Justification of the a priori assumption

5.1 Let us call a riemannian manifold $(M,g_{ij})$ normalized if $M$ is a closed oriented 3-manifold, the sectional curvatures of $g_{ij}$ do not exceed one in absolute value, and the volume of every metric ball of radius one is at least half the volume of the euclidean unit ball. For smooth Ricci flow with normalized initial data we have, by [H 4, 4.1], at any time $t>0$ the pinching estimate

$$Rm\ge-\phi(R(t+1))R,$$ (5.1)

where $\phi$ is a decreasing function, which behaves at infinity like $\frac{1}{\mathrm{log}}.$ As explained in 4.4, this pinching estimate can be preserved for Ricci flow with $\delta$-cutoff. Justification of the canonical neighborhood assumption requires additional arguments. In fact, we are able to construct solutions satisfying this assumption only allowing $r$ and $\delta$ be functions of time rather than constants; clearly, the arguments of the previous section are valid in this case, if we assume that $r(t),\ \delta (t)$ are non-increasing, and bounded away from zero on every finite time interval.

Proposition. There exist decreasing sequences $0<r_j<\epsilon ^2, \kappa _j>0, 0<\bar{\delta }_j<\epsilon ^2, j=1,2,...,$ such that for any normalized initial data and any function $\delta (t),$ satisfying $0<\delta (t)<\bar{\delta }_j$ for $t\in [2^{j-1}\epsilon ,2^j\epsilon ],$ the Ricci flow with $\delta (t)$-cutoff is defined for $t\in[0,+\infty]$ and satisfies the $\kappa _j$-noncollapsing assumption and the canonical neighborhood assumption with parameter $r_j$ on the time interval $[2^{j-1}\epsilon ,2^j\epsilon ].$( Recall that we have excluded from the list of canonical neighborhoods the closed manifolds, $\epsilon$-close to metric quotients of the round sphere. Complete extinction of the solution in finite time is not ruled out.)

The proof of the proposition is by induction: having constructed our sequences for $1\le j\le i,$ we make one more step, defining $r_{i+1}, \kappa _{i+1}, \bar{\delta }_{i+1},$ and redefining $\bar{\delta }_i=\bar{\delta }_{i+1};$ each step is analogous to the proof of Theorem I.12.1.

First we need to check a $\kappa$-noncollapsing condition.

5.2 Lemma. Suppose we have constructed the sequences, satisfying the proposition for $1\le j\le i.$ Then there exists $\kappa >0,$ such that for any $r, 0<r<\epsilon ^2,$ one can find $\bar{\delta }=\bar{\delta }(r)>0,$ which may also depend on the already constructed sequences, with the following property. Suppose we have a solution to the Ricci flow with $\delta (t)$-cutoff on a time interval $[0,T],$ with normalized initial data, satisfying the proposition on $[0,2^i\epsilon ],$ and the canonical neighborhood assumption with parameter $r$ on $[2^i\epsilon ,T],$ where $2^i\epsilon \le T\le 2^{i+1}\epsilon ,\ \ 0<\delta (t)<\bar{\delta }\ \ for \ \ t\in [2^{i-1}\epsilon ,T] .$ Then it is $\kappa$-noncollapsed on all scales less than $\epsilon .$

Proof. Consider a neighborhood $P(x_0,t_0,r_0,-r_0^2), 2^i\epsilon <t_0\le T, 0<r_0<\epsilon ,$ where the solution is defined and satisfies $\vert Rm\vert\le r_0^{-2}.$ We may assume $r_0\ge r,$ since otherwise the lower bound for the volume of the ball $B(x_0,t_0,r_0)$ follows from the canonical neighborhood assumption. If the solution was smooth everywhere, we could estimate from below the volume of the ball $B(x_0,t_0,r_0)$ using the argument from [I.7.3]: define $\tau(t)=t_0-t$ and consider the reduced volume function using the $\mathcal{L}$-exponential map from $x_0;$ take a point $(x,\epsilon )$ where the reduced distance $l$ attains its minimum for $\tau=t_0-\epsilon ,$ $\ \ l(x,\tau)\le 3/2;$ use it to obtain an upper bound for the reduced distance to the points of $B(x,0,1),$ thus getting a lower bound for the reduced volume at $\tau=t_0,$ and apply the monotonicity formula. Now if the solution undergoes surgeries, then we still can measure the $\mathcal{L}$-length, but only for admissible curves, which stay in the region, unaffected by surgery. An inspection of the constructions in [I,§7] shows that the argument would go through if we knew that every barely admissible curve, that is a curve on the boundary of the set of admissible curves, has reduced length at least $3/2+\kappa '$ for some fixed $\kappa '>0.$ Unfortunately, at the moment I don't see how to ensure that without imposing new restrictions on $\delta (t)$ for all $t\in [0,T],$ so we need some additional arguments.

