and radius $\frac{1}{6}r_{v}(x)$ in $(2^{-(k+1)},2^{-k}]$. For any $k\geq 0$,
the set

$M^{[>\epsilon]}\cap\bigcup_{B\in\tilde{\mathcal{F}}_{k}}B$

can be covered by a family of balls
$\tilde{G}_{k}\subset\tilde{\mathcal{F}}_{k}$ with uniformly bounded
multiplicity by Bishop-Gromov (each point belongs to at most $c_{\text{univ}}$
balls of $\tilde{G}_{k}$ for a universal constant $c_{\text{univ}}$). Consider
the family of balls $\bigcup_{k}\tilde{G}_{k}$, and remove from it any ball
$B$ that is contained in $2B^{\prime}$ for some ball $B^{\prime}\neq B$ in
$\bigcup_{k}\tilde{G}_{k}$. By doubling the size of each remaining ball, this
yields a new family $\tilde{\mathcal{H}}$ of balls of the form
$B_{g}(x,\frac{1}{3}r_{v}(x))$. Let $\tilde{\mathcal{H}}^{\prime}$ be the
family of balls $3B$, where $B\in\tilde{\mathcal{H}}$. Balls of
$\tilde{\mathcal{H}}^{\prime}$ are thus of the form $B_{g}(x,r_{v}(x))$.

Claim: $\tilde{\mathcal{H}}$ is a covering of $M^{[>\epsilon]}$ and
$\tilde{\mathcal{H}}^{\prime}$ has bounded multiplicity.

Let us check this claim, which would conclude the proof. The fact that
$\tilde{\mathcal{H}}$ is a cover of $M^{[>\epsilon]}$ is clear from its
construction and the identity

$M^{[>\epsilon]}=\bigcup_{k}\Big{(}M^{[>\epsilon]}\cap\bigcup_{B\in\tilde{\mathcal{F}}_{k}}B\Big{)}.$

Suppose that the multiplicity of $\tilde{\mathcal{H}}^{\prime}$ cannot be
chosen uniformly bounded, i.e. it cannot be chosen bounded from above
independently of $(M,g)$. Then for any integer $N>0$, there is such a $(M,g)$
such that the family $\tilde{\mathcal{H}}^{\prime}$ constructed above has
multiplicity at least $N$ at some point $x\in M^{[>\epsilon]}$, in other words
there are different balls $B_{1},...,B_{N}\in\tilde{\mathcal{H}}^{\prime}$
containing $x$. Let $y_{1},...y_{N}$ be the respective centers of
$B_{1},...,B_{N}$. We assume without loss of generality that the radii are in
non-decreasing order

$\operatorname{rad}B_{1}\leq\operatorname{rad}B_{2}...\leq\operatorname{rad}B_{N}.$

Since by construction, no $\frac{1}{6}B_{i}$ is contained in another
$\frac{1}{3}B_{j}$ ($i\neq j$), by Bishop-Gromov inequality and by extracting
a subsequence of $\\{B_{i}\\}$ if necessary, we can assume that

$2^{-1}\operatorname{dist}_{g}(y_{1},y_{2})<2^{-2}\operatorname{dist}_{g}(y_{1},y_{3})...<2^{-(N-1)}\operatorname{dist}_{g}(y_{1},y_{N}),$
(35)
$\operatorname{rad}B_{1}<2^{-1}\operatorname{rad}B_{2}...<2^{-(N-1)}\operatorname{rad}B_{N},$

and moreover

$C_{v}^{-1}\operatorname{rad}B_{j}\leq\operatorname{dist}_{g}(y_{1},y_{j})\leq
C_{v}\operatorname{rad}B_{j}\quad\forall j\in\\{1,..,N\\}.$

We can also make sure that $\operatorname{rad}B_{j_{N}}\to 0$ as $N\to\infty$.
But since

$r_{v}(y_{j})=\operatorname{rad}B_{j},$

$B_{g}(y_{1},\operatorname{dist}_{g}(y_{1},y_{N}))$ is not too collapsed:

$\frac{\operatorname{Vol}(B_{g}(y_{1},\operatorname{dist}_{g}(y_{1},y_{N})),g)}{\operatorname{dist}_{g}(y_{1},y_{N})^{4}}\geq
C_{v}^{-1}>0,$

therefore by applying Bishop-Gromov monotonicity formula again, the difference
of the volume ratios

$\frac{\operatorname{Vol}(B_{g}(y_{1},\operatorname{dist}_{g}(y_{1},y_{j_{N-1}})),g)}{\operatorname{dist}_{g}(y_{1},y_{j_{N-1}})^{4}}-\frac{\operatorname{Vol}(B_{g}(y_{1},\operatorname{dist}_{g}(y_{1},y_{j_{N+1}})),g)}{\operatorname{dist}_{g}(y_{1},y_{j_{N+1}})^{4}}$

is arbitrarily small for some $j_{N}\in\\{3,...,N-1\\}$ if $N$ is chosen large
enough. By a blow-up and compactness argument, by the “volume cone implies
metric cone” theorem [26, Theorem 4.91], and since in dimension $4$ the
singular set has codimension $4$ [36, Theorem 1.4], the ball

$\frac{1}{4}B_{g}(y_{j_{N}},\operatorname{dist}_{g}(y_{1},y_{j_{N}}))$

endowed with the metric $\operatorname{dist}_{g}(y_{1},y_{j_{N}})^{-2}g$ would
be arbitrarily close (in the $C^{\infty}$ topology by the Einstein condition)
to a smooth open domain of a metric cone $\mathfrak{C}$ which is smooth
outside its tip if $N$ is large enough. The smooth part of this
$4$-dimensional cone $\mathfrak{C}$ is Ricci flat so actually it is flat.
Since $y_{j_{N}}\in M^{[>\epsilon]}$, we have

$\int_{B_{g}(y_{j_{N}},\frac{1}{24}r_{v}(y_{j_{N}}))}|\operatorname{Rm}_{g}|^{2}>\epsilon.$

But for large $N$, since by construction
$\frac{1}{6}B_{g}(y_{1},r_{v}(y_{1}))$ is not included inside
$\frac{1}{3}B_{g}(y_{j_{N}},r_{v}(y_{j_{N}}))$ and since (35), we have

$B_{g}(y_{j_{N}},\frac{1}{24}r_{v}(y_{j_{N}}))\subset\frac{1}{4}B_{g}(y_{j_{N}},\operatorname{dist}_{g}(y_{1},y_{j_{N}}))$

so that would eventually contradict the flatness of the cone $\mathfrak{C}$ as
$N\to\infty$, by the $\epsilon$-regularity theorem.

∎

Let $\\{x_{i}\\}_{i=1}^{L^{\prime}}$ be as in the previous lemma. In following
lemma we find annuli with bounded curvature inside
$B_{g}(x_{i},r_{v}(x_{i}))$.

