regions for Riemannian manifolds with some GH-prescriptions at infinity,
completing the proof of 1.3. We also derive some rigidity properties for
minimizing (for the perimeter) sequences of finite perimeter sets with fixed
volume.

###  5.1 Existence theorems

The following result, due to Morgan–Johnson [55, Theorem 3.5], constitutes the
key for the existence, since it asserts that a geodesic ball lying on a
manifold with $\mathrm{Ric}\geq(n-1)k$ is isoperimetrically more convenient
then the ball in the model of curvature $k$ (having the same volume). The
centrality of this result in such context was already pointed out in [51].

######  Theorem 5.1.

Let $(M^{n},g)$ be a complete Riemannian manifold such that
$\mathrm{Ric}\geq(n-1)k$ on some open set $\Omega\subset M^{n}$, for
$k\in\mathbb{R}$. Then

$P(B)\leq P_{k}(\mathbb{B}_{k}(\operatorname{vol}(B))),\quad\text{for every
geodesic ball $B\subset\Omega$},$

where $\mathbb{B}_{k}(\operatorname{vol}(B))$ is a geodesic ball on the simply
connected model of sectional curvature $k$ and dimension $n$ having volume
equal to $\operatorname{vol}(B)$.

Moreover, equality holds if and only if $(B,g)$ is isometric to
$(\mathbb{B}_{k}(\operatorname{vol}(B)),g_{k})$, where $g_{k}$ is the metric
on the simply connected model of sectional curvature $k$ and dimension $n$.

We can now state and prove our main existence result.

######  Theorem 5.2.

Let $k\in(-\infty,0]$ and let $(M^{n},g)$ be a complete noncompact Riemannian
manifold such that $\mathrm{Ric}\geq(n-1)k$ on $M\setminus\mathcal{C}$, where
$\mathcal{C}$ is compact.

Suppose that $(M^{n},g)$ is GH-asymptotic to the simply connected model of
constant sectional curvature $k$ and dimension $n$.

Then for any $V>0$ there exists an isoperimetric region of volume $V$ on
$(M^{n},g)$.

###### Proof.

Since $(M^{n},g)$ is GH-asymptotic to the simply connected model of constant
sectional curvature $k$ and dimension $n$, then $(M^{n},g)$ is noncollapsed.
Indeed, if there is a sequence of balls $B_{1}(y_{i})$ with
$\lim_{i}\operatorname{vol}(B_{1}(y_{i}))=0$, then $y_{i}$ must diverge to
infinity, hence by assumption $(M^{n},\mathsf{d},y_{i})$ converges in the pGH-
sense to the simply connected model of constant sectional curvature $k$ and
dimension $n$, which we denote here by $\mathbb{M}^{n}_{k}$, with its own
geodesic distance and volume measure, and pointed at some fixed
$o\in\mathbb{M}^{n}_{k}$. Hence item (b) in 2.10 occurs, and thus $n={\rm
dim}_{H}\,\mathbb{M}^{n}_{k}\leq n-1$, which is impossible.

By Remark 2.3, let $\Omega_{i}\subset M^{n}$ be a minimizing sequence (for the
perimeter) of volume $V>0$ such that $\Omega_{i}$ is bounded and smooth for
any $i$. Let $\Omega_{i}^{c},\Omega_{i}^{d}$ be as in 4.1. If
$\operatorname{vol}(\Omega_{i}^{d})\to 0$, then the set $\Omega$ given by 4.1
is an isoperimetric region of the volume $V$ and the proof ends. So suppose
instead that $\lim_{i}\operatorname{vol}(\Omega_{i}^{d})=W>0$. Then we can
apply 4.6. We employ the notation of 4.6. By assumption and from 2.10, for any
$j<\overline{N}+1$ the pmGH limit space
$(X_{j},\mathsf{d}_{j},\mathfrak{m}_{j},p_{j})$ is $\mathbb{M}^{n}_{k}$ with
its own geodesic distance and volume measure, and pointed at some fixed
$o\in\mathbb{M}^{n}_{k}$. Moreover, since in $\mathbb{M}_{k}^{n}$ balls are
isoperimetric regions for their own volume, we have that, for any
$j<\overline{N}+1$,

$P_{k}(Z_{j})\geq P_{k}(\mathbb{B}_{k}(\operatorname{vol}_{k}(Z_{j}))),$ (5.1)

where $\mathbb{B}_{k}(\operatorname{vol}_{k}(Z_{j}))$ is a geodesic ball in
$\mathbb{M}_{k}^{n}$ having volume equal to $\operatorname{vol}_{k}(Z_{j})$,
while $P_{k}$ is the perimeter functional on $\mathbb{M}^{n}_{k}$.

Now observe that for any compact set $\mathcal{K}\subset M^{n}$, we have that

$\sup\left\\{\operatorname{vol}(B_{r}(p))\ :\ r>0\,\text{ and
}\,B_{r}(p)\Subset M\setminus\mathcal{K}\right\\}=+\infty.$ (5.2)

Indeed, suppose by contradiction the above supremum is bounded by a constant
$S<+\infty$. Take $R>0$ such that
$\operatorname{vol}_{k}(B^{\mathbb{M}_{k}^{n}}_{R}(o))>10S$, where
$B^{\mathbb{M}_{k}^{n}}_{R}(o)$ is a ball of radius $R$ and center $o$ in
$\mathbb{M}_{k}^{n}$. Consider a sequence of balls $B_{R}(x_{i})\subset M^{n}$
of radius $R$ with $\mathsf{d}(x_{i},\mathcal{K})\to+\infty$. Then, up to
passing to a subsequence, 2.10 and the absurd hypothesis imply that

$S\geq\lim_{i}\operatorname{vol}(B_{R}(x_{i}))>10S,$

that is impossible.

Hence (5.2), together with the continuity of the volume with respect to the
radius of balls, imply that, for any compact set $\mathcal{K}\subset M^{n}$
and any assigned finite volume $v$, we can find a ball of volume $v$ compactly
contained in the end $M\setminus\mathcal{K}$. So let
$v_{j}\vcentcolon=\operatorname{vol}_{k}(Z_{j})$ for $j<\overline{N}+1$. Since
$\Omega=\lim_{i}\Omega_{i}^{c}$ is bounded by 4.2, there is a first compact
set $\mathcal{K}_{1}$ such that $\Omega\cup\mathcal{C}\subset\mathcal{K}_{1}$.
Then by (5.2) there is a ball $B_{r_{1}}(q_{1})\Subset
M^{n}\setminus\mathcal{K}_{1}$ such that
$\operatorname{vol}(B_{r_{1}}(q_{1}))=v_{1}$. Inductively, for any $2\leq
j<\overline{N}+1$ we find a compact set $\mathcal{K}_{j}$ such that

$\mathcal{K}_{j}\Supset B_{r_{j-1}}(q_{j-1})\cup\mathcal{K}_{j-1},$

and balls $B_{r_{j}}(q_{j})\Subset M^{n}\setminus\mathcal{K}_{j}$ having
volume $\operatorname{vol}(B_{r_{j}}(q_{j}))=v_{j}$. Hence the balls
$\\{B_{r_{j}}(q_{j})\ :\ j<\overline{N}+1\\}$ are pairwise located at positive
distance and we can define the set

$\widetilde{\Omega}\vcentcolon=\Omega\cup\bigcup_{j=1}^{\overline{N}}B_{r_{j}}(q_{j}).$

By (4.7) we have that

$\operatorname{vol}(\widetilde{\Omega})=\operatorname{vol}(\Omega)+\sum_{j=1}^{\overline{N}}\operatorname{vol}(B_{r_{j}}(q_{j}))=\operatorname{vol}(\Omega)+\sum_{j=1}^{\overline{N}}v_{j}=\operatorname{vol}(\Omega)+W=V.$

Moreover, combining (4.7), (5.1), and 5.1, since all the constructed balls
$B_{r_{j}}(q_{j})$ are contained in an open set of $M^{n}$ on which
$\mathrm{Ric}\geq(n-1)k$, we obtain

$\begin{split}I(V)&=P(\Omega)+\sum_{j=1}^{\overline{N}}P_{k}(Z_{j})\geq
P(\Omega)+\sum_{j=1}^{\overline{N}}P_{k}(\mathbb{B}_{k}(\operatorname{vol}_{k}(Z_{j})))\\\
&\geq
P(\Omega)+\sum_{j=1}^{\overline{N}}P(B_{r_{j}}(q_{j}))=P(\widetilde{\Omega}).\end{split}$
(5.3)

Therefore $\widetilde{\Omega}$ is an isoperimetric region for the volume $V$.
∎

######  Remark 5.3.

We observe that a posteriori $\overline{N}$ is a finite natural number in
$\mathbb{N}$ in the proof of 5.2. Indeed, if $\overline{N}=+\infty$, then the
countably many constructed balls $B_{r_{j}}(q_{j})$ can be easily taken so
that the resulting $\widetilde{\Omega}$ is unbounded. But as
$\widetilde{\Omega}$ turns out to be an isoperimetric region, it must be
bounded by 4.2.

