We will fix $\epsilon({\rm v})>0$ later. For the moment let any $\epsilon_{1}>0$ be arbitrary with $\delta_{1}({\rm v},\epsilon_{1})>0$ the corresponding number from Theorem 8.3. If $T^{\delta_{1}}_{\alpha_{1}}(x)=0$ then there exists a diffeomorphism

where $0\in\mathds{R}^{4}/\Gamma_{\alpha_{1}}$ and $A_{r_{\alpha_{1}}/2,2r_{\alpha_{1}}}(x)\subseteq U_{\alpha}\subseteq A_{(1-\epsilon)r_{\alpha_{1}}/2,(1+\epsilon)r_{\alpha_{1}}}(x)$, such that

In particular, if $\epsilon>0$ is fixed and $2\delta(n,\epsilon)$ is the corresponding number from Theorem 8.3, then we can choose $\epsilon_{1}=\epsilon_{1}(\epsilon,{\rm v})$ sufficiently small so that

Thus, if $\alpha_{2}$ is such that

then for all $\alpha_{1}\leq\alpha\leq\alpha_{2}$ we have $T^{2\delta}_{\alpha}(x)=0$.

By Theorem 8.3, there exists for each $\alpha_{1}\leq\alpha\leq\alpha_{2}$, a diffeomorphism

where $0\in\mathds{R}^{4}/\Gamma_{\alpha}$ and $A_{r_{\alpha}/2,2r_{\alpha}}(x)\subseteq U_{\alpha}\subseteq A_{(1-\epsilon)r_{\alpha}/2,2(1+\epsilon)r_{\alpha}}(x)$, such that

In particular this implies that $\Gamma_{\alpha}=\Gamma$ is independent of $\alpha$.

Next we focus on the inverse maps

Observe that by (8.36), after possibly composing $\Phi_{\alpha}$ with a rotation of $\mathds{R}^{4}/\Gamma$ we can assume for $x\in U_{\alpha}\cap U_{\beta}$ that

Now let $\epsilon<\epsilon({\rm v})$ be sufficiently small, so that if $x\in\mathds{R}^{4}/\Gamma$, then $B_{\epsilon|x|}(x)\subseteq\mathds{R}^{4}/\Gamma$ is isometric to the standard Euclidean ball $B_{\epsilon|x|}(0^{4})\subseteq\mathds{R}^{4}$. Note in particular that if $\{x_{i}\}\in B_{\epsilon|x|}(x)$ is a collection of points, then any convex combination is well defined.

For each $\alpha$ let $\varphi^{\prime}_{\alpha}:U_{\alpha}\to\mathds{R}$ be a smooth cutoff function such that

and such that $|\nabla\varphi^{\prime}_{\alpha}|\leq 10r_{\alpha}^{-1}$. If we set $\varphi^{\prime}(x)=\sum_{\alpha}\varphi^{\prime}_{\alpha}(x)$ then $1\leq\varphi^{\prime}(x)\leq 4$. In particular,

sarisfies $\sum\varphi_{\alpha}(x)=1$, and so, is a partition of unity, with $|\nabla\varphi_{\alpha}|\leq 40r_{\alpha}^{-1}$.

Define the map

given by

(As previously noted, the convex combination is well defined since the $\Phi^{-1}_{\alpha}(x)$ all live in a ball which is isometric to a Euclidean ball.) On each domain, $U_{\alpha}$, we have by (8.32) and (8.38) that $\Phi^{-1}$ and $\Phi^{-1}_{\alpha}$ are $C^{1}$-close. Hence, $\Phi^{-1}$ is a diffeomorphism, and a quick computation using (8.32) and (8.38) verifies the desired estimates:

By choosing $\epsilon$ appropriately small, we complete the proof. ∎

First note by Theorem 8.3 that if $\delta$ is fixed, then there exists $\delta^{\prime}({\rm v},\delta)$ such that if $|{\rm Ric}|\leq 3\delta^{\prime}$ and if $T^{\delta^{\prime}}_{0}=0$, then

By rescaling this inequality, we see that in the context of this lemma, the following holds. If $x\in B_{1}(p)$, $\alpha>\bar{\alpha}(v,\delta)$ and

then we have

In particular, for $x\in B_{1}(p)$, we can apply Lemma 8.5 to see that there exists a scale $\alpha\leq\bar{\alpha}(v,\delta)$ such that

and hence

However, if

this implies $|\Gamma_{\alpha}|<N$, which completes the proof. ∎

Let $\delta>0$ be chosen with $\delta^{\prime}({\rm v},\delta)$ to be chosen later. Note that by Lemma 8.5, for each $x\in B_{1}(p)$ there exists $\alpha_{x}\leq\bar{\alpha}(v,\delta^{\prime})$ such that $T^{\delta^{\prime}}_{\alpha_{x}}(x)=0$. Consider the covering $\{B_{r_{\alpha_{x}}}(x)\}$ of $B_{1}(p)$, and choose an efficient subcovering $\{B_{r_{j}}(x^{\prime}_{j})\}_{1}^{N}$, where $r_{j}=r_{\alpha_{x^{\prime}_{j}}}$ and the balls in $\{B_{r_{j}/4}(x^{\prime}_{j})\}$ are disjoint. The usual doubling arguments imply that $N\leq N(v,\delta^{\prime})$.