######  Lemma 26.

There are constants $\frac{1}{1000}>\delta_{1}>\delta_{2}>0$ and $C_{v}$
depending only on $v$ such that one can choose for each
$i\in\\{1,\dots,L^{\prime}\\}$ a radius

$r_{i}\in((\frac{1}{2}-\delta_{1})r_{v}(x_{i}),(\frac{1}{2}+\delta_{1})r_{v}(x_{i}))$

satisfying the following: at any point

$x\in B_{g}(x_{i},r_{i}+\delta_{2}r_{v}(x_{i}))\backslash
B_{g}(x_{i},r_{i}-\delta_{2}r_{v}(x_{i})),$

we have

(36) $r_{v}(x_{i})^{2}|\operatorname{Rm}_{g}(x)|\leq C_{v},$ (37)
$\operatorname{injrad}_{g}(x)\geq\delta_{2}r_{v}(x_{i}).$

###### Proof.

We argue by contradiction: assume that there is a sequence of $4$-manifolds of
the form $B_{g_{k}}(y_{k},R_{k})$ such that $g_{k}$ is a metric with
$|\operatorname{Ric}_{g_{k}}|\leq n-1$, $R_{k}\leq 1$ and
$\operatorname{Vol}_{g_{k}}(B_{g_{k}}(y_{k},R_{k}))\geq vR_{k}^{4}$, such that
the conclusion of the lemma fails. By rescaling one can assume that $R_{k}=1$,
and so we are supposing that one cannot find $\delta_{1}>\delta_{2}>0$ and
$C_{v}$ such that for some radius

$\bar{r}\in(\frac{1}{2}-\delta_{1},\frac{1}{2}+\delta_{1}),$

at any point

$x\in B_{g_{k}}(y_{k},\bar{r}+\delta_{2})\backslash
B_{g_{k}}(y_{k},\bar{r}-\delta_{2}),$

the bounds $|\operatorname{Rm}_{g_{k}}(x)|\leq C_{v}$ and
$\operatorname{injrad}_{g_{k}}(x)\geq\delta_{2}$ hold.

By Cheeger-Naber [36, Theorem 1.4, Theorem 8.6, Corollary 8.85] however, a
subsequence of the pointed balls $(B_{g_{k}}(y_{k},\frac{3}{4}),g_{k},y_{k})$
(which are all $v$-noncollapsed) converge in the Gromov-Hausdorff topology to
a metric space $(X,g_{\infty},y_{\infty})$ smooth outside of finitely many
points $u_{1},...,u_{K}$. It is then possible to find
$\delta_{1}>\delta_{2}>0$ such that for some radius

$\bar{r}\in(\frac{1}{2}-\delta_{1},\frac{1}{2}+\delta_{1}),$

the region

$B_{g_{k}}(y_{k},\bar{r}+\delta_{2})\backslash
B_{g_{k}}(y_{k},\bar{r}-\delta_{2}),$

is $\delta_{2}$-far from $\\{u_{1},...,u_{K}\\}$ as measured with the distance
$g_{\infty}$. By the $\epsilon$-regularity theorem in [4][36, (2.8)], we get
the desired bounds

$|\operatorname{Rm}_{g_{k}}(x)|\leq C_{v}\text{ and
}\operatorname{injrad}_{g_{k}}(x)\geq\delta_{2}$

on $B_{g_{k}}(y_{k},\bar{r}+\delta_{2})\backslash
B_{g_{k}}(y_{k},\bar{r}-\delta_{2})$ for $k$ large enough. This contradiction
finishes the proof.

∎

Let $r_{i}$, $\delta_{2}$ be as in Lemma 26. By Lemma 25, the balls
$\\{B_{g}(x_{i},r_{i}-\frac{\delta_{2}}{2}r_{v}(x_{i}))\\}_{i=1}^{L^{\prime}}$
cover $M^{[>\epsilon]}$ and the covering multiplicity of
$\\{B_{g}(x_{i},r_{i})\\}_{i=1}^{L^{\prime}}$ is bounded by $C_{v}$. By
removing some balls from this family, we can assume (after renumbering) that
there is an integer $L\leq L^{\prime}$ so that the points $x_{1},...,x_{L}$
satisfy:

(38)
$\begin{split}&\text{$\\{B_{g}(x_{i},r_{i}-\frac{\delta_{2}}{2}r_{v}(x_{i}))\\}_{i=1}^{L}$
cover $M^{[>\epsilon]}$ with covering multiplicity bounded by $C_{v}$,}\\\
&\text{and for any $i\neq j\in\\{1,...,L\\}$, $B_{g}(x_{i},r_{i})$ is not
included inside}\\\
&B_{g}(x_{j},r_{j}-\frac{\delta_{2}}{2}r_{v}(x_{j})).\end{split}$

Each $x_{i}$ ($i\in\\{1,...,L\\}$) is a point of $M^{[>\epsilon]}$, so
$\int_{B_{g}(x_{i},r_{i}-\frac{\delta_{2}}{2}r_{v}(x_{i}))}|\operatorname{Rm}_{g}|^{2}>\epsilon$.
Thus, as a consequence of the multiplicity bound on the covering
$\\{B_{g}(x_{i},r_{i}-\frac{\delta_{2}}{2}r_{v}(x_{i}))\\}_{i=1}^{L}$, there
is $C_{v}>0$ so that

(39) $L\leq C_{v}\int_{M}|\operatorname{Rm}_{g}|^{2}dvol_{g}.$

The region
$\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i}+\frac{\delta_{2}}{2}r_{v}(x_{i}))\backslash\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i}-\frac{\delta_{2}}{2}r_{v}(x_{i}))$
is contained in $M^{[\leq\epsilon]}$ and we want to cover it with nice balls
capturing the collapsed geometry around that region:

######  Lemma 27.

There are finitely many points

$y_{1},...,y_{Q}\in\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i}+\frac{\delta_{2}}{2}r_{v}(x_{i}))\backslash\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i}-\frac{\delta_{2}}{2}r_{v}(x_{i}))$

and corresponding radii $\bar{r}_{1},...,\bar{r}_{Q}$ such that for some
$C_{v}$ depending only on $v$:

  1. (i)

$\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i}+\frac{\delta_{2}}{2}r_{v}(x_{i}))\backslash\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i}-\frac{\delta_{2}}{2}r_{v}(x_{i}))\subset\bigcup_{q=1}^{Q}B_{g}(y_{q},\frac{\bar{r}_{q}}{4}),$

  2. (ii)

$\frac{1}{48}r_{v}(y_{q})\leq\bar{r}_{q}\leq 1,$

  3. (iii)

$\sup\\{|\sec_{x}|^{1/2};\quad x\in
B_{g}(y_{q},\bar{r}_{q})\\}\leq\frac{1}{\bar{r}_{q}}\text{ with equality if
}\bar{r}_{q}<1,$

  4. (iv)

$Q\leq C_{v}L.$

Moreover for a universal constant $c_{\text{univ}}$, if
$q,q^{\prime}\in\\{1,...,Q\\}$ and

$B_{g}(y_{q},\frac{\bar{r}_{q}}{2})\cap
B_{g}(y_{q^{\prime}},\frac{\bar{r}_{q^{\prime}}}{2})\neq\varnothing,$

then a universal Harnack property holds:

(40) $c_{\text{univ}}^{-1}\bar{r}_{q}\leq\bar{r}_{q^{\prime}}\leq
c_{\text{univ}}\bar{r}_{q},$

and the multiplicity of the covering
$\\{B_{g}(y_{q},\frac{\bar{r}_{q}}{2})\\}_{q=1}^{Q}$ is bounded by
$c_{\text{univ}}$.