######  Remark 5.4 (About the GH-asymptoticity hypothesis in 5.2).

Assuming only that a complete manifold $(M^{n},g)$ has a uniform lower bound
on the Ricci tensor and that it is noncollapsed is not sufficient to get the
existence of isoperimetric regions (cf. 5.2). In fact, in Example 3.6 we gave
some examples of complete noncollapsed manifolds $(M^{n},g)$ having even
asymptotically vanishing sectional curvature, i.e., such that for any
$\varepsilon>0$ there is a compact $\mathcal{C}\subset M^{n}$ s.t.
$|\mathrm{Sect}|<\varepsilon$ on $M\setminus\mathcal{C}$, such that no
isoperimetric regions of any assigned volume exist. Some of these examples
were already known in the literature, see the introduction of [59].

On the other hand, the above mentioned examples do not have nonnegative Ricci
curvature. It will be target of future investigations to understand whether
complete manifolds $(M^{n},g)$ with $\mathrm{Ric}\geq 0$ and
$\mathrm{AVR}(M^{n},g)>0$ have isoperimetric regions. This could be related
also to the geometry of the asymptotic cones to $(M^{n},g)$.

The existence of isoperimetric regions in nonnegatively Ricci curved manifolds
is very interestingly linked to the concavity of the isoperimetric profile.
This relation relies on the study contained in [13]. Similarly as in [51], the
concavity of the isoperimetric profile is then related to the
indecomposability of the isoperimetric regions as well as to rigidity
statements on the behavior of the minimizing sequences, that will be studied
in the next section. We conclude this section by mentioning these anticipated
results about the concavity of the profile and the indecomposability of
isoperimetric sets.

######  Proposition 5.5.

Let $(M^{n},g)$ be a complete Riemannian manifold such that $\mathrm{Ric}\geq
0$ everywhere. Let us assume the isoperimetric profile $I_{(M^{n},g)}$ is a
continuous function, and that for every $V>0$ there exists an isoperimetric
region of volume $V$. Then the function $I_{(M^{n},g)}^{n/(n-1)}$ is concave,
and thus the function $I_{(M^{n},g)}$ is strictly concave.

###### Proof.

Let $I$ be an open interval of $\mathbb{R}$. For an arbitrary function
$f:I\to\mathbb{R}$ let us denote

$\overline{D}^{2}f(x):=\limsup_{u\to 0^{+}}\frac{f(x+u)+f(x-u)-2f(x)}{u^{2}}.$

From [13, Theorem 2.1] it follows that whenever $(M^{n},g)$ is a compact
Riemannian manifold with $\mathrm{Ric}\geq 0$ in the sense of quadratic forms,
then $\overline{D}^{2}I_{(M^{n},g)}^{n/(n-1)}(x)\leq 0$ for every
$x\in(0,\operatorname{vol}(M^{n}))$. As noticed by Bayle in [13, Remark 2.4],
the same result holds for an arbitrary complete Riemannian manifold
$(M^{n},g)$ with $\mathrm{Ric}\geq 0$ such that for every $V>0$ there exists
an isoperimetric region of volume $V$. Indeed, the computations in [13] only
rely on the existence of an isoperimetric region for any volume $V>0$ and the
regularity result in [66, Proposition 2.4] (see also [54]). It is now an
elementary consequence (see [14, Proposition B.2.1]) that, since
$I_{(M^{n},g)}$ is continuous and satisfies
$\overline{D}^{2}I_{(M^{n},g)}^{n/(n-1)}\leq 0$, then
$I_{(M^{n},g)}^{n/(n-1)}$ is concave, and thus $I_{(M^{n},g)}$ is strictly
concave. ∎

######  Proposition 5.6.

Let $(M^{n},g)$ be a complete Riemannian manifold such that its isoperimetric
profile is a strictly concave function. Then, any nonempty isoperimetric
region $\Omega$ is indecomposable, i.e., if $E,F$ are finite perimeter sets
such that $\operatorname{vol}(\Omega\Delta(E\cup F))=0$ and
$P(\Omega)=P(E)+P(F)$, then $\operatorname{vol}(E)=0$ or
$\operatorname{vol}(F)=0$.

###### Proof.

It is sufficient to notice that since $I_{(M^{n},g)}$ is strictly concave and
$I_{(M^{n},g)}(0)=0$, then $I_{(M^{n},g)}$ is strictly subadittive. Hence one
can argue exactly as at the end of [51, Corollary 3.4] to prove the statement.
∎

######  Corollary 5.7.

Let $(M^{n},g)$ be a complete noncompact Riemannian manifold with
$\mathrm{Ric}\geq 0$ and such that it is GH-asymptotic to $\mathbb{R}^{n}$.
Then the isoperimetric profile $I_{(M^{n},g)}$ is strictly concave and the
isoperimetric regions are indecomposable.

###### Proof.

By the reasoning at the beginning of the proof of 5.2 we get that $M^{n}$ is
noncollapsed. Hence $I_{(M^{n},g)},$ is continuous, thanks to [58, Theorem 2],
since $M^{n}$ satisfies $\mathrm{Ric}\geq 0$ and it is noncollapsed. Then the
conclusion is reached by joining together 5.5 and 5.6. ∎

The statement of 1.3 is just the combination of 5.2 and 5.7. Hence, it is
proved.

###  5.2 Rigidity properties of the minimizing sequences

In the argument leading to the proof of the main isoperimetric existence
result (5.2), we essentially showed that whenever a minimizing sequence does
not provide an isoperimetric set of the desired volume, then the union of a
smaller isoperimetric set with a suitable union of geodesic balls recovers the
desired volume while mantaining the optimality in perimeter. This happens in
case part of the minimizing sequence is diverging at infinity. The results
below roughly assert that, in case this happens, the ambient manifold has,
outside every compact set, some open set with constant sectional curvature.

We adopt the following terminology.

######  Definition 5.8 (Finite perimeter sets diverging at infinity).

Let $(M^{n},g)$ be a complete Riemannian manifold. Let $\Omega_{i}\subset
M^{n}$ be a sequence of finite perimeter sets such that
$0<c_{1}\leq\operatorname{vol}(\Omega_{i})\leq c_{2}<+\infty$ for any $i$.

  * •

We say that $\Omega_{i}$ _diverges at infinity_ if
$\limsup_{i\to+\infty}\operatorname{vol}(\Omega_{i}\cap\mathcal{C})=0$ for any
compact set $\mathcal{C}\subset M^{n}$.

  * •

Suppose that $\Omega_{i}\to\Omega$ in the sense of finite perimeter sets. Then
we say that $\Omega_{i}$ _loses mass at infinity_ if
$\operatorname{vol}(\Omega)<\liminf_{i\to+\infty}\operatorname{vol}(\Omega_{i})$.

We are now ready for proving the results about rigidity of perimeter-
minimizing sequences.

######  Theorem 5.9.

Let $(M^{n},g)$ be a complete Riemannian manifold satisfying the hypothesis of
5.2, with $k\in(-\infty,0]$. Then, for every $V>0$, every minimizing (for the
perimeter) sequence of finite perimeter sets
$\\{\Omega_{i}\\}_{i\in\mathbb{N}}$ of volume $V>0$

  * (i)

either does not lose mass at infinity,

  * (ii)

or loses mass at infinity and in this case for every compact set
$\mathcal{K}\subset M^{n}$ there exists a geodesic ball in
$M^{n}\setminus\mathcal{K}$ that has constant sectional curvature $k$.

