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.

∎