###### Proof.

This lemma follows from an argument appearing in [31, Lemma 5.4, Theorem 5.5].
For

$x\in\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i}+\frac{\delta_{2}}{2}r_{v}(x_{i}))\backslash\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i}-\frac{\delta_{2}}{2}r_{v}(x_{i}))$

we set

$\bar{k}(x):=\inf\\{\frac{1}{R};R\leq 1,\quad\sup_{x\in
B_{g}(x,R)}|\sec_{g}(x)|^{1/2}\leq\frac{1}{R}\\}.$

In the statements and proofs of [31, Lemma 5.4, Theorem 5.5] one should
replace $\bar{k}$ by $\bar{k}^{-1}$ except in conformally changed metrics of
the form “$\bar{k}^{2}(.)g$”. Then their arguments imply our lemma, and give
points $y_{1},...,y_{Q}$ with radii

$\bar{r}_{q}:=\bar{k}(y_{q})^{-1}\in(0,1]$

so that the family of balls
$\\{B_{g}(y_{q},\frac{\bar{r}_{q}}{4})\\}_{q=1}^{Q}$ covers

$\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i}+\frac{\delta_{2}}{2}r_{v}(x_{i}))\backslash\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i}-\frac{\delta_{2}}{2}r_{v}(x_{i})).$

In particular,

(41) $\sup\\{|\sec_{x}|^{1/2};\quad x\in
B_{g}(y_{q},\bar{r}_{q})\\}\leq\frac{1}{\bar{r}_{q}}\text{ with equality if
}\bar{r}_{q}<1,$

and the multiplicity of $\\{B_{g}(y_{q},\frac{\bar{r}_{q}}{2})\\}_{q=1}^{Q}$
is universally bounded (see proof of [31, Theorem 5.5]). Moreover we have a
Harnack property: the sizes of intersecting balls of the form
$B_{g}(y_{q},\frac{\bar{r}_{q}}{2})$ are universally comparable. The
additional properties we need to check in order to finish the proof are:

$\frac{1}{48}r_{v}(y_{q})\leq\bar{r}_{q}\leq 1$ $\text{and }Q\leq C_{v}L.$

The first line comes from (34). By Lemma 26, by (41) and the fact that
$x_{i}\in M^{[>\epsilon]}$, if $B_{g}(y_{q},\frac{\bar{r}_{q}}{4})$ intersects

$B_{g}(x_{i},r_{i}+\frac{\delta_{2}}{2}r_{v}(x_{i})\backslash
B_{g}(x_{i},r_{i}-\frac{\delta_{2}}{2}r_{v}(x_{i}))$

then $r_{v}(x_{i})$ and $\bar{r}_{q}$ are comparable up to a factor $C_{v}$
and so Bishop-Gromov inequality yields the second line.

∎

###  7.5. Metric deformations

Given $s>0$, given a subset $S$ of a manifold $\mathcal{W}$ and a metric $h$
on $\mathcal{W}$, we recall the following notation for the $s$-neighborhood of
$S$:

$T_{h,s}(S):=\\{y\in\mathcal{W};\operatorname{dist}_{h}(y,S)<s\\}.$

Given a compact Riemannian manifold with boundary $(\Omega,h)$, we say that
$h$ is a product metric near $\partial\Omega\subset\Omega$ if there is a
tubular neighborhood of $\partial\Omega$ isometric to

$\big{(}\partial\Omega\times[0,1),h|_{\Omega}\oplus dt^{2}\big{)},$

in particular $\partial\Omega$ is totally geodesic in that case.

Let $r_{i}$, $\delta_{2}$ be as in Lemma 26. Let $\\{x_{i}\\}_{i=1}^{L}$ be as
in (38) and let $\\{y_{q}\\}_{q=1}^{Q}$, $\\{\bar{r}_{q}\\}_{q=1}^{Q}$ be as
in Lemma 27. Recall that $\epsilon$ only depends on $v$ (see Subsection 7.3).

######  Lemma 28.

There exists a universal constant $c_{univ}$ such that the following is true.

For any $\eta>0$, if $v>0$ is taken small enough depending on $\eta$, then
there is an open neighborhood $\mathcal{W}$ of

$\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i})$

and compact subdomains
$\mathcal{R}_{1},...,\mathcal{R}_{K_{0}}\subset\mathcal{W}$ with smooth
boundaries so that the following holds: there is a constant $C_{v}$ depending
only on $v$ and a metric $h_{0}$ on $\mathcal{W}$ such that

  1. (i)

$\mathcal{W}\subset\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i})\cup\bigcup_{q=1}^{Q}B_{g}(y_{q},\frac{\bar{r}_{q}}{3})$,

  2. (ii)

for each $k\in\\{1,...,K_{0}\\}$, $h_{0}$ is a product metric near
$\partial\mathcal{R}_{k}\subset\mathcal{R}_{k}$ and there is
$i\in\\{1,...,L\\}$ with $\mathcal{R}_{k}\subset B_{g}(x_{i},r_{i})$,

  3. (iii)

$\operatorname{Vol}(\mathcal{R}_{k},h_{0})\geq 1/C_{v}$,
$\operatorname{diam}_{h_{0}}\mathcal{R}_{k}\leq C_{v}$ and
$\operatorname{injrad}_{h_{0}}\geq 1/C_{v}$ on
$T_{h_{0},1}(\partial\mathcal{R}_{k})$ for all $k\in\\{1,...,K_{0}\\}$,

  4. (iv)

$|\operatorname{Ric}_{h_{0}}|\leq 3$ on $\mathcal{W}$, and $|\sec_{h_{0}}|\leq
C_{v}$ on
$T_{h_{0},1}\big{(}\mathcal{W}\backslash\bigcup_{k=1}^{K_{0}}\mathcal{R}_{k}\big{)}$,

  5. (v)

$\operatorname{Vol}\big{(}T_{h_{0},1}\big{(}\mathcal{W}\backslash\bigcup_{k=1}^{K_{0}}\mathcal{R}_{k}\big{)},h_{0}\big{)}\leq
C_{v}L$,

  6. (vi)

$h_{0}$ is a product metric near $\partial\mathcal{W}\subset\mathcal{W}$,
$|\sec_{h_{0}}|\leq c_{univ}$ near $\partial\mathcal{W}$ and
$\operatorname{injrad}_{h_{0}}(x)\leq\eta$ for $x\in\partial\mathcal{W}$.

###### Proof.

Let $r_{i}$, $\delta_{2}$ be as in Lemma 26. Let $\\{x_{i}\\}_{i=1}^{L}$ be as
in (38) and let $\\{y_{q}\\}_{q=1}^{Q}$, $\\{\bar{r}_{q}\\}_{q=1}^{Q}$ as in
Lemma 27. As before, we denote by $C_{v}$ a constant depending only on $v$,
and by $c_{\text{univ}}$ a universal constant. These constants can change from
line to line.

Since $g$ is an Einstein metric and because of Lemma 26, for all
$i\in\\{1,...,L\\}$ and for all $x\in
B_{g}(x_{i},r_{i}+\delta_{2}r_{v}(x_{i}))\backslash
B_{g}(x_{i},r_{i}-\delta_{2}r_{v}(x_{i}))$:

(42)
$\begin{split}\operatorname{injrad}_{g}(x)&\geq\delta_{2}r_{v}(x_{i})\geq\frac{r_{i}}{C_{v}},\\\
r_{i}^{k+1}|D^{k}_{g}\operatorname{Rm}_{g}|&\leq C_{v}^{(k)}\text{ for all
}k\geq 0,\end{split}$

where $D_{g}$ denotes the covariant derivative with respect to $g$. Similarly
by Lemma 27, for all $q\in\\{1,...,Q\\}$ and all $y\in
B_{g}(y_{q},\bar{r}_{q})$:

(43) $\bar{r}_{q}^{k+1}|D^{k}_{g}\operatorname{Rm}_{g}|\leq
c_{\text{univ}}^{(k)}\text{ for any }k\geq 0.$

Note the following Harnack type property about the relative size of radii.
Consider first $i\neq j\in\\{1,...,L\\}$ and suppose that

$B_{g}(x_{i},r_{i})\cap B_{g}(x_{j},r_{j})\neq\varnothing.$

Then by (38), the intersection of

$B_{g}(x_{i},r_{i})\backslash
B_{g}(x_{i},r_{i}-\frac{\delta_{2}}{2}r_{v}(x_{i}))$

and

$B_{g}(x_{j},r_{j})\backslash
B_{g}(x_{j},r_{j}-\frac{\delta_{2}}{2}r_{v}(x_{j}))$

is non-empty. By the curvature bound of Lemma 26 and the definition of
$M^{[>\epsilon]}$,

$C_{v}^{-1}r_{i}\leq r_{j}\leq C_{v}r_{i}.$

The analogous statement for the balls
$\\{B_{g}(y_{q},\frac{\bar{r}_{q}}{2})\\}_{q=1}^{Q}$ (with a universal
constant) was shown in Lemma 27 $(iv)$: if for $q,q^{\prime}\in\\{1,...,Q\\}$