Recall that for a curve $\gamma,$ parametrized by $t,$ with $\gamma(t_0)=x_0,$ we have $\mathcal{L}(\gamma ,\tau)= \int_{t_0-\tau}^{t_0}{\sqrt{t_0-t}(R(\gamma(t),t)+\vert\dot{\gamma}(t)\vert^2)dt}.$ We can also define $\mathcal{L}_+(\gamma,\tau)$ by replacing in the previous formula $R$ with $R_+=\mathrm{max}(R,0).$ Then $\mathcal{L}_+\le \mathcal{L}+4T\sqrt{T}$ because $R\ge -6$ by the maximum principle and normalization. Now suppose we could show that every barely admissible curve with endpoints $(x_0,t_0)$ and $(x,t),$ where $t\in [2^{i-1}\epsilon ,T),$ has $\mathcal{L}_+> 2\epsilon ^{-2}T\sqrt{T};$ then we could argue that either there exists a point $(x,t), t\in [2^{i-1}\epsilon ,2^i\epsilon ],$ such that $R(x,t)\le r_i^{-2}$ and $\mathcal{L}_+\le \epsilon ^{-2}T\sqrt{T},$ in which case we can take this point in place of $(x,\epsilon )$ in the argument of the previous paragraph, and obtain (using Claim 1 in 4.2) an estimate for $\kappa$ in terms of $r_i,\kappa _i,T,$ or for any $\gamma,$ defined on $[2^{i-1}\epsilon ,t_0], \gamma(t_0)=x_0,$ we have $\mathcal{L}_+\ge \mathrm{min}(\epsilon ^{-2}T\sqrt{T}, \frac{2}{3}(2^{i-1}\epsilon )^{\frac{3}{2}} r_i^{-2})>\epsilon ^{-2}T\sqrt{T},$ which is in contradiction with the assumed bound for barely admissible curves and the bound $\mathrm{min}\ l(x,t_0-2^{i-1}\epsilon )\le 3/2,$ valid in the smooth case. Thus, to conclude the proof it is sufficient to check the following assertion.

5.3 Lemma. For any $\mathcal{L}<\infty$ one can find $\bar{\delta }=\bar{\delta }(\mathcal{L},r_0)>0$ with the following property. Suppose that in the situation of the previous lemma we have a curve $\gamma,$ parametrized by $t\in[T_0,t_0], 2^{i-1}\epsilon \le T_0<t_0,$ such that $\gamma(t_0)=x_0,$ $T_0$ is a surgery time, and $\gamma(T_0)\in B(p,T_0,\epsilon ^{-1}h),$ where $p$ corresponds to the center of the cap, and $h$ is the radius of the $\delta$-neck. Then we have an estimate $\int_{T_0}^{t_0}{\sqrt{t_0-t}(R_+(\gamma(t),t)+\vert\dot{\gamma}(t)\vert^2)dt}\ge\mathcal{L}.$

Proof. It is clear that if we take $\triangle t=\epsilon r_0^4\mathcal{L}^{-2},$ then either $\gamma$ satisfies our estimate, or $\gamma$ stays in $P(x_0,t_0,r_0,-\triangle t)$ for $t\in [t_0-\triangle t,t_0].$ In the latter case our estimate follows from Corollary 4.6, for $l=\mathcal{L}(\triangle t)^{-\frac{1}{2}},$ since clearly $T_{\gamma}<t_0-\triangle t$ when $\delta$ is small enough.