###### Proof.

Let $V,\Omega_{i}$ be as in the statement. Let $\Omega$ be the limit of
$\Omega_{i}^{c}$ in $L^{1}_{\rm loc}(M^{n},g)$, where $\Omega_{i}^{c}$ is as
in 4.1. Assume that $\\{\Omega_{i}\\}_{i\in\mathbb{N}}$ loses mass at
infinity, i.e. $V-\operatorname{vol}(\Omega)>0$, and let $\mathcal{K}$ be a
fixed compact set. From the argument in the proof of 5.2, the balls recovering
the lost volume $V-\operatorname{vol}(\Omega)$ can be taken outside
$\mathcal{K}$. Hence there exists a finite number $\overline{N}$ (cf. Remark
5.3) of balls such that in (5.3) the equality holds, because there we also
obviously have $P(\widetilde{\Omega})\geq I(V)$. Then, employing the notation
of the proof of 5.2, we deduce that for every $j<\overline{N}+1$,
$P(B_{r_{j}}(q_{j}))=P_{k}(\mathbb{B}_{k}(\operatorname{vol}_{k}(Z_{j})))$.
Hence, for every $j<\overline{N}+1$, $(B_{r_{j}}(q_{j}),g)$ is isometric to
$(\mathbb{B}_{k}(\operatorname{vol}_{k}(Z_{j})),g_{k})$ by the rigidity part
of 5.1. Thus, outside every compact set $\mathcal{K}$ there exists at least
one geodesic ball on which the sectional curvature is constantly equal to $k$.
∎

In case in the hypotheses of 5.2 we have that $\mathrm{Ric}\geq 0$ everywhere,
we can use the strict concavity of the isoperimetric profile to deduce the
following stronger rigidity result.

######  Theorem 5.10.

Let $(M^{n},g)$ be a complete noncompact Riemannian manifold such that
$\mathrm{Ric}\geq 0$ on $M^{n}$. Suppose that $(M^{n},g)$ is GH-asymptotic to
$\mathbb{R}^{n}$. Then, for every $V>0$, every minimizing (for the perimeter)
sequence of finite perimeter sets $\\{\Omega_{i}\\}_{i\in\mathbb{N}}$ of
volume $V$

  * (i)

either does not lose mass at infinity,

  * (ii)

or diverges at infinity and in this case $\mathrm{Sect}\equiv 0$ on $M^{n}$,
and thus $(M^{n},g)$ is isometric to a quotient of $\mathbb{R}^{n}$ with
respect to a proper discrete subgroup of the Euclidean isometries.

###### Proof.

Let $V,\Omega_{i}$ be as in the statement. Let $\Omega$ be the limit of
$\Omega_{i}^{c}$ in $L^{1}_{\rm loc}(M^{n},g)$, where $\Omega_{i}^{c}$ is as
in 4.1. Let $\Omega_{i}^{d}=\Omega_{i}\setminus\Omega_{i}^{c}$ be as in 4.1.
Since $M^{n}$ is noncollapsed, see the first part of the proof of 5.2, and
satisfies $\mathrm{Ric}\geq 0$ everywhere, an application of [58, Theorem 2]
tells us that the isoperimetric profile $I_{(M^{n},g)}$ is continuous. Then,
since we have also the existence of isoperimetric regions of any volume from
5.2, we can use 5.5 and conclude that $I_{(M^{n},g)}$, in what follows simply
denoted by $I$, is continuous and strictly subadditive.

Assume that $\\{\Omega_{i}\\}_{i\in\mathbb{N}}$ loses mass at infinity, in
other words that $\operatorname{vol}(\Omega)<V$. Then,
$W:=\lim_{i}\operatorname{vol}(\Omega^{d}_{i})>0$. We first prove that
$\operatorname{vol}(\Omega)=0$. Indeed, supposing by contradiction that
$\operatorname{vol}(\Omega)>0$, we get, using also the identities in 4.1 and
the subadditivity of the isoperimetric profile

$\begin{split}I(V)&=\lim_{i}\left(P(\Omega_{i}^{c})+P(\Omega_{i}^{d})\right)=P(\Omega)+\lim_{i}P(\Omega_{i}^{d})\\\
&\geq
I(\operatorname{vol}(\Omega))+\limsup_{i}I(\operatorname{vol}(\Omega_{i}^{d}))=I(\operatorname{vol}(\Omega))+I(W)\\\
&>I(\operatorname{vol}(\Omega)+W)=I(V),\end{split}$

that is impossible.

We can then assume that $W>0$ and $\Omega=\emptyset$, and so since
$\Omega_{i}^{c}=\Omega_{i}\cap B_{r_{i}}(o)$ for some diverging sequence
$r_{i}>0$, for any compact set $\mathcal{K}$ we have

$\limsup_{i}\operatorname{vol}(\Omega_{i}\cap\mathcal{K})=\limsup_{i}\operatorname{vol}(\Omega_{i}^{c}\cap\mathcal{K})=\operatorname{vol}(\Omega\cap\mathcal{K})=0.$

Hence $\Omega_{i}$ diverges at infinity.

Since we also have $\mathrm{Ric}\geq 0$ everywhere, the finite number
$\overline{N}$ (see Remark 5.3) of balls $B_{r_{j}}(q_{j})$ with volume
$v_{j}$ in the proof of 5.2 can be taken anywhere on the manifold. Thus, for
example, for every point $q\in M^{n}$ on the manifold we can choose
$B_{r_{1}}(q_{1})$ in the proof of 5.2 to be some ball $B_{r_{1}}(q)$ with the
radius $r_{1}$ chosen in such a way the volume of $B_{r_{1}}(q)$ is $v_{1}$.
Thus, since in (5.3) equality holds, this implies that $P(B_{r_{1}}(q))=P_{\rm
eu}(\mathbb{B}^{n}(v_{1}))$, and thus the rigidity part of 5.1 implies that
$(B_{r_{1}}(q),g)$ is isometric to $(\mathbb{B}^{n}(v_{1}),g_{\rm eu})$, and
thus the sectional curvature is constantly equal to zero on $B_{r_{1}}(q)$.
Exploiting the arbitrariness of $q$ we conclude that $\mathrm{Sect}\equiv 0$
on $M^{n}$, and thus the proof is concluded. ∎

######  Remark 5.11 (A remark on the proof of 5.10).

It is useful to notice that in the proof of 5.10 we actually proved the
following result: if $(M^{n},g)$ is a complete Riemannian manifold such that
the isoperimetric profile $I_{(M^{n},g)}$ is continuous and strictly
subadditive, then a minimizing (for the perimeter) sequence of finite
perimeter sets $\\{\Omega_{i}\\}_{i\in\mathbb{N}}$ of volume $V>0$ either
diverges at infinity, or converges to a finite perimeter set $\Omega$ (in the
sense of finite perimeter sets) without losing mass at infinity. In the latter
case, $\Omega$ is an isoperimetric region of volume $V$ and $P(\Omega_{i})\to
P(\Omega)$.

Indeed, such a statement follows by repeating the arguments in the proof of
5.10 employing 4.1. In fact, this results is completely analogous to [51,
Corollary 3.5], whose proof indeed relies on the assumptions that the
isoperimetric profile is continuous and strictly subadditive only.

##  6 Applications and examples

In this section we give effective conditions that imply the hypotheses of 5.2.
We start by recalling some definitions about convergence of manifolds. The
following definition is taken from [62, Section 11.3.2].

######  Definition 6.1 ($C^{0}$-convergence of manifolds).

Given $(M^{n},g,p)$ a pointed Riemannian manifold, and
$\\{(M_{i}^{n},g_{i},p_{i})\\}_{i\in\mathbb{N}}$ a sequence of pointed
Riemannian manifolds, we say that $(M_{i}^{n},g_{i},p_{i})$ converge to
$(M^{n},g,p)$ in the $C^{0}$-sense if for every $R>0$ there exists a domain
$\Omega\subset M^{n}$ containing $B_{R}(p)$ and, for large $i$, embeddings
$F_{i}:\Omega\to M_{i}^{n}$ such that $F_{i}(p)=p_{i}$, $F_{i}(\Omega)$
contains $B_{R}(p_{i})$, and the pull-back metrics $F_{i}^{*}g_{i}$ converge
to $F$ in the $C^{0}$-sense on $\Omega$, i.e., all the components of the
metric tensors converge in the $C^{0}$ norm in a finite covering of coordinate
patches on $\Omega$.