$B_{g}(y_{q},\frac{\bar{r}_{q}}{2})\cap
B_{g}(y_{q^{\prime}},\frac{\bar{r}_{q^{\prime}}}{2})\neq\varnothing,$

then

$c_{\text{univ}}^{-1}\bar{r}_{q}\leq\bar{r}_{q^{\prime}}\leq
c_{\text{univ}}\bar{r}_{q}.$

Finally, if $q\in\\{1,...,Q\\}$, $i\in\\{1,.,L\\}$ and if

$B_{g}(y_{q},\frac{\bar{r}_{q}}{2})\cap B_{g}(x_{i},r_{i})\neq\varnothing,$

then because $y_{q}$ is not in
$\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i}-\frac{\delta_{2}}{2}r_{v}(x_{i}))$ by
Lemma 27, we have

$B_{g}(y_{q},\frac{\bar{r}_{q}}{2})\cap\big{(}B_{g}(x_{i},r_{i})\backslash
B_{g}(x_{i},r_{i}-\frac{\delta_{2}}{2}r_{v}(x_{i}))\big{)}\neq\varnothing,$

and so

$C_{v}^{-1}r_{i}\leq\bar{r}_{q}\leq C_{v}r_{i}.$

Indeed the curvature bound of Lemma 26 and Lemma 27 imply the first
inequality. For the second inequality, recall that the curvature inside
$B_{g}(y_{q},{\bar{r}_{q}})$ is bounded by $\frac{1}{\bar{r}^{2}_{q}}$ (Lemma
27), and that
$\int_{B_{g}(x_{i},\frac{r_{v}(x_{i})}{24})}|\operatorname{Rm}_{g}|^{2}>\epsilon$
since $x_{i}\in M^{[>\epsilon]}$.

We try to look for a conformal metric of the form “$h_{0}=\rho^{2}g$” for some
function $\rho$. Let us choose the conformal factor $\rho$ as follows. Let
$\\{\varphi_{i}\\}_{i=1}^{L}$ and $\\{\phi_{q}\\}_{q=1}^{Q}$ be functions
satisfying the following for some constant $C_{v}>0$ and universal constant
$c_{\text{univ}}$:

$\varphi_{i},\phi_{q}:M\to[0,1],$ $\varphi_{i}=0\text{ outside of
$B_{g}(x_{i},r_{i})$},$ $\varphi_{i}=1\text{ inside of
$B_{g}(x_{i},r_{i}-\frac{\delta_{2}}{2}r_{v}(x_{i}))$},$ $\phi_{q}=0\text{
outside of $B_{g}(y_{q},\frac{\bar{r}_{q}}{2})$},$ $\phi_{q}=1\text{ inside of
$B_{g}(y_{q},\frac{\bar{r}_{q}}{3})$},$

(44) $\begin{split}r_{i}|D_{g}\varphi_{i}|\leq C_{v},&\quad
r_{i}^{2}|\operatorname{Hess}_{g}\varphi_{i}|\leq C_{v},\\\
\bar{r}_{q}|D_{g}\phi_{q}|\leq
c_{\text{univ}},&\quad\bar{r}_{q}^{2}|\operatorname{Hess}_{g}\phi_{q}|\leq
c_{\text{univ}}.\end{split}$

where $\operatorname{Hess}_{g}$ denotes as usual the Hessian with respect to
the metric $g$. The existence of such functions follows from the properties of
$g$ listed at the beginning of the proof.

Set for all $i\in\\{1,...,L\\}$ or $q\in\\{1,...Q\\}$ and for any

$x\in\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i})\cup\bigcup_{q=1}^{Q}B_{g}(y_{q},\frac{\bar{r}_{q}}{3})$

the following functions:

$\psi_{i}(x):=\frac{\varphi_{i}(x)}{\sum_{i=1}^{L}\varphi_{i}(x)+\sum_{q=1}^{Q}\phi_{q}(x)}$
$\psi^{\prime}_{q}(x):=\frac{\phi_{q}(x)}{\sum_{i=1}^{L}\varphi_{i}(x)+\sum_{q=1}^{Q}\phi_{q}(x)}$
$\rho(x):=\sum_{i=1}^{L}\frac{\psi_{i}(x)}{r_{i}}+\sum_{q=1}^{Q}\frac{\psi^{\prime}_{q}(x)}{\bar{r}_{q}}.$

For any point

$x\in\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i})\cup\bigcup_{q=1}^{Q}B_{g}(y_{q},\frac{\bar{r}_{q}}{3}),$

we have

$\sum_{i=1}^{L}\varphi_{i}(x)+\sum_{q=1}^{Q}\phi_{q}(x)\geq 1.$

Moreover, by the bound on the multiplicity of the families

$\\{B_{g}(x_{i},{r_{i}})\\}_{i=1}^{L},\quad\\{B_{g}(y_{q},\frac{\bar{r}_{q}}{2})\\}_{q=1}^{Q},$

(see Lemma 25, Lemma 27), for any

$x\in\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i})\cup\bigcup_{q=1}^{Q}B_{g}(y_{q},\frac{\bar{r}_{q}}{3}),$

there is a small neighborhood $U_{x}$ of $x$ such that there are at most
$C_{v}$ functions $\varphi_{i}$, and $c_{\text{univ}}$ functions $\phi_{q}$
which are non-vanishing in $U_{x}$. The Harnack property of the second
paragraph ensures that the radii of nearby balls are comparable. Then observe
that by (44), the derivatives of $\psi_{i}$ and $\psi^{\prime}_{q}$ are
locally bounded: at any point in $B_{g}(x_{i},r_{i})$ or in
$B_{g}(y_{q},\frac{\bar{r}_{q}}{3})$, we have

$|D_{g}\psi_{i}|\leq C_{v}r_{i}^{-1},\quad|D^{2}_{g}\psi_{i}|\leq
C_{v}r_{i}^{-2},$ $\text{or }|D_{g}\psi^{\prime}_{q}|\leq
C_{v}\bar{r}_{q}^{-1},\quad|D^{2}_{g}\psi^{\prime}_{q}|\leq
C_{v}\bar{r}_{q}^{-2}$

and in fact

$|D_{g}\psi^{\prime}_{q}|\leq
c_{\text{univ}}\bar{r}_{q}^{-1},\quad|D^{2}_{g}\psi^{\prime}_{q}|\leq
c_{\text{univ}}\bar{r}_{q}^{-2}\text{ at points which are outside of
$\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i})$}.$

Since $r_{i}\leq 1$, $\bar{r}_{q}\leq 1$, at any point in
$B_{g}(x_{i},{r_{i}})$ or $B_{g}(y_{q},\frac{\bar{r}_{q}}{3})$ we have again
by the Harnack properties:

(45) $\begin{split}&C_{v}^{-1}r_{i}^{-1}\leq\rho\leq C_{v}r_{i}^{-1},\\\
&|D_{g}\rho|\leq C_{v}r_{i}^{-2},\quad|D^{2}_{g}\rho|\leq C_{v}r_{i}^{-3}\\\
&\text{or }\\\ &C_{v}^{-1}\bar{r}_{q}^{-1}\leq\rho\leq
C_{v}\bar{r}_{q}^{-1},\\\ &|D_{g}\rho|\leq
C_{v}\bar{r}_{q}^{-2},\quad|D^{2}_{g}\rho|\leq C_{v}\bar{r}_{q}^{-3},\\\
&\text{and in fact at points outside of $\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i})$
we have:}\\\ &c_{\text{univ}}^{-1}\bar{r}_{q}^{-1}\leq\rho\leq
c_{\text{univ}}\bar{r}_{q}^{-1},\\\ &|D_{g}\rho|\leq
c_{\text{univ}}\bar{r}_{q}^{-2},\quad|D^{2}_{g}\rho|\leq
c_{\text{univ}}\bar{r}_{q}^{-3}.\end{split}$