By Theorem 8.3, if we are given $\epsilon>0$, then we can choose $\delta^{\prime}({\rm v},\epsilon,\delta)$ such that for each $x\in B_{\epsilon r_{j}}(x^{\prime}_{j})$ we have $T^{\delta}_{\alpha_{j}}(x)=0$, while for each $x\in A_{\epsilon r_{j},2r_{j}}(x_{j})$ we have $r_{h}(x)>\bar{r}(v,\epsilon)r_{j}\geq r_{0}(v,\epsilon,\delta^{\prime})$. Let $\Gamma_{j}$ be the group associated to $B_{r_{j}}(x^{\prime}_{j})$, and for each $x\in B_{\epsilon r_{j}}(x^{\prime}_{j})$ let $\beta_{j}(x)$ be the largest integer such that $V^{\delta}_{r_{\beta_{j}}/4}(x)\geq\ln|\Gamma_{j}|-\delta$. Let $\beta_{j}=\max\beta_{j}(x)$ with $x_{j}$ the corresponding point. Note that for $\epsilon(v,\delta)$ sufficiently small, we have $B_{2r_{\beta_{j}}}(x_{j})\subseteq B_{\epsilon r_{j}}(x^{\prime}_{j})$, and in particular, for every $x\in B_{2r_{\beta_{j}}}(x_{j})$

Consider the collection of balls $\{B_{r_{j}}(x_{j})\}$. Clearly, by construction, conditions (1) and (3) are satisfied. If $x\in B_{1}(p)\setminus\{B_{r_{j}}(x_{j})\}$ then since $\{B_{2r_{j}}(x_{j})\}$ cover $B_{1}(p)$ we have that for some $x_{j}$ that $x\in A_{r_{j},2r_{j}}(x_{j})$, which implies $r_{h}(x)\geq r_{0}(v,\delta)$, as claimed. ∎

Let us remark first, that if $p\in M^{n}$, then by volume ratio monotonicity, we have for every $r\leq 1$ that

Let $\epsilon<\epsilon({\rm v}^{\prime})$ from Lemma 8.8 with $\delta({\rm v},D,\epsilon)$ sufficiently small to satisfy Theorem 8.3 and Lemmas 8.8, 8.9, 8.10. After rescaling, it is sufficient to consider a Riemannian manifold $(M^{4},g)$ with $|{\rm Ric}_{M^{4}}|\leq 3\delta$, ${\rm diam}(M)\leq D\delta^{-2}=D^{\prime}$ and ${\rm Vol}(B_{1}(p))>{\rm v}^{\prime}>0$ for every $p\in M$.

Let us begin by efficiently covering $M^{4}$ by balls $\{B_{1}(x^{0}_{j})\}$ such that the balls in $\{B_{1/4}(x^{0}_{j})\}$ are disjoint. By the usual doubling argument, there are at most $N(n,D,{\rm v})$ such balls. For each such ball, we apply Lemma 8.10 in order to produce a collection of balls $\{B_{r^{1}_{j}}(x^{1}_{j})\}_{1}^{N_{1}}$ such that $r^{1}_{j}=r_{\alpha^{1}_{j}}>\bar{r}({\rm v},D)$, $N_{1}\leq N({\rm v}^{\prime},D^{\prime})$, $T^{\delta}_{\alpha^{1}_{j}}(x^{1}_{j})=0$, and such that if $x\in M^{4}\setminus\bigcup_{j}B_{r^{1}_{j}}(x^{1}_{j})$ then $r_{h}(x)>\bar{r}$. Furthermore, if we denote by $\Gamma^{2}_{j}$, the group associated to $B_{r^{1}_{j}}(x^{1}_{j})$, then if $\beta^{1}_{j}$ is the largest integer such that $V^{\delta}_{r_{\beta^{1}_{j}}/2}(x^{1}_{j})\geq\ln|\Gamma_{j}^{2}|-\delta$, then for all $x\in B_{2r_{\beta^{1}_{j}}}(x^{1}_{j})$ we have

Define

as the first body region. Then we can write

where by using Theorem 8.3, we have that $B_{2r^{1}_{j}}(x^{1}_{j})\cap\mathcal{B}^{1}$ is diffeomorphic to $\mathds{R}\times S^{3}/\Gamma^{1}_{j}$.

Now to prove the theorem, let us inductively build a decomposition of $M^{4}$

with the following properties:

1. (1)

   If $x\in\mathcal{B}^{\ell}_{j}$ then $r_{h}(x)>r_{0}(n,{\rm v},D)\cdot{\rm diam}(\mathcal{B}^{\ell}_{j})$.

2. (2)

   Each neck $\mathcal{N}^{\ell}_{j}$ is diffeomorphic to $\mathds{R}\times S^{3}/\Gamma^{\ell}_{j}$ for some $\Gamma^{\ell}_{j}<O(4)$.