By using the previous notion of convergence, we can define what means for a
Riemannian manifold to be $C^{0}$-asymptotic to the simply connected model
$\mathbb{M}^{n}_{k}$ of dimension $n\in\mathbb{N}$ and constant sectional
curvature $k\in\mathbb{R}$. The forthcoming notion has been investigated in
[51], see in particular [51, Theorem 1.2].

######  Definition 6.2 ($C^{0}$-local asymptoticity).

We say that a Riemannian manifold $(M^{n},g)$ is $C^{0}$-locally asymptotic to
the simply connected model $\mathbb{M}^{n}_{k}$ of dimension $n\in\mathbb{N}$
and constant sectional curvature $k\in\mathbb{R}$ if for every diverging
sequence of points $p_{i}$ in $M^{n}$ we have that $(M^{n},g,p_{i})$ converge
to $(\mathbb{M}^{n}_{k},g_{k},o)$ in the $C^{0}$-sense, where $g_{k}$ is the
Riemannian metric on $\mathbb{M}^{n}_{k}$ and $o$ is a fixed origin.

######  Remark 6.3 (GH-asymptoticity and $C^{0}$-local asymptoticity).

We remark that our 5.2 implies one of the main theorems in [51], namely [51,
Theorem 1.2]. Indeed, the notion of being $C^{0}$-locally asymptotic to the
simply connected model $\mathbb{M}^{n}_{k}$ of constant sectional curvature
$k\in\mathbb{R}$ and dimension $n\in\mathbb{N}$, see [51, Definition 2.2,
Definition 2.4], is readily stronger than being GH-asymptotic to
$\mathbb{M}^{n}_{k}$, cf. [62, Section 11.3.2].

As a consequence, all the examples in [51, Remark 1.1], namely the ALE
gravitational instantons, the asymptotically hyperbolic Einstein manifolds,
and the Bryant type solitons satisfy the hypotheses of 5.2. It would be
interesting to show an example of a Riemannian manifold that is GH-asymptotic
to the model $\mathbb{M}^{n}_{k}$ but that is not $C^{0}$-locally asymptotic
to it.

As an easy consequence of Lemma A.2 we get a criterion to check that a
Riemannian manifold is $C^{0}$-locally asymptotic, and hence GH-asymptotic
(see Remark 6.3), to the simply connected model of constant sectional
curvature $k\in\mathbb{R}$ and dimension $n\in\mathbb{N}$. Let us introduce
our notions of _sectional curvature asymptotically equal to $k$_ and of
_asymptotically diverging injectivity radius_.

######  Definition 6.4 (Sectional curvature asymptotically equal to $k$).

Given $k\in(-\infty,0]$, we say that a noncompact Riemannian manifold
$(M^{n},g)$ has sectional curvature asymptotically equal to $k$ if there
exists $o\in M^{n}$ such that for every $0<\varepsilon<1$ there exists
$R_{\varepsilon}>0$ for which

$\begin{split}\lvert\mathrm{Sect}_{x}(\pi)-k\rvert\leq\varepsilon&\qquad\text{for
all}\,x\in M\setminus\overline{B}_{R_{\varepsilon}}(o),\,\,\text{for all
2-planes $\pi$ in $T_{x}M^{n}$}.\end{split}$ (6.1)

If $k=0$ we say that $(M^{n},g)$ has asymptotically vanishing sectional
curvature.

We stress that, from now on, when we write $\lvert\mathrm{Sect}\rvert\leq c$
everywhere on some set $\Omega$, we mean that
$\lvert\mathrm{Sect}_{x}(\pi)\rvert\leq c$ for every $x\in\Omega$ and every
2-plane $\pi\in T_{x}M^{n}$.

######  Definition 6.5 (Asymptotically diverging injectivity radius).

We say that a noncompact Riemannian manifold $(M^{n},g)$ has asymptotically
diverging injectivity radius if there exists $o\in M^{n}$ such that for every
$S>1$ there exists $R_{S}>0$ for which

$\begin{split}\mathrm{inj}(x)\geq S&\qquad\text{for all}\,\,x\in
M\setminus\overline{B}_{R_{S}}(o).\end{split}$ (6.2)

In the following statement we record how the coupling of the two conditions
above suffices to infer the GH-asymptoticity to space forms. The following
statement is a consequence of Lemma A.2.

######  Proposition 6.6.

Let $(M^{n},g)$ be a complete noncompact Riemannian manifold with _sectional
curvature asymptotically equal to $k$_, for some $k\in(-\infty,0]$, and with
_asymptotically diverging injectivity radius_. Then, $(M^{n},g)$ is
$C^{0}$-locally asymptotic, and hence GH-asymptotic, to the simply connected
model $\mathbb{M}^{n}_{k}$ of dimension $n\in\mathbb{N}$ and constant
sectional curvature $k\in(-\infty,0]$.

By means of a classical compactness theorem, we can actually prove more than
the $C^{0}$-local convergence above, as explained in the following Remark.

######  Remark 6.7 (A stronger thesis in 6.6).

We remark that under the same hypotheses of 6.6 we can prove that, for every
$0<\alpha<1$, $(M^{n},g)$ is $C^{1,\alpha}$-asymptotic to the simply connected
model of dimension $n\in\mathbb{N}$ and constant sectional curvature
$k\in(-\infty,0]$, see [62, Section 11.3.1 and Section 11.3.2] for the main
definitions about $C^{m,\alpha}$ convergence, which we give for granted here.
Indeed, under the hypotheses of 6.6, it is readily seen that there exist
constants $K,R>0$ such that $\lvert\mathrm{Sect}\rvert\leq K$ on $M^{n}$, and
$\operatorname{inj}\geq R$ on $M^{n}$. Hence, from the compactness result in
[62, Theorem 11.4.7] we get that for every diverging sequence of points
$p_{i}\in M^{n}$ we have that $(M^{n},g,p_{i})$ converge in the
$C^{1,\alpha}$-pointed topology, up to subsequences, to a Riemannian manifold
$(M_{\infty}^{n},g_{\infty},p_{\infty})$. But since the $C^{0}$-limit of
$(M^{n},g,p_{i})$ is $\mathbb{M}^{n}_{k}$, we get that every subsequence of
$(M^{n},g,p_{i})$ must converge, up to subsequences, to $\mathbb{M}^{n}_{k}$
in the $C^{1,\alpha}$-pointed topology. Hence, the entire sequence
$(M^{n},g,p_{i})$ converges in the $C^{1,\alpha}$-pointed topology to
$\mathbb{M}^{n}_{k}$.

We are now committed to link the two above defined notions of asymptotically
constant sectional curvature and asymptotically diverging injectivity radius,
in order to discuss some effective and basic conditions that on a complete
noncompact Riemannian manifold ultimately imply the existence of isoperimetric
regions of any volume. The following is essentially a consequence of a
fundamental injectivity radius estimate in [24].

######  Lemma 6.8.

Let $(M^{n},g)$ be a complete Riemannian manifold with asymptotically
vanishing sectional curvature. Let us assume there exists a compact set
$\mathcal{C}\subset M^{n}$, a real number $\alpha<1$, and a constant $C>0$
such that

$\mathrm{vol}{(B_{r}(p))}\geq Cr^{n-\alpha},$ (6.3)

for any ball $B_{r}(p)\subset M^{n}\setminus\mathcal{C}$ with $r>1$. Then
$(M^{n},g)$ has asymptotically diverging injectivity radius.

###### Proof.

Let $0<\varepsilon<1/100$, and fix $o\in M^{n}$. By the asymptotic vanishing
of the sectional curvature, there exists a radius $R_{\varepsilon}$ such that
$\lvert\mathrm{Sect}\rvert<\varepsilon$ on
$M^{n}\setminus\overline{B}_{R_{\varepsilon}}(o)$ and $\mathcal{C}\subset
B_{R_{\varepsilon}}(o)$. Let then $p\in
M^{n}\setminus\overline{B}_{R_{\varepsilon}+\pi/\sqrt{\varepsilon}}(o)$, and
observe that
$B_{\pi/\sqrt{\varepsilon}}(p)\Subset(M^{n}\setminus\overline{B}_{R_{\varepsilon}}(o))$.