Consider the metric $h_{0}$

$h_{0}:=\rho^{2}g.$

By the formula for the Riemann curvature tensor under a conformal change of
metric, see (14), and by (45), for any point in the union of

$\bigcup_{q=1}^{Q}B_{g}(y_{q},\frac{\bar{r}_{q}}{3})$

and

$\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i})\backslash
B_{g}(x_{i},r_{i}-\frac{\delta_{2}}{2}r_{v}(x_{i})),$

the following curvature bound holds:

(46) $|\sec_{h_{0}}(x)|\leq C_{v}.$

For points in

$\bigcup_{q=1}^{Q}B_{g}(y_{q},\frac{\bar{r}_{q}}{3})\backslash\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i})$

the sectional curvature of $h_{0}$ is actually bounded by a universal constant
$c_{\text{univ}}$ (in particular not depending on $v$), because the
multiplicity bounds and Harnack property for balls in
$\\{B_{g}(y_{q},\frac{\bar{r}_{q}}{2})\\}_{q=1}^{Q}$ are relying on a
universal constant, see (45).

Since by Lemma 27 the balls in
$\\{B_{g}(y_{q},\frac{\bar{r}_{q}}{4})\\}_{q=1}^{Q}$ already covers

$\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i})\backslash
B_{g}(x_{i},r_{i}-\frac{\delta_{2}}{2}r_{v}(x_{i})),$

and because of (40), with respect to the metric $h_{0}$, the set

$\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i})\cup\bigcup_{q=1}^{Q}B_{g}(y_{q},\frac{\bar{r}_{q}}{3})$

contains a complete $\frac{1}{c_{\text{univ}}}$-neighborhood of
$\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i})$. Thus by the chopping theorem [34], we
can find a domain $\mathcal{W}$ such that

$\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i})\subset\mathcal{W}\subset\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i})\cup\bigcup_{q=1}^{Q}B_{g}(y_{q},\frac{\bar{r}_{q}}{3}),$

and the boundary $\mathcal{W}$ has bounded second fundamental form. By Lemma
10, we modify the metric $h_{0}$ near $\partial\mathcal{W}$ so that the metric
$h_{0}$ is now a product metric near $\partial\mathcal{W}\subset\mathcal{W}$
and the sectional curvature of $h_{0}$ is still universally bounded near
$\partial\mathcal{W}$.

By (34), the injectivity radius of $h_{0}$ near $\partial\mathcal{W}$ is less
than the constant $\eta>0$ if $v$ is chosen small enough.

Similarly there is a neighborhood $\mathbf{U}$ with smooth boundary of the set
of points which are in

$\mathcal{W}\cap\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i}-\frac{\delta_{2}}{2}r_{v}(x_{i}))$

but not in

$\bigcup_{i=1}^{L}B_{g}(x_{i},r_{i})\backslash
B_{g}(x_{i},r_{i}-\frac{\delta_{2}}{2}r_{v}(x_{i})),$

and whose boundary $\partial\mathbf{U}$ has bounded second fundamental form
and the curvature is bounded by $C_{v}$ in a $\frac{1}{C_{v}}$-neighborhood
neighborhood of it. We can again modify $h_{0}$ so that it is a product metric
near $\partial\mathbf{U}\subset\mathbf{U}$. The bound (46) means that

$\sup_{x\in\mathcal{W}\backslash\mathbf{U}}|\sec_{h_{0}}(x)|^{1/2}\leq C_{v}.$

Recall that the curvature of $h_{0}$ near $\partial\mathcal{W}$ is bounded by
a universal constant, and we chose $v$ small enough so that the injectivity
radius of $h_{0}$ near $\partial\mathcal{W}$ is less than $\eta$. (While the
sectional curvature of $h_{0}$ near $\partial\mathcal{W}$ is bounded
independently of the choice of $v$, the curvature bound elsewhere is
potentially very large if $v$ is chosen small.)

Define $\mathcal{R}_{1},...,\mathcal{R}_{K_{0}}$ to be the connected
components of $\mathbf{U}$. Note that by definition, each $\mathcal{R}_{k}$ is
contained in a subset of the form $B_{g}(x_{i},r_{i})$. In our constructions,
the constant $v$ is chosen according to $\eta$. Once $v$ is fixed, the
constants $C_{v}$ is fixed. The properties in the statement of the lemma then
follows from the constructions. For instance the volume bound of
$T_{h_{0},1}(\mathcal{W}\backslash\mathbf{U})$ comes from the covering
multiplicity bounds, the Harnack property and the fact that $Q\leq C_{v}L$
(Lemma 27).

∎

Let $\mathcal{W}$, let
$\mathcal{R}_{1},...,\mathcal{R}_{K_{0}}\subset\mathcal{W}$, and let $h_{0}$
be given by Lemma 28. Recall that by Lemma 28 $(iii)$ $(iv)$, each
$(\mathcal{R}_{k},h_{0})$ has volume lower bound, diameter upper bound and
$|\operatorname{Ric}_{h_{0}}|\leq 3$. Note that by Lemma 28 $(ii)$ $(iii)$,
there is a dimensional constant $C_{v}$ so that:

(47) $K_{0}\leq C_{v}L$

where $K_{0}$ is the number of regions $\mathcal{R}_{k}$.

######  Lemma 29.

There is a constant $C_{v}$ depending only on $v$, such that for each
$\mathcal{R}_{k}$ ($k\in\\{1,...,K_{0}\\}$), one can find a metric $h_{k}$
which coincides with $h_{0}$ on a neighborhood of $\mathcal{R}_{k}$, such that

  1. (i)

$|\sec_{h_{k}}|\leq C_{v}$ on $\mathcal{R}_{k}$,

  2. (ii)

$\operatorname{injrad}_{h_{k}}\geq C_{v}^{-1}$ on $\mathcal{R}_{k}$,

  3. (iii)

$\operatorname{Vol}(\mathcal{R}_{k},h_{k})\leq C_{v}$.

###### Proof.

We can argue by contradiction. Consider a sequence of Riemannian manifolds
with boundary $\\{(\hat{S}_{j},g_{j})\\}_{j\geq 0}$ such that for some $c>0$,

  * •

$T_{g_{j},1}(\partial\hat{S}_{j})$ has injectivity radius bounded below by
$c^{-1}$ and sectional curvature bounded by $c$,

  * •

$g_{j}$ is a metric product near $\partial\hat{S}_{j}\subset\hat{S}_{j}$,

  * •

$\operatorname{diam}_{g_{j}}(\hat{S}_{j})\leq c$,
$\operatorname{Vol}_{g_{j}}(\hat{S}_{j})\geq c^{-1}$,

  * •

and $|\operatorname{Ric}_{g_{j}}|\leq 3$,

but such that the conclusion of the lemma fails. Then a subsequence converges
in the Gromov-Hausdorff distance, and the proof of [36, Theorem 1.12] shows
that the possible diffeomorphism types of $\hat{S}_{j}$ is actually finite.
Hence, it is possible to put a metric on $\hat{S}_{j}$ with uniform control on
curvature, injectivity radius and volume, which coincides with $g_{j}$ on
$T_{g_{j},{1}}(\partial\hat{S}_{j})$. This is the desired contradiction. ∎

Let $\mathcal{V}$ be the complement of $\mathcal{W}$ in $M$. Given $\eta$, let
$v$ and $\epsilon$ be fixed by Lemma 28. We replace the metric $h_{0}$ of
Lemma 28 by $h_{i}$ of Lemma 29 on each ${R}_{i}$ ($i\in\\{1,...,{K_{0}}\\}$),
and call the new metric on $\mathcal{W}$ by $\underline{\mathbf{g}}_{1}$.
Clearly, by Lemmas 29 and 28,

$|\sec_{{\underline{\mathbf{g}}_{1}}}|\leq C_{v},$
$|\sec_{{\underline{\mathbf{g}}_{1}}}|\leq c_{univ}\text{ near
$\partial\mathcal{W}$}$

and the volume of $(\mathcal{W},\underline{\mathbf{g}}_{1})$ is well
controlled:

(48) $\operatorname{Vol}(\mathcal{W},\underline{\mathbf{g}}_{1})\leq C_{v}L.$

Moreover the injectivity radius of $\underline{\mathbf{g}}_{1}$ near
$\partial\mathcal{W}$ is at most $\eta$

The next lemma produces a metric on $\mathcal{V}$, the complement of
$\mathcal{W}$ in $M$.

