1.1) we get $X\cdot f(\gamma(\bar{s}))\ge \frac{\bar{s}}{2}+\mathrm{const}, \vert Y\cdot f(\gamma(\bar{s}))\vert\le \mathrm{const} (\sqrt{\bar{s}}+1).$ We conclude that at large distances from $x_0$ the function $f$ has no critical points, and its gradient makes small angle with the gradient of the distance function from $x_0.$

Now from (1.2) we see that $R$ is increasing along the gradient curves of $f,$ in particular, $\bar{R}=\mathrm{lim \ sup \ }R>0.$ If we take a limit of our soliton about points $(x^{\alpha}, -1)$ where $R(x^{\alpha})\to\bar{R},$ then we get an ancient $\kappa$-solution, which splits off a line, and it follows from I.11.3, that this solution is the shrinking round infinite cylinder with scalar curvature $\bar{R}$ at time $t=-1.$ Now comparing the evolution equations for the scalar curvature on a round cylinder and for the asymptotic scalar curvature on a shrinking soliton we conclude that $\bar{R}=1.$ Hence, $R(x)<1$ when the distance from $x$ to $x_0$ is large enough, and $R(x)\to 1$ when this distance tends to infinity.

Now let us check that the level surfaces of $f,$ sufficiently distant from $x_0,$ are convex. Indeed, if $Y$ is a unit tangent vector to such a surface, then $\nabla_Y\nabla_Y f= \frac{1}{2}-\mathrm{Ric}(Y,Y)\ge \frac{1}{2}-\frac{R}{2}>0.$ Therefore, the area of the level surfaces grows as $f$ increases, and is converging to the area of the round sphere of scalar curvature one. On the other hand, the intrinsic scalar curvature of a level surface turns out to be less than one. Indeed, denoting by $X$ the unit normal vector, this intrinsic curvature can be computed as

$$R-2\mathrm{Ric}(X,X)+2\frac{\mathrm{det}(\mathrm{Hess} f)}{\vert\... ...R-2\mathrm{Ric}(X,X)+\frac{(1-R+\mathrm{Ric}(X,X))^2}{2\vert\nabla f\vert^2}<1$$

when $R$ is close to one and $\vert\nabla f\vert$ is large. Thus we get a contradiction to the Gauss-Bonnet formula.

1.3 Now, having listed all the asymptotic solitons, we can classify the ancient $\kappa$-solutions. If such a solution has a compact asymptotic soliton, then it is itself a metric quotient of the round 3-sphere, because the positive curvature pinching can only improve in time [H 1]. If the asymptotic soliton contains the one-sided projective plane, then the solution has a $\mathbb{Z}_2$ cover, whose asymptotic soliton is the round infinite cylinder. Finally, if the asymptotic soliton is the cylinder,then the solution can be either noncompact (the round cylinder itself, or the Bryant soliton, for instance), or compact. The latter possibility, which was overlooked in the first paragraph of [I.11.7], is illustrated by the example below, which also gives the negative answer to the question in the very end of [I.5.1].

1.4 Example. Consider a solution to the Ricci flow, starting from a metric on $\mathbb{S}^3$ that looks like a long round cylinder $\mathbb{S}^2\times\mathbb{I}$ (say, with radius one and length $L>>1$), with two spherical caps, smoothly attached to its boundary components. By [H 1] we know that the flow shrinks such a metric to a point in time, comparable to one (because both the lower bound for scalar curvature and the upper bound for sectional curvature are comparable to one) , and after normalization, the flow converges to the round 3-sphere. Scale the initial metric and choose the time parameter in such a way that the flow starts at time $t_0=t_0(L)<0,$ goes singular at $t=0,$ and at $t=-1$ has the ratio of the maximal sectional curvature to the minimal one equal to $1+\epsilon .$ The argument in [I.7.3] shows that our solutions are $\kappa$-noncollapsed for some $\kappa>0$ independent of $L.$ We also claim that $t_0(L)\to -\infty$ as $L\to\infty.$ Indeed, the Harnack inequality of Hamilton [H 3] implies that $\ \ R_t\ge\frac{R}{t_0-t},\ \ \ \$ hence $\ \ R\le\frac{2(-1-t_0)}{t-t_0}$ for $t\le -1,\ \ \ \ \ \ \ \$ and then the distance change estimate $\ \ \ \frac{d}{dt}\mathrm{dist}_t(x,y)\ge -\mathrm{const}\sqrt{R_{\mathrm{max}}(t)}\ \ \ \$ from [H 2,§17] implies that the diameter of $g_{ij}(t_0)$ does not exceed $-\mathrm{const}\cdot t_0$, which is less than $L\sqrt{-t_0}$ unless $t_0$ is large enough. Thus, a subsequence of our solutions with $L\to\infty$ converges to an ancient $\kappa$-solution on $\mathbb{S}^3,$ whose asymptotic soliton can not be anything but the cylinder.

1.5 The important conclusion from the classification above and the proof of Proposition I.11.2 is that there exists $\kappa_0>0,$ such that every ancient $\kappa$-solution is either $\kappa_0$-solution, or a metric quotient of the round sphere. Therefore, the compactness theorem I.11.7 implies the existence of a universal constant $\eta,$ such that at each point of every ancient $\kappa$-solution we have estimates

$$\vert\nabla R\vert<\eta R^{\frac{3}{2}}, \vert R_t\vert<\eta R^2$$ (1.3)

Moreover, for every sufficiently small $\epsilon >0$ one can find $C_{1,2}=C_{1,2}(\epsilon ),$ such that for each point $(x,t)$ in every ancient $\kappa$-solution there is a radius $r, 0<r<C_1R(x,t)^{-\frac{1}{2}},$ and a neighborhood $B, B(x,t,r)\subset B\subset B(x,t,2r),$ which falls into one of the four categories:

(a) $B$ is a strong $\epsilon$-neck (more precisely, the slice of a strong $\epsilon$-neck at its maximal time), or

(b) $B$ is an $\epsilon$-cap, or

(c) $B$ is a closed manifold, diffeomorphic to $\mathbb{S}^3$ or $\mathbb{RP}^3,$ or

(d) $B$ is a closed manifold of constant positive sectional curvature;

furthermore, the scalar curvature in $B$ at time $t$ is between $C_2^{-1}R(x,t)$ and $C_2R(x,t),$ its volume in cases (a),(b),(c) is greater than $C_2^{-1}R(x,t)^{-\frac{3}{2}},$ and in case (c) the sectional curvature in $B$ at time $t$ is greater than $C_2^{-1}R(x,t).$