Assume that $\mathrm{inj}(p)<\pi/\sqrt{\varepsilon}$. Then, there exists
$q\in\mathrm{Cut}(p)$ such that $d(p,q)=\mathrm{inj}(p)$, and that in
particular still belongs to $M^{n}\setminus\overline{B}_{R_{\varepsilon}}(o)$.
Then, by [18, Proposition 2.12, Chapter 13], either there exists a geodesic
$\gamma$ joining $p$ and $q$ such that $q$ is conjugate to $p$ along $\gamma$,
or there exists a geodesic loop $\sigma$ based at $p$ passing through $q$ with
length equal to $2\operatorname{inj}(p)$, so that in particular $\sigma$ is
still contained in $M^{n}\setminus\overline{B}_{R_{\varepsilon}}(o)$. In the
first case, a straightforward application of Rauch’s Comparison Theorem [18,
Proposition 2.4, Chapter 10] implies that $d(p,q)\geq\pi/\sqrt{\varepsilon}$,
a contradiction with the assumption above.

In the second case, we have
$2\,\mathrm{inj}(p)=\ell:=\mathrm{length}\,(\sigma)$, and we can thus estimate
$\mathrm{inj}(p)$ in terms of volumes of balls by means of [24, Theorem 4.3],
that yields

$\mathrm{inj}(p)\geq\frac{\pi}{8\sqrt{\varepsilon}}\left(1+\frac{v(n,-\varepsilon,\frac{7\pi}{16\sqrt{\varepsilon}})}{\mathrm{vol}\left(B_{\frac{3\pi}{16\sqrt{\varepsilon}}}\left(p\right)\right)}\right)^{-1}\geq\frac{\pi}{8\sqrt{\varepsilon}}\left(1+\frac{v(n,-\varepsilon,\pi/\sqrt{\varepsilon})}{\mathrm{vol}\left(B_{\frac{3\pi}{16\sqrt{\varepsilon}}}\left(p\right)\right)}\right)^{-1},$
(6.4)

where we applied [24, Theorem 4.3] with $r=\pi/\sqrt{\varepsilon}$,
$r_{0}=r/4$, and $s=\tfrac{3}{16}r$ in the notation therein.

We have, for a dimensional constant $C(n)$, the following equality

$v(n,-\varepsilon,\pi/\sqrt{\varepsilon})=\int\limits_{0}^{\pi/\sqrt{\varepsilon}}\left(\frac{\sinh(s\sqrt{\varepsilon})^{n-1}}{\sqrt{\varepsilon}}\right)^{n-1}\,\mathrm{d}s=\frac{1}{(\sqrt{\varepsilon})^{n}}\int\limits_{0}^{\pi}\sinh(t)^{n-1}\,\mathrm{d}t=C(n)\frac{1}{(\sqrt{\varepsilon})^{n}}.$
(6.5)

Plugging (6.3) and (6.5) into (6.4) then yields

$\mathrm{inj}(p)\geq\frac{\pi}{8\sqrt{\varepsilon}}\left(1+\overline{C}\varepsilon^{-\frac{\alpha}{2}}\right)^{-1},$

where $\overline{C}=\overline{C}(C(n),C)$. All in all, we proved

$\mathrm{inj}(p)\geq\mathrm{min}\left\\{\frac{\pi}{\sqrt{\varepsilon}},\frac{\pi}{8\sqrt{\varepsilon}}\left(1+\overline{C}\varepsilon^{-\frac{\alpha}{2}}\right)^{-1}\right\\}.$

Since $\alpha<1$, the right-hand-side of the previous inequality diverges at
infinity when $\varepsilon\to 0$, implying that $(M^{n},g)$ has asymptotically
diverging injectivity radius. ∎

######  Remark 6.9 (A weakened hypothesis in Lemma 6.8).

From the proof of Lemma 6.8 it is readily noticed that the condition (6.3) can
be slightly weakened into the following

$\liminf_{r\to+\infty}\liminf_{d(o,p)\to+\infty}\frac{\operatorname{vol}(B_{r}(p))}{r^{n-\alpha}}>0,$
(6.6)

where $o\in M^{n}$ is some fixed origin.

As a consequence of 6.6, Lemma 6.8, and 5.2 we get the following isoperimetric
existence result under curvature and volume conditions.

######  Corollary 6.10.

Let $(M^{n},g)$ be a complete Riemannian manifold with asymptotically
vanishing sectional curvature. Moreover, assume that there exists a compact
set $\mathcal{C}$ such that that $\mathrm{Ric}\geq 0$ on
$M\setminus\mathcal{C}$, and moreover there exist $\alpha<1$ and $C>0$ such
that $\operatorname{vol}(B_{r}(p))\geq Cr^{n-\alpha}$ for any ball
$B_{r}(p)\Subset M\setminus\mathcal{C}$ with $r>1$. Then, for every $V>0$
there exists an isoperimetric region of volume $V$.

######  Remark 6.11 (A weakened hypothesis in 6.10).

As already discussed in Remark 6.9, we can slightly weaken the hypothesis on
the bound of the volumes in the previous 6.10 with (6.6).

The volume condition in the statement of 6.10 is automatically satisfied on
manifolds $(M^{n},g)$ with nonnegative Ricci curvature, asymptotically
vanishing sectional curvature, and Euclidean volume growth, that is,
$\mathrm{AVR}(M^{n},g)>0$.

Indeed, in such a case, by Bishop-Gromov we have
$\mathrm{AVR}(M^{n},g)\omega_{n}r^{n}\leq\mathrm{vol}(B_{r}(p))\leq\omega_{n}r^{n}$
for any $p\in M^{n}$ and any $r>0$. We observe also that, since such a
condition on the volume of balls is needed to hold just outside some compact
set, we actually get the existence of isoperimetric regions of any volume on
any compact perturbation of a complete Riemannian manifold with asymptotically
vanishing sectional curvature, $\mathrm{Ric}\geq 0$, and Euclidean volume
growth.

######  Corollary 6.12.

Let $(M^{n},\tilde{g})$ be a complete Riemannian manifold with nonnegative
Ricci curvature, asymptotically vanishing sectional curvature, and Euclidean
volume growth. Let $(M^{n},g)$ be a compact perturbation of
$(M^{n},\tilde{g})$, that is, there exists a compact set $\mathcal{C}$ such
that $\tilde{g}=g$ on $M^{n}\setminus\mathcal{C}$. Then, for every $V>0$ there
exists an isoperimetric region of volume $V$.

It is interesting to observe that 6.12 applies to Perelman’s example
constructed in [61] (see also [21, Section 8]), that is a complete Riemannian
manifold $(M^{n},g)$ with nonnegative Ricci curvature, Euclidean volume
growth, asymptotically vanishing sectional curvature (it satisfies a quadratic
decay), and admitting non-isometric asymptotic cones. We recall that an
asymptotic cone to a manifold $(M^{n},g)$ at some $x\in M^{n}$ is the pGH
limit of the sequence of metric spaces $(M^{n},r_{i}^{-1}\mathsf{d},x)$ for
some diverging sequence $r_{i}\to+\infty$. In particular, we deduce that the
non-uniqueness of asymptotic cones is not an obstruction to the existence of
isoperimetric regions, even in the case of $\mathrm{Ric}\geq 0$ and Euclidean
volume growth.

######  Remark 6.13 (Asymptotically Euclidean and conical manifolds).

A direct application of 6.12 implies that every compact perturbation of an ALE
manifold with $\mathrm{Ric}\geq 0$, see e.g. [1, Definition 4.13], has
isoperimetric regions for any volume. Indeed, it is immediately checked that
an ALE manifold with $\mathrm{Ric}\geq 0$ has Euclidean volume growth and
asymptotically vanishing sectional curvature.

An application of 6.12 implies also that every compact perturbation of a
$C^{2}$-asymptotically conical manifold (in the sense of [26]) with
$\mathrm{Ric}\geq 0$ has isoperimetric regions for every volume. Indeed, every
$C^{2}$-asymptotically conical manifold has asymptotically vanishing sectional
curvature and Euclidean volume growth. We remark that in [26, Theorem 3] the
authors prove that a $C^{1,\alpha}$-asymptotically conical manifold (without
further bounds on $\mathrm{Ric}$) has isoperimetric regions for large volumes,
and they describe the structure of isoperimetric regions with large volumes
for $C^{2,\alpha}$-asymptotically conical manifolds.