######  Lemma 30.

There is a universal constant $c_{univ}$ such that the following is true.

Given $\eta>0$, there is a metric $\underline{\mathbf{g}}_{2}$ on a
neighborhood of $\mathcal{V}=M\backslash\mathcal{W}$ such that

  1. (i)

$|\sec_{\underline{\mathbf{g}}_{2}}|\leq c_{univ}$,

  2. (ii)

$\operatorname{injrad}_{\underline{\mathbf{g}}_{2}}\leq\eta$ everywhere on
$\mathcal{V}$,

  3. (iii)

$\underline{\mathbf{g}}_{2}=\underline{\mathbf{g}}_{1}$ near
$\partial\mathcal{V}$.

###### Proof.

Since $\mathcal{V}\subset M^{[\leq\epsilon]}$, similarly to the proof of Lemma
27, we get a cover of $\mathcal{V}$ by balls with controlled curvature
relative to their size and satisfying a universal Harnack inequality. By a
conformal modification of the metric as in the proof of Lemma 28, if $v$ was
chosen small enough, we get a metric $\underline{\mathbf{g}}_{2}$ on
$\mathcal{V}$ of curvature bounded by a universal constant $c_{univ}$ and with
injectivity radius at most $\eta$. In fact, we can make sure that
$\underline{\mathbf{g}}_{2}=\underline{\mathbf{g}}_{1}$ near
$\partial\mathcal{V}$ by construction of $\underline{\mathbf{g}}_{1}$ near
$\partial\mathcal{V}=\partial\mathcal{W}$.

∎

###  7.6. End of the proof

###### Proof of Theorem 23.

Let $\eta>0$ and let $\underline{\mathbf{g}}_{1}$ and
$\underline{\mathbf{g}}_{2}$ be the metrics obtained in the previous lemmas.
For a large $A_{v}>0$, consider
$\mathcal{V}\cup(\partial\mathcal{V}\times[0,A_{v}])\cup\mathcal{W}$ where we
naturally identify $\partial\mathcal{V}\times\\{0\\}$ with
$\partial\mathcal{V}$ (resp. $\partial\mathcal{V}\times\\{A_{v}\\}$ with
$\partial W$). This manifold is nothing else than $M$. We put the following
metric $\underline{\mathbf{g}}_{3}$ on it:

$\underline{\mathbf{g}}_{3}=\underline{\mathbf{g}}_{1}\text{ on
$\mathcal{W}$},$ $\underline{\mathbf{g}}_{3}=\underline{\mathbf{g}}_{2}\text{
on $\mathcal{V}$},$
$\underline{\mathbf{g}}_{3}=\underline{\mathbf{g}}_{1}\big{|}_{\partial
V}\oplus dt^{2}\text{ on $[0,A_{v}]$}.$

This smooth metric satisfies on
$\mathcal{V}\cup(\partial\mathcal{V}\times[0,A_{v}])$:

$|\sec_{\underline{\mathbf{g}}_{3}}|\leq c_{univ},$
$\operatorname{injrad}_{\underline{\mathbf{g}}_{3}}\leq\eta,$

and satisfies

$|\sec_{\underline{\mathbf{g}}_{3}}|\leq C_{v}$

everywhere. Now we can rescale it by the constant $2c_{univ}>0$ so that
$|\sec_{\underline{\mathbf{g}}_{3}}|\leq 1/2$ and
$\operatorname{injrad}_{\underline{\mathbf{g}}_{3}}\leq\sqrt{2c_{univ}}\eta$
on $\mathcal{V}\cup(\partial\mathcal{V}\times[0,A_{v}])$. Then if $A_{v}$ was
taken large enough, by conformally changing that new metric with a function
with controlled Hessian equal to $1$ on $\mathcal{V}$ and equal to a large
constant on $\mathcal{W}$ depending only on $v$, we get a final metric
$\underline{\mathbf{g}}$ with

$|\sec_{\underline{\mathbf{g}}}|\leq 1,$
$\operatorname{injrad}_{\underline{\mathbf{g}}}\leq\sqrt{2c_{univ}}\eta\text{
on $\mathcal{V}$}.$

From (48), we can also ensure that

(49)
$\operatorname{Vol}((\partial\mathcal{V}\times[0,A_{v}])\cup\mathcal{W},\underline{\mathbf{g}})\leq
C_{v}L$

for some $C_{v}$ depending only on $v$.

Let $\hat{\epsilon}$ be as in Proposition 21. Set

(50) $\eta=\frac{\hat{\epsilon}}{2\sqrt{2c_{univ}}}$

(now $v,\epsilon$ are finally fixed as in Lemma 28). The metric
$\underline{\mathbf{g}}$ on $M$ constructed just above then satisfies
$|\sec_{\underline{\mathbf{g}}}|\leq 1$ and

(51) $\operatorname{injrad}_{\underline{\mathbf{g}}}<\hat{\epsilon}\text{ on
$\mathcal{V}$.}$

From our construction, and by respectively (51), (49) and (39), we estimate

$\displaystyle\operatorname{Vol}(M^{(\underline{\mathbf{g}})}_{>\hat{\epsilon}},\underline{\mathbf{g}})$
$\displaystyle\leq
C_{v}\operatorname{Vol}((\partial\mathcal{V}\times[0,A_{v}])\cup\mathcal{W},\underline{\mathbf{g}})$
$\displaystyle\leq C_{v}L$ $\displaystyle\leq
C_{v}\int_{M}|\operatorname{Rm}_{g}|^{2}dvol_{g}$

where $C_{v}$ is a constant only depending on $v$, which was fixed by the
choice of $\eta$ in (50), independently of $(M,g)$. Moreover (if we identify
$(\partial\mathcal{V}\times[0,A_{v}])\cup\mathcal{W}$ with $\mathcal{W}$) by
Lemma 25, Lemma 28 we have $M^{[>\epsilon]}\subset\mathcal{W}$ and by (51)
$\mathcal{V}\subset M^{({\underline{\mathbf{g}}})}_{\leq\hat{\epsilon}}$. This
proves Theorem 24, which finishes the proof of Theorem 23.

∎

###  7.7. An extension to $4$-orbifolds with isolated singularities

In the next Section, we will need the following generalization of Theorem 23
for closed $4$-orbifolds with isolated singular points. Let $M$ be a closed
$4$-orbifold with finitely many singular points endowed with a smooth Einstein
metric $g$. Let

$\\{p_{1},...,p_{K}\\}$

be the singular points of $M$, with corresponding groups $\Gamma_{i}\subset
O(4)$, then for each $i=1,...,K$ there is a trivializing neighborhood
$\hat{B}_{i}$ of $p_{i}$, that is $\hat{B}_{i}$ is a small geodesic ball of
radius $\hat{r}_{i}$ centered at $p_{i}$ such that
$(2\hat{B}_{i},\frac{1}{\hat{r}_{i}^{2}}g)$ is, say, $C^{2}$-close to the
quotient of the Euclidean ball ${B}_{\text{Eucl}}(0,2)$ by $\Gamma_{i}$. By
Chern-Gauss-Bonnet for compact $4$-manifolds with boundary, we have

$\int_{M}|\operatorname{Rm}_{g}|^{2}dvol_{g}\leq{c}\operatorname{\textbf{e}}(M)$
$\text{and }K\leq c\operatorname{\textbf{e}}(M)$

for some universal constant $c$.