5.4 Proof of proposition. Assume the contrary, and let the sequences $r^{\alpha }, \bar{\delta }^{\alpha \beta }$ be such that $r^{\alpha }\to 0$ as $\alpha \to\infty,$ $\ \ \bar{\delta }^{\alpha \beta }\to 0$ as $\beta \to\infty$ with fixed $\alpha ,$ and let $(M^{\alpha \beta },g_{ij}^{\alpha \beta })$ be normalized initial data for solutions to the Ricci flow with $\delta (t)$-cutoff, $\delta (t)<\bar{\delta }^{\alpha \beta }$ on $[2^{i-1}\epsilon ,2^{i+1}\epsilon ],$ which satisfy the statement on $[0,2^i\epsilon ],$ but violate the canonical neighborhood assumption with parameter $r^{\alpha }$ on $[2^i\epsilon ,2^{i+1}\epsilon ].$ Slightly abusing notation, we'll drop the indices $\alpha ,\beta$ when we consider an individual solution.

Let $\bar{t}$ be the first time when the assumption is violated at some point $\bar{x};$ clearly such time exists, because it is an open condition. Then by lemma 5.2 we have uniform $\kappa$-noncollapsing on $[0,\bar{t}].$ Claims 1,2 in 4.2 are also valid on $[0,\bar{t}];$ moreover, since $h<<r,$ it follows from Claim 1 that the solution is defined on the whole parabolic neighborhood indicated there in case $R(x_0,t_0)\le r^{-2}.$

Scale our solution about $(\bar{x},\bar{t})$ with factor $R(\bar{x},\bar{t})\ge r^{-2}$ and take a limit for subsequences of $\alpha ,\beta \to\infty.$ At time $\bar{t},$ which we'll shift to zero in the limit, the curvature bounds at finite distances from $\bar{x}$ for the scaled metric are ensured by Claim 2 in 4.2. Thus, we get a smooth complete limit of nonnegative sectional curvature, at time zero. Moreover, the curvature of the limit is uniformly bounded, since otherwise it would contain $\epsilon$-necks of arbitrarily small radius.

Let $Q_0$ denote the curvature bound. Then, if there was no surgery, we could, using Claim 1 in 4.2, take a limit on the time interval $[-\epsilon \eta^{-1}Q_0^{-1},0].$ To prevent this, there must exist surgery times $T_0\in [\bar{t}-\epsilon \eta^{-1}Q_0^{-1}R^{-1}(\bar{x},\bar{t}),\bar{t}]$ and points $x$ with $\mathrm{dist}^2_{T_0}(x,\bar{x})R^{-1}(\bar{x},\bar{t})$ uniformly bounded as $\alpha ,\beta \to\infty ,$ such that the solution at $x$ is defined on $[T_0,\bar{t}],$ but not before $T_0.$ Using Claim 2 from 4.2 at time $T_0,$ we see that $R(\bar{x},\bar{t})h^2(T_0)$ must be bounded away from zero. Therefore, in this case we can apply Corollary 4.7, Lemma 4.5 and Claim 5 in section 2 to show that the point $(\bar{x},\bar{t})$ in fact has a canonical neighborhood, contradicting its choice. (It is not excluded that the strong $\epsilon$-neck neighborhood extends to times before $T_0,$ where it is a part of the strong $\delta$-neck that existed before surgery.)

Thus we have a limit on a certain time interval. Let $Q_1$ be the curvature bound for this limit. Then we either can construct a limit on the time interval $[-\epsilon \eta^{-1}(Q_0^{-1}+Q_1^{-1}),0],$ or there is a surgery, and we get a contradiction as before. We can continue this procedure indefinitely, and the final part of the proof of Theorem I.12.1 shows that the bounds $Q_k$ can not go to infinity while the limit is defined on a bounded time interval. Thus we get a limit on $(-\infty,0],$ which is $\kappa$-noncollapsed by Lemma 5.2, and this means that $(\bar{x},\bar{t})$ has a canonical neighborhood by the results of section 1 - a contradiction.