Our results allow to generalize the applications on asymptotically conical
manifolds considered in Remark 6.13 to manifolds which are suitably asymptotic
to warped products with $\mathrm{Ric}\geq 0$ and asymptotically vanishing
sectional curvature. We discuss this observation in the next remark.

######  Remark 6.14 (Warped products with $\mathrm{Ric}\geq 0$ and
asymptotically vanishing sectional curvature).

Let $(W,\widetilde{g})$ be an arbitrary warped product defined by

$W:=(0,+\infty)\times L,\qquad\widetilde{g}:=\,\mathrm{d}r^{2}+f(r)^{2}g_{L},$

where $(L,g_{L})$ is a compact Riemannian manifold and
$f:(0,+\infty)\to(0,+\infty)$ is a smooth function. Let $(M^{n},g)$ be a
complete Riemannian manifold such that there exists a compact set
$\mathcal{K}\subset M^{n}$ such that $(M^{n}\setminus\mathcal{K},g)$ is
isometric to $((a,+\infty)\times L,\widetilde{g})$ for some $a\geq 0$.

We want to show here that if

$\lim_{r\to+\infty}f(r)=+\infty,\qquad\text{and $W$ has asymptotically
vanishing sectional curvature},$ (6.7)

then $(M^{n},g)$ has asymptotically diverging injectivity radius. Observe
that, by a direct computation of the Riemann tensor, the asymptotic vanishing
of the sectional curvature is ensured every time $f^{\prime}=o(f)$ and
$f^{\prime\prime}=o(f)$ as $r\to+\infty$.

Let us now prove the latter claim after (6.7). Indeed, since the sectional
curvature is asymptotically vanishing, arguing as in the proof of Lemma 6.8,
for a given $\varepsilon\in(0,1)$ it suffices to estimate from below the
length $\ell$ of a geodesic loop based at $p\in
M\setminus\mathcal{K}_{\varepsilon}$ by $\ell\geq C/\varepsilon$ for a
constant $C$ independent of $\varepsilon$, and for some compact set
$\mathcal{K}_{\varepsilon}\supset\mathcal{K}$. We identify
$M\setminus\mathcal{K}$ with $((a,+\infty)\times L,\widetilde{g})$. Take
$\widetilde{\mathcal{K}}_{\varepsilon}$ such that
$|\mathrm{Sect}|<\varepsilon$ on
$M\setminus\widetilde{\mathcal{K}}_{\varepsilon}$. As in Lemma 6.8, we can
consider
$\mathcal{K}_{\varepsilon}\supset\widetilde{\mathcal{K}}_{\varepsilon}$ such
that for every $p\in M\setminus\mathcal{K}_{\varepsilon}$ we have
$\mathsf{d}(p,\widetilde{\mathcal{K}}_{\varepsilon})>\pi/\sqrt{\varepsilon}$.
Also, without loss of generality, up to eventually enlarging
$\mathcal{K}_{\varepsilon}$, we can just estimate a geodesic loop $\gamma$
based at $p$ such that $\gamma:[0,1]\to((a_{\varepsilon},+\infty)\times
L,\widetilde{g})$ and $a_{\varepsilon}$ is such that $f(r)\geq 1/\varepsilon$
for $r\geq a_{\varepsilon}$. We have $\gamma=(\gamma_{1},\gamma_{2})$, and
$\gamma_{2}^{\prime}(0)\neq 0$, for otherwise $\gamma$ would be tangent to
$(a_{\varepsilon},+\infty)$ and $\gamma$ would not be closed. Then
$\gamma_{2}$ is a nonconstant continuous curve in $L$. For $S\subset L$, it
can be shown by the direct computation of the second fundamental form of the
isometric embedding $(a^{\prime},+\infty)\times
S\hookrightarrow((a^{\prime},+\infty)\times L,\widetilde{g})$ that $S$ is a
totally geodesic submanifold of $(L,g_{L})$ if and only if so is
$(a^{\prime},+\infty)\times S$ in $((a^{\prime},+\infty)\times
L,\widetilde{g})$, for any $a^{\prime}$. This implies that $\gamma_{2}$ is a
geodesic loop in $(L,g_{L})$, up to reparametrization. Hence the length of
$\gamma$ is estimated from below by

$\begin{split}\ell&=\int_{0}^{1}\left(|\gamma_{1}^{\prime}(t)|^{2}+f^{2}(\gamma_{1}(t))g_{L}(\gamma_{2}^{\prime}(t),\gamma_{2}^{\prime}(t))\right)^{1/2}\,\mathrm{d}t\\\
&\geq\frac{\sqrt{2}}{2}\left(\int_{0}^{1}|\gamma_{1}^{\prime}(t)|\,\mathrm{d}t+L(\gamma_{2})\min_{[0,1]}f(\gamma_{1}(t))\right)\\\
&\geq\frac{\sqrt{2}}{2}\left(\int_{0}^{1}|\gamma_{1}^{\prime}(t)|\,\mathrm{d}t+{\rm
syst}(L)\,\min_{[0,1]}f(\gamma_{1}(t))\right),\end{split}$

where ${\rm syst}(L)>0$ denotes the systole of $(L,g_{L})$, that is the length
of the shortest geodesic loop in $(L,g_{L})$. By construction, the above
estimate implies that

$\ell\geq\frac{\sqrt{2}}{2}{\rm syst}(L)\frac{1}{\varepsilon}.$

Hence we conclude that the injectivity radius is asymptotically diverging.

Therefore, if in addition to (6.7), we have that $\mathrm{Ric}\geq 0$ outside
a compact set of $M^{n}$, then 6.6 applies, $(M^{n},g)$ is GH-asymptotic to
the Euclidean space $\mathbb{R}^{n}$, and by 5.2 there exist isoperimetric
regions of any volume.

It is clear that the very same conclusion holds true if we assumed that
$(M^{n},g)$ satisfies $\mathrm{Ric}\geq 0$ outside a compact set and that
$M^{n}$ is just $C^{2}$-asymptotic to $(W,\widetilde{g})$.

We observe that, for examples, the Bryant type solitons mentioned in Remark
6.3 fit in this setting of warped products.

##  Appendix A Comparison results in Riemannian Geometry

We write down a complete statement of a rather classical comparison result in
Geometric Analysis, i.e., the Bishop–Gromov–Günther volume and area comparison
under Ricci curvature lower bounds and sectional curvature upper bounds. The
conclusions (A.1), (A.2), (A.4), (A.5), and the rigidity part of A.1 are
consequences, e.g., of [35, Theorem 3.101], [67, Theorem 1.2 and Theorem 1.3],
and the arguments within their proofs. We stress that in the case of a Ricci
lower bound, the balls are not required to stay within the cut-locus as first
realized by Gromov, see, e.g., [27, Theorem 1.132]. Finally, the conclusion
(A.3) follows from [63, Corollary 2.22, item (i)] and the coarea formula,
while (A.6) follows verbatim from the proof of [63, Corollary 2.22, item (i)]
by using (A.4) and concluding again with the coarea formula.

We also stress that in the forthcoming A.1 we do not actually assume the
curvature bounds on all $M^{n}$, but just on an open subset $\Omega\subset
M^{n}$. Consequently the conclusions hold for balls contained inside $\Omega$:
indeed, the proofs of the classical geometric comparison theorems leading to
A.1 can be localized, see, e.g., [63, Remark 2.6]. For a comparison result
assuming more general Ricci lower bounds, we also refer the reader to [63,
Theorem 2.14].

######  Theorem A.1 (Volume and perimeter comparison).

Let $(M^{n},g)$ be a complete Riemannian manifold, and let $\Omega\subset
M^{n}$ be a on open subset such that $\mathrm{Ric}\geq(n-1)k$ on $\Omega$ in
the sense of quadratic forms for some $k\in\mathbb{R}$. Let us set
$T_{k}:=+\infty$ if $k\leq 0$, and $T_{k}:=\pi/\sqrt{k}$ if $k>0$. Then, for
every $p\in\Omega$ and for $r\leq T_{k}$ such that $B_{r}(p)\Subset\Omega$ the
following hold

$\displaystyle\frac{\operatorname{vol}(B_{r}(p))}{v(n,k,r)}\to 1\,\text{as
$r\to 0$ and it is nonincreasing},$ (A.1)
$\displaystyle\frac{P(B_{r}(p))}{s(n,k,r)}\to 1\,\text{as $r\to 0$ and it is
almost everywhere nonincreasing},$ (A.2)
$\displaystyle\frac{P(B_{r}(p))}{s(n,k,r)}\leq\frac{\operatorname{vol}(B_{r}(p))}{v(n,k,r)}\,\text{almost
everywhere}.$ (A.3)

Moreover, if one has
$\operatorname{vol}(B_{\overline{r}}(p))=v(n,k,\overline{r})$ for some
$\overline{r}\leq T_{k}$ such that $B_{\overline{r}}(p)\Subset\Omega$, then
$B_{\overline{r}}(p)$ is isometric to the ball of radius $\overline{r}$ in the
simply connected model of constant sectional curvature $k$ and dimension $n$.