Let $\hat{\epsilon}>0$ be as in Proposition 21.

######  Theorem 31.

Let $M$ be a closed $4$-orbifold with finitely many singular points endowed
with a smooth Einstein metric $g$ as above. There is a metric
$\underline{\mathbf{g}}$ on

$N:=M\backslash\\{p_{1},...,p_{K}\\}$

such that

  1. (i)

$|\sec_{\underline{\mathbf{g}}}|\leq 1$,

  2. (ii)

near the end corresponding to $p_{i}$, $\underline{\mathbf{g}}$ is a product
metric of the form

$g_{\text{round}}\oplus dt^{2}$

on $\partial\hat{B}_{i}\times[0,1)$ where $g_{\text{round}}$ is the quotient
of the round metric on $S^{3}$ with volume $8\pi$, and $dt^{2}$ is the
standard metric on $[0,\infty)$,

  3. (iii)

the volume of the $\hat{\epsilon}$-thick part satisfies

$\operatorname{Vol}(N^{(\underline{\mathbf{g}})}_{>\hat{\epsilon}},\underline{\mathbf{g}})\leq
C\operatorname{\textbf{e}}(M)$

where $C$ is a universal constant.

###### Proof.

We only sketch the proof, since it is almost identical to that of Theorem 23.
Informally we treat the singular points as points with infinite curvature.
Take $v,\epsilon>0$ small constants.

As before, define for all $p\in M$:

$r_{v}(p):=\sup\\{r\in(0,1];\operatorname{Vol}(B(p,r),g)\geq vr^{4}\\}.$

If for some $i_{0}\in\\{1,...,K\\}$, the limit volume ratio at $p_{i_{0}}$
satisfies

(52) $\lim_{r\to\infty}\frac{\operatorname{Vol}(B(p_{i_{0}},r))}{r^{4}}\leq
v,$

then there is a small region $U_{i_{0}}$ near $p_{i_{0}}$ which is a cone over
a spherical quotient, and since $v$ is small, the geometry is locally
collapsed and there is an $N$-structure on
$U_{i_{0}}\backslash\\{p_{i_{0}}\\}$. We remove from $M$ small disjoint
neighborhoods of all points $p_{i}$ such that (52) holds at $p_{i}$. We obtain
a compact orbifold with smooth boundary called $M^{\prime}$, and we can make
sure that a neighborhood of the boundary $\partial M^{\prime}$ is saturated by
an $N$-structure. At any point $p\in M^{\prime}$, $r_{v}(p)$ is strictly
positive and we define

$\displaystyle M^{\prime[>\epsilon]}:=\\{x\in M^{\prime};\quad$
$\displaystyle\int_{B_{g}(x,\frac{r_{v}(x)}{24})}|\operatorname{Rm}_{g}|^{2}dvol_{g}>\epsilon\text{
or }B_{g}(x,\frac{r_{v}(x)}{24})\cap\\{p_{1},...,p_{K}\\}\neq\varnothing\\},$

$M^{\prime[\leq\epsilon]}:=M^{\prime}\backslash M^{\prime[>\epsilon]}.$

By definition and by the $\epsilon$-regularity theorem, at any point $x$ of
the thin part $M^{\prime[\leq\epsilon]}$, the ball
$B_{g}(x,\frac{r_{v}(x)}{24})$ is smooth and locally collapsed, in particular
it is covered by an $N$-structure.

We can basically follow the proof of Theorem 23 step by step, applied to
$M^{\prime},M^{\prime[>\epsilon]},M^{\prime[\leq\epsilon]}$. The only
difference lies in the treatment of the singular points of $M^{\prime}$. For
this we can use the generalization of Cheeger-Naber [36] for orbifolds with
isolated singularities described in Subsection 7.1, which implies that the
analogues of Lemmas 25, 26 and 29 hold. In the proof of Lemma 27, we need to
replace the definition of $\bar{k}$ by the following:

$\bar{k}:=\inf\\{\frac{1}{R};R\leq 1,\quad\sup_{x\in
B_{g}(x,R)}|\sec_{g}(x)|^{1/2}\leq\frac{1}{R}\text{ and
}B_{g}(x,R)\cap\\{p_{1},...,p_{K}\\}=\varnothing\\}.$

It follows from the arguments of the previous subsection that we get a metric
on $M^{\prime}$ with bounded curvature, such that the metric has bounded
geometry near $\\{p_{1},...,p_{K}\\}\cap M^{\prime}$. Moreover the volume of
the $\hat{\epsilon}$-thick part of $M^{\prime}$ is bounded by
$C_{v}\operatorname{\textbf{e}}(M)$. Then after modifying once more that
metric near $\\{p_{1},...,p_{K}\\}\cap M^{\prime}$ and $\partial M^{\prime}$,
we get the desired metric $\underline{\mathbf{g}}$ on
$N=M\backslash\\{p_{1},...,p_{K}\\}$, which proves the theorem. ∎

###  7.8. A conjectural obstruction and further remarks

Since manifolds with finite essential minimal volume have simple descriptions
in terms of $F$-structures (see Subsection 6), Theorem 23 could be relevant in
finding new topological obstructions for existence of Einstein manifolds (see
[64] for a survey). Indeed, we conjecture that for some constant $C>0$, for
any surface $\Sigma_{\gamma}$ of genus $\gamma>1$ and any closed oriented
$4$-manifold $N$, if $M$ is homeomorphic to the connected sum
$(S^{2}\times\Sigma_{\gamma})\sharp N$, one should have

$\operatorname{ess-MinVol}(M) \geq C\gamma.$

If this is true and if $N$ is chosen so that
$\operatorname{\textbf{e}}(M)<C^{\prime\prime}\gamma$ for a small enough
constant $C^{\prime\prime}$, then $M$ would not carry any Einstein metric by
Theorem 23.

Theorem 23 is also related to a question of [63], about the number of smooth
structures carrying Einstein metrics, on a given topological $4$-manifold.
What Theorem 23 and Lemma 22 mean is that, at least, the minimal number of
$F$-representatives (see Section 6) for the set of such smooth structures is
quantatively bounded only in terms of the Euler characteristic, and thus those
smooth structures are in finite number modulo some $F$-structures.

We think that Theorem 23 should hold for $\operatorname{MinVol}$ instead of
$\operatorname{ess-MinVol}$, but a proof would require a better understanding
of the thin part of Einstein $4$-manifolds.

Some results analogous to Theorem 23 were proved in the context of minimal
surfaces [85, Corollary 3, Section 4]. The sheeted/non-sheeted decomposition,
the Morse index and curvature bounds for stable surfaces replace respectively
the thin-thick decomposition, the $L^{2}$-norm of the curvature and
$\epsilon$-regularity theorems. The combinatorial argument of [85] could be
adapted to give a proof of Theorem 23 not relying on the codimension 4 theorem
of Cheeger-Naber [36].

##  8\. Essential minimal volume of complex surfaces

This section is about the asymptotic behavior of the essential minimal volume
on complex surfaces. The main theorem gives, for most complex surfaces, an
asymptotic bound for $\operatorname{ess-MinVol}$ in terms of topology. Before
stating and proving this result, we review some elements pertaining to the
classification of complex surfaces.