3. (3)

   $\mathcal{N}^{\ell}_{j}\cap\mathcal{B}^{\ell}_{j}$ are diffeomorphic to $\mathds{R}\times S^{3}/\Gamma^{\ell}_{j}$. $\mathcal{N}^{\ell}_{j}\cap\mathcal{B}^{\ell-1}_{j^{\prime}}$ are either empty or diffeomorphic to $\mathds{R}\times S^{3}/\Gamma^{\ell}_{j}$.

4. (4)

   $N_{\ell}\leq N(n,{\rm v},D)$.

5. (5)

   If $\mathcal{N}^{\ell+1}_{a}\cap\mathcal{B}^{\ell}_{j}\neq\emptyset$, then $|\Gamma^{\ell}_{a}|\leq|\Gamma^{\ell}_{j}|-1$.

6. (6)

   We have $r^{k}_{a}=r_{\alpha^{k}_{a}}$ with $T^{\delta}_{\alpha^{k}_{a}}=0$, and $\mathcal{B}^{k}_{j}\cap B_{r^{k}_{a}}(x_{a})\subseteq A_{r^{k}_{a}/2,r^{k}_{a}}(x_{a})$.

7. (7)

   If $\beta^{k}_{a}$ is the largest integer such that $\mathcal{V}^{\delta}_{r_{\beta^{k}_{a}}/4}(x_{a})\geq\ln|\Gamma^{k}_{a}|-\delta$, then for every $x\in B_{2r_{\beta^{k}_{a}}}(x_{a})$ we have $\mathcal{V}^{\delta}_{r_{\beta^{k}_{a}}/8}(x)<\ln|\Gamma^{k}_{a}|-\delta$.

Before building the inductive decomposition, let us note that once we have it, we will have finished the proof. In fact, all we really need to see is that for some $k\leq k(n,{\rm v},D)$, there are no balls $\{B_{r^{k}_{a}}(x_{a})\}$ in the decomposition. To see this, observe that by the lower volume bound we have the upper order bound $|\Gamma^{2}_{j}|\leq C({\rm v},D)$. By condition (5) above we have by iteration that for each $j$ that there is some $j_{2}$ such that

and in particular this immediately implies the upper bound

To prove the inductive decomposition, we begin by noting that (8.55) provides the basic case. So let us assume that the decomposition has been constructed for some $k$, and let us build the decomposition for $k+1$.

First, we use condition (7) and Lemma 8.8 to see that there exists an open set

and a diffeomorphism $\Phi^{k+1}_{a}:\mathcal{N}^{k+1}_{a}\to A_{r_{\beta^{k}_{a}}/2,2r_{\alpha^{k}_{a}}}(0)$ with $0\in\mathds{R}^{4}/\Gamma^{k}_{a}$. By Lemma 8.9, there exists a radius $r_{a}=\bar{r}({\rm v},\delta)r_{\beta^{k}_{a}}$ such that

for every $x\in B_{2r_{\beta^{k}_{a}}}(x_{a})$.

Pick some efficient covering $\{B_{r_{a}}(x_{aj}\}$ of $B_{2r_{\beta^{k}_{a}}}(x_{a})$ such that the balls in $\{B_{r_{a}/4}(x_{aj}\}$ disjoint. Now apply Lemma 8.10 to each ball $\{B_{r_{a}}(x_{aj}\}$ in order to construct a collection of balls $\{B_{r^{k+1}_{ab}}(x^{k+1}_{ab})\}$ with $r^{k+1}_{ab}=r_{\alpha^{k+1}_{ab}}>\bar{r}({\rm v},\delta)r_{\beta^{k}_{a}}$. Observe that since there are at most $N({\rm v},D)$ balls in the collection $\{B_{r_{a}/4}(x_{aj}\}$, and the application of Lemma 8.10 produces at most $N({\rm v},D)$ balls for each of these, we have at most $N({\rm v},D)$ such balls in total.

If we put

we see that $\mathcal{B}^{k+1}_{a}$ and the collection $\{B_{r^{k+1}_{ab}}(x^{k+1}_{ab})\}$ satisfy the inductive conditions. Specifically, what is left to check is condition (5). However, by construction, we have

which for $\delta({\rm v})$ sufficiently small implies $|\Gamma^{k+1}_{aj}|<|\Gamma^{k}_{a}|$. In particular, the decomposition

satisfies the inductive hypothesis as well, which completes the proof.

∎

Let $M^{4}$ satisfy $|{\rm Ric}_{M^{4}}|\leq 3$, ${\rm Vol}(M)>{\rm v}>0$ and ${\rm diam}(M^{4})\leq D$. Then using Theorem 8.11, we can write

First we will analyze each body region $\mathcal{B}^{k}_{j}$. Indeed, by (1) and theorem 8.2, it follows that there are at most $C({\rm v},D)$-diffeomorphism types for each $\mathcal{B}^{k}_{j}$. By (4), there are at most $C({\rm v},D)$ such body regions, and by (2) and (3), there are at most $C({\rm v},D)$ diffeomorphism types that can arise by gluing them together, which proves the theorem. ∎