Conversely, let $(M^{n},g)$ be a complete Riemannian manifold, and let
$\Omega\subset M^{n}$ be an open subset such that $\mathrm{Sect}\leq k$ on
$\Omega$, for some $k\in\mathbb{R}$. Then, for every $p\in\Omega$ and for
$r\leq\min\\{T_{k},\operatorname{inj}(p)\\}$ such that $B_{r}(p)\Subset\Omega$
the following hold

$\displaystyle\frac{\operatorname{vol}(B_{r}(p))}{v(n,k,r)}\to 1\,\text{as
$r\to 0$ and it is nondecreasing},$ (A.4)
$\displaystyle\frac{P(B_{r}(p))}{s(n,k,r)}\to 1\,\text{as $r\to 0$ and it is
nondecreasing},$ (A.5)
$\displaystyle\frac{P(B_{r}(p))}{s(n,k,r)}\geq\frac{\operatorname{vol}(B_{r}(p))}{v(n,k,r)}.$
(A.6)

Moreover, if one has
$\operatorname{vol}(B_{\overline{r}}(p))=v(n,k,\overline{r})$ for some
$\overline{r}\leq\min\\{T_{k},\operatorname{inj}(p)\\}$ such that
$B_{\overline{r}}(p)\Subset\Omega$, then $B_{\overline{r}}(p)$ is isometric to
the ball of radius $\overline{r}$ in the simply connected model of constant
sectional curvature $k$ and dimension $n$.

In the case of sectional curvature bounds, the above result strengthens and
yields a comparison between metric tensors. Again, this is well known, but
since we were not able to find a complete proof in the literature, we provide
one.

######  Lemma A.2 (Comparison of metrics).

Let $(M^{n},g)$ be a complete Riemannian manifold and fix $p\in M^{n}$. For
every $k\in\mathbb{R}$, let $T_{k}$ be as in A.1. Denote by $r$ the distance
from $p$, let $R=\operatorname{inj}(p)$, and let $\\{x^{i}\\}_{i=1}^{n}$ be
geodesic normal coordinates at the point $p$. Through the latter coordinates,
let us identify $B_{R}(p)$ with the Euclidean ball $\mathbb{B}^{n}_{R}$, and
let us denote by $g_{1}$ the canonical metric on $\mathbb{S}^{n-1}$. Then the
following statements hold true

  * (i)

if $\mathrm{Sect}(\nabla r\wedge X)\leq k$ for any $X\perp\nabla r$ with
$g(X,X)=1$, then the inequality $g\geq
g_{k}:=\mathrm{d}r^{2}+\operatorname{sn}_{k}(r)^{2}g_{1}$ holds in the sense
of quadratic forms on $\mathbb{B}^{n}_{\rho}$, where
$\rho:=\min\\{R,T_{k}\\}$,

  * (ii)

if $\mathrm{Sect}(\nabla r\wedge X)\geq k$ for any $X\perp\nabla r$ with
$g(X,X)=1$, then the inequality $g\leq
g_{k}:=\mathrm{d}r^{2}+\operatorname{sn}_{k}(r)^{2}g_{1}$ holds in the sense
of quadratic forms on $\mathbb{B}^{n}_{\rho}$, where
$\rho:=\min\\{R,T_{k}\\}$.

###### Proof.

We prove the first item, the second one being completely analogous. On
$\mathbb{B}^{n}_{\rho}\cap\\{x_{1}\neq 0\\}$ we consider the polar frame given
by $\partial_{r}:=(x^{i}/r)\partial_{i},v_{2},\ldots,v_{n}$, where
$r=\sqrt{\sum_{j}(x^{j})^{2}}$ and
$v_{j}\vcentcolon=x^{1}\partial_{j}-x^{j}\partial_{1}$ for $j=2,\ldots,n$.

We also consider on $\mathbb{B}^{n}_{\rho}\setminus
S\simeq(0,\rho)\times\mathbb{S}^{n-1}\setminus\Sigma$ polar coordinates
$(r,\theta)$, which are defined out of a negligible set $S$ which is the cone
over some $\Sigma\subset\mathbb{S}^{n-1}$ (that is negligible in
$\mathbb{S}^{n-1}$) in $\mathbb{R}^{n}$. When considering polar coordinates,
we shall write that the following computations hold on $\mathbb{B}^{n}_{\rho}$
meaning that they are well defined on $\mathbb{B}^{n}_{\rho}\setminus S$ but
then the resulting conclusions extend on $S$ by continuity.

So, since $\mathrm{d}r(v_{i})=g(\nabla r,v_{i})=0$ for all $i=2,\dots,n$ (see
the proof of [62, Lemma 5.5.5]) in the latter frame we can rewrite the metric
as $g=\mathrm{d}r^{2}+h_{ij}(r,\theta)w^{i}\otimes w^{j}$, where $\\{w^{i}\\}$
is the dual coframe corresponding to $\\{v_{i}\\}$.

Moreover, since $\mathrm{d}r(v_{i})=g(\nabla r,v_{i})=0$ for any
$i=2,\ldots,n$, we can also rewrite
$g_{k}=\mathrm{d}r^{2}+\operatorname{sn}_{k}(r)^{2}\overline{h}_{ij}(\theta)w^{i}\otimes
w^{j}$. We remark that $\overline{h}_{ij}=g_{1}(v_{i},v_{j})$ does not depend
on $r$ as each $v_{i}$ is a tangent vector along $\mathbb{S}^{n-1}$
independent of $r$: this is readily checked by writing $v_{i}$ in polar
coordinates.

We claim that $h_{ij}(r,\theta)w^{i}\otimes
w^{j}\geq\operatorname{sn}_{k}(r)^{2}\overline{h}_{ij}(\theta)w^{i}\otimes
w^{j}$ in the sense of quadratic forms on
$\mathbb{B}^{n}_{\rho}\cap\\{x_{1}\neq 0\\}$, which implies the lemma true on
$\mathbb{B}^{n}_{\rho}\cap\\{x_{1}\neq 0\\}$. It eventually follows, by
considering each $x^{k}$ in place of $x^{1}$, that the analogous argument
implies that the assertion we want to prove is true on
$\mathbb{B}^{n}_{\rho}\setminus\\{0\\}$, and then on $\mathbb{B}^{n}_{\rho}$
by continuity.

One can easily check that $[\partial_{r},v_{i}]=0$ for any $i=2,\ldots,n$, and
$\partial_{r}=\nabla r$ on $\mathbb{B}^{n}_{\rho}\setminus\\{0\\}$, see the
proof of [62, Lemma 5.5.5]. Thus we have

$\partial_{r}(g(v_{i},v_{j}))=g(\nabla_{\partial_{r}}v_{i},v_{j})+g(\nabla_{\partial_{r}}v_{j},v_{i})=g(\nabla_{v_{i}}\nabla
r,v_{j})+g(\nabla_{v_{j}}\nabla r,v_{i})=2\nabla^{2}r(v_{i},v_{j}),$

for any $i,j=2,\ldots,n$. Hence, as $\mathrm{d}r(v_{i})=0$ for any
$i=2,\ldots,n$, the Hessian comparison theorem [62, Theorem 6.4.3] implies

$\partial_{r}h_{ij}=\partial_{r}(g(v_{i},v_{j}))\geq
2\frac{\operatorname{sn}_{k}^{\prime}(r)}{\operatorname{sn}_{k}(r)}g(v_{i},v_{j})=2\frac{\operatorname{sn}_{k}^{\prime}(r)}{\operatorname{sn}_{k}(r)}h_{ij},$

in the sense of quadratic forms on $\mathbb{B}^{n}_{\rho}\cap\\{x_{1}\neq
0\\}$, so that
$\partial_{r}\left(\frac{h_{ij}}{\operatorname{sn}_{k}(r)^{2}}\right)\geq 0$
in the sense of quadratic forms on $\mathbb{B}^{n}_{\rho}\cap\\{x_{1}\neq
0\\}$.