###  8.1. Preliminaries

Closed complex surfaces are described via the Enriques-Kodaira classification
[11, 46], which divides them according to their Kodaira dimension
$\kappa\in\\{-\infty,0,1,2\\}$. For a complex surface $S$, let $K_{S}$ be its
canonical line bundle. Let $P_{m}(S)$ be the dimension of the space of
holomorphic sections of $K_{S}^{m}$, the $m$th tensor power of $K_{S}$. If
$P_{m}(S)=0$ for all $m\geq 1$, then set $\kappa:=-\infty$. Otherwise, set
$\kappa:=\limsup_{m\to\infty}\frac{\log(P_{m}(S))}{\log(m)}$. A blow-up of $S$
corresponds in terms of diffeomorphism type to performing the connected sum
$S\sharp\overline{\mathbb{C}P}^{2}$. The surface $S$ is called minimal if it
is not the blow-up of another complex surface.

Surfaces of Kodaira dimension $2$ are said to be of general type. In some
sense, most surfaces fall into that category (for instance smooth complete
intersections of degree at least $5$ in $\mathbb{C}P^{3}$ are of general
type). They have positive Euler characteristic.

Surfaces of Kodaira dimension $1$ or $0$ are all diffeomorphic to elliptic
surfaces, and they have nonnegative Euler characteristic. Any minimal elliptic
surface of Euler characteristic $0$ is diffeomorphic to a torus bundle over a
surface on which one performs some logarithmic transforms. Any minimal
elliptic surface is diffeomorphic to a fiber sum of an elliptic surface with
Euler characteristic $0$ and copies of the rational elliptic surface
(diffeomorphic to $\mathbb{C}P^{2}\sharp 9\overline{\mathbb{C}P^{2}}$).

Sufaces of Kodaira dimension $-\infty$ are not completely classified. The
Kähler case is understood: these are just blow-ups of $\mathbb{C}P^{2}$ or of
$\mathbb{C}P^{1}$-bundles over a Riemann surface. Non-Kähler surfaces of
Kodaira dimension $-\infty$ are called surfaces of class VII, and there are
examples but no general classification so far.

###  8.2. The linear bound

Quite similarly to real $2$-dimensional surfaces, the essential minimal volume
of many complex surfaces is comparable to the Euler characteristic.

######  Theorem 32.

There is a universal constant $C>0$ such that for any closed complex surface
$S$ of nonnegative Kodaira dimension,

$C^{-1}\operatorname{\textbf{e}}(S)\leq\operatorname{ess-MinVol}(S)\leq
C\operatorname{\textbf{e}}(S).$

###### Proof.

The lower bound is always true by Lemma 8.

Before continuing, note that since surfaces with nonnegative Kodaira dimension
have nonnegative Euler characteristic, it is enough to prove the theorem for
surfaces that are minimal by Lemma 18. The upper bound is proved in two parts,
depending on the Kodaira dimension of $S$.

If $S$ is a minimal surface of Kodaira dimension $0$ or $1$, then it is
diffeomorphic to an elliptic surface. If it has vanishing Euler
characteristic, [67, Theorem 5] implies that
$\operatorname{MinVol}(S)=\operatorname{ess-MinVol}(S)=0$. Otherwise, $S$ is
the fiber sum of a zero Euler characteristic elliptic surface with some
rational elliptic surfaces and elementary considerations about collapsing
theory suffice to show the desired linear bound, see proof of [67, Theorem 4].

If $S$ is a minimal surface of general type, then its Euler characteristic is
strictly positive. Moreover there is an orbifold Kähler-Einstein metric
$g^{\prime}$ on an orbifold $S^{\prime}$ obtained from $S$ by contracting
finitely many $(-2)$-curves to points inside $S$ [8, 90, 62, 89]. One recovers
$S$ from $S^{\prime}$ by gluing some Ricci-flat ALE spaces [65]

$(X_{1},h_{1}),..,(X_{K},h_{K}),$

to the finitely many orbifold singularities $\\{p_{1},...,p_{K}\\}$ of
$S^{\prime}$ (see also proof of Theorem 4.3 in [68]). These spaces are
asymptotically conical, with limiting flat cone
$(\mathcal{C}_{i},g_{\text{Eucl}})$ and conical section a quotient of the
round unit $3$-sphere. By using the Chern-Gauss-Bonnet formula and the fact
that the $L^{2}$-norm of the Riemann curvature tensor on the non-flat Ricci-
flat ALE spaces $X_{i}$ are lower bounded by a universal positive constant, we
have

(53) $\begin{split}&K\leq\sum_{i=1}^{K}\operatorname{\textbf{e}}(X_{i})\leq
C\operatorname{\textbf{e}}(S)\\\ &\text{and
}\operatorname{\textbf{e}}(S^{\prime})\leq\operatorname{\textbf{e}}(S)+K\leq
C\operatorname{\textbf{e}}(S)\end{split}$

for some universal constant $C>0$.

By Theorem 31, there is a metric $\underline{\mathbf{g}}$ on
$N:=S^{\prime}\backslash\\{p_{1},...,p_{K}\\}$ with
$|\sec_{\underline{\mathbf{g}}}|\leq 1$, which is a nice product metric near
each $p_{i}$, and whose $\hat{\epsilon}$-thick part
$N^{(\underline{\mathbf{g}})}_{>\hat{\epsilon}}$ has controlled volume:

(54)
$\begin{split}\operatorname{Vol}(N^{(\underline{\mathbf{g}})}_{>\hat{\epsilon}},\underline{\mathbf{g}})&\leq
C\operatorname{\textbf{e}}(S^{\prime})\\\ &\leq
C\operatorname{\textbf{e}}(S)\end{split}$

where the universal constant $C$ changes from line to line.

Next, to fix ideas, we can imagine by rescaling that for each $i$ there is
$x_{i}\in X_{i}$ such that $X_{i}\backslash B_{h_{i}}(x_{i},1)$ is
$C^{2}$-close to $(\mathcal{C}_{i}\backslash
B_{g_{\text{Eucl}}}(O,1),g_{\text{Eucl}})$ where $O$ is the tip of
$\mathcal{C}_{i}$. By the proof of Theorem 23, for each $i=1,...,K$, there is
similarly a metric $\underline{\mathbf{g}}_{i}$ on $B_{h_{i}}(x_{i},1)\subset
X_{i}$ with $|\sec_{\underline{\mathbf{g}}_{i}}|\leq 1$, which is a nice
product metric near the boundary $\partial B_{h_{i}}(x_{i},1)$ and whose
$\hat{\epsilon}$-thick part $\mathbf{B}_{i,\geq\hat{\epsilon}}$ satisfies:

(55)
$\operatorname{Vol}(\mathbf{B}_{i,\geq\hat{\epsilon}},{\underline{\mathbf{g}}_{i}})\leq
C\operatorname{\textbf{e}}(X_{i}).$

By gluing the new metrics $(X_{i},\underline{\mathbf{g}}_{i})$ back to
$(S^{\prime},\underline{\mathbf{g}})$ along their boundaries, we get the
desired metric $\underline{\mathbf{h}}$ on $S$, since by (53), (54), (55):

  * •

$|\sec_{\underline{\mathbf{h}}}|\leq 1$,

  * •

$\operatorname{Vol}(S^{(\underline{\mathbf{h}})}_{>\hat{\epsilon}},\underline{\mathbf{h}})\leq
C\operatorname{\textbf{e}}(S)$.

We conclude the proof thanks to Proposition 21.

∎

Since the essential minimal volume quantifies smooth structures modulo
$F$-structures (see end of Subsection 6.1), Theorem 32 can be compared to the
known fact that the set of complex surfaces of general type homeomorphic to a
given $4$-manifold contains finitely many smooth structures, see [46, Theorem
2.8].

Surfaces of the form

$(\mathbb{C}P^{1}\times\Sigma_{\gamma})\sharp k\overline{\mathbb{C}P}^{2},$

where $k$ is an integer and $\Sigma_{\gamma}$ is a Riemann surface of genus
$\gamma>1$, are of Kodaira dimension $-\infty$, and we think that Theorem 32
probably does not generalize to these surfaces even when the Euler
characteristic is positive, see Subsection 7.8.

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