For every $\theta$ and every $i,j=1,\dots,n$ we have

$\lim_{r\to
0^{+}}\frac{g(v_{i},v_{j})}{\operatorname{sn}_{k}(r)^{2}}=\lim_{r\to
0^{+}}\frac{h_{ij}(r,\theta)}{\operatorname{sn}_{k}(r)^{2}}=\overline{h}_{ij}(\theta).$

Indeed, for $\theta$ fixed,
$g(v_{i},v_{j})=r^{2}\overline{h}_{ij}(\theta)(1+O(r^{2}))$ as $r\to 0^{+}$,
since in normal coordinates $g_{ij}=\delta_{ij}(1+O(r^{2}))$ as $r\to 0^{+}$.
Hence classical ODE comparison implies that
$h_{ij}(r,\theta)=g(v_{i},v_{j})\geq\operatorname{sn}_{k}(r)^{2}\overline{h}_{ij}(\theta)$
in the sense of quadratic forms, thus giving the conclusion. ∎

##  Appendix B Boundedness of isoperimetric regions

In this part we prove that having at disposal a Euclidean-like isoperimetric
inequality for merely _small_ volumes suffices to imply that isoperimetric
regions on a complete Riemannian manifold are bounded. This is a technical
fact that we will employ several times. The proof is based on a rather
classical argument already appearing in [66, Proposition 3.7] and in [53,
Lemma 13.6] in the Euclidean setting, and in [59, Theorem 3] on Riemannian
manifolds. However, we present here a rather self-contained proof for the
convenience of the reader, pointing out that the weak assumption of a
Euclidean-like isoperimetric inequality for small volumes is sufficient for
the assertion.

######  Theorem B.1.

Let $(M^{n},g)$ be a complete Riemannian manifold. Assume that there is
$v_{0}>0$ such that the isoperimetric inequality

$c_{0}\operatorname{vol}(\Omega)^{(n-1)/n}\leq P(\Omega),$

holds true with some $c_{0}>0$ for any finite perimeter set $\Omega\subset
M^{n}$ with $\operatorname{vol}(\Omega)<v_{0}$. Then the isoperimetric regions
of $(M^{n},g)$ are bounded.

###### Proof.

Let $E$ be an isoperimetric region and fix a point $p_{0}\in M^{n}$. Let, for
every $r>0$,

$V(r)\vcentcolon=\operatorname{vol}(E\setminus B_{r}(p_{0})),\qquad
A(r)\vcentcolon=P(E,M\setminus B_{r}(p_{0})).$

By hypothesis there exists $r_{0}>0$ such that for any $r\geq r_{0}$ the
volume $V(r)$ is sufficiently small to apply the isoperimetric inequality. In
particular, for almost every $r\geq r_{0}$ we can write

$|V^{\prime}(r)|+A(r)=\mathcal{H}^{n-1}(\partial B_{r}(p_{0})\cap
E)+P(E,M\setminus B_{r}(p_{0}))=P(E\setminus B_{r}(p_{0}))\geq
c_{0}V(r)^{\frac{n-1}{n}}.$

We want to prove that

$A(r)\leq|V^{\prime}(r)|+CV(r),$ (B.1)

for some constant $C$, and for almost every $r$ sufficiently big. Combining
with the previous inequality, in this way we would get

$c_{0}V(r)^{\frac{n-1}{n}}\leq
CV(r)+2|V^{\prime}(r)|\leq\frac{c_{0}}{2}V(r)^{\frac{n-1}{n}}-2V^{\prime}(r),$

because $|V^{\prime}(r)|=-V^{\prime}(r)$ and
$CV(r)\leq\tfrac{c_{0}}{2}V(r)^{\frac{n-1}{n}}$ for almost every sufficiently
big radius. Hence ODE comparison implies that $V(r)$ vanishes at some
$r=\overline{r}<+\infty$, i.e., $E$ is bounded as a set of finite perimeter.

So we are left to prove (B.1). Let $R>0$ be fixed such that
$P(E,B_{R}(p_{0}))>0$. There exists $\varepsilon_{0}=\varepsilon_{0}(R,E)>0$
and $C=C(R,E)>0$ such that for any
$\varepsilon\in(-\varepsilon_{0},\varepsilon_{0})$ there is a finite perimeter
set $F$ with

$F\Delta E\Subset
B_{R}(p_{0}),\qquad\operatorname{vol}(F)=\operatorname{vol}(E)+\varepsilon,\qquad
P(F,B_{R}(p_{0}))\leq P(E,B_{R}(p_{0}))+C|\varepsilon|.$ (B.2)

Indeed, this follows by the fact that, since the gradient of the
characteristic function $\chi_{E}$ is represented by a measure
$\nu|D\chi_{E}|$ where $\nu:M\to TM^{n}$ with $|\nu|=1$ at $|D\chi_{E}|$-a.e.
point, and $P(E,\Omega)=\sup\left\\{\int\langle
X,\nu\rangle\,\mathrm{d}|D\chi_{E}|\ :\
X\in\mathfrak{X}(\Omega),\operatorname{spt}X\Subset\Omega,|X|\leq 1\right\\}$
for any open set $\Omega$, we can take a field $X$ with $|X|\leq 1$ and
compact support in $B_{R}(p_{0})$ such that $\int\langle
X,\nu\rangle\,\mathrm{d}|D\chi_{E}|\geq\tfrac{1}{2}P(E,B_{R}(p_{0}))>0$. Then,
for small $t$, there is a smooth family of diffeomorphisms $\phi_{t}$ such
that $\phi_{0}={\rm id}$ and $\partial_{t}\phi_{t}|_{0}=X$. So the sets
$F_{t}\vcentcolon=\phi_{t}(E)$ verify the expansions

$\begin{split}\operatorname{vol}(F_{t})&=\operatorname{vol}(E)+t\int\langle
X,\nu\rangle\,\mathrm{d}|D\chi_{E}|+O(t^{2}),\\\
P(F_{t},B_{R}(p_{0}))&=P(E,B_{R}(p_{0}))+t\int(\operatorname{div}X-\langle\nabla_{\nu}X,\nu\rangle)\,\mathrm{d}|D\chi_{E}|+O(t^{2}).\end{split}$

The above formulas for the variations of volume and perimeter are easily
checked to hold on $M^{n}$ by the same computations carried out in the
Euclidean space in [47, Theorem 17.5 & Proposition 17.8]. Since $\int\langle
X,\nu\rangle\,\mathrm{d}|D\chi_{E}|>0$ and
$\left|\int(\operatorname{div}X-\langle\nabla_{\nu}X,\nu\rangle)\,\mathrm{d}|D\chi_{E}|\right|\leq
C(R,E)$, (B.2) follows by taking $\varepsilon_{0}(R,E)$ sufficiently small and
then $F=F_{t_{\varepsilon}}$ for the suitable $t_{\varepsilon}$.

Now consider $r>R$ such that $V(r)<\varepsilon_{0}$, and set
$\varepsilon=V(r)$. Then there is $F$ satisfying (B.2). Define also
$\widetilde{F}=F\cap B_{r}(p_{0})$, so that

$\operatorname{vol}(\widetilde{F})=\operatorname{vol}(F)-\operatorname{vol}(F\setminus
B_{r}(p_{0}))=\operatorname{vol}(F)-\operatorname{vol}(E\setminus
B_{r}(p_{0}))=\operatorname{vol}(E)+\varepsilon-\varepsilon=\operatorname{vol}(E).$

Moreover, for almost every such $r$ we can additionally require that
$P(F,\partial B_{r}(p_{0}))\vcentcolon=\int_{\partial
B_{r}(p_{0})}\,\mathrm{d}|D\chi_{F}|=0$, as $|D\chi_{F}|$ is a finite Radon
measure, see [47, Proposition 2.16]. In this way (see [47, Theorem 16.3]) we
have

$\begin{split}P(\widetilde{F})&=P(F,B_{r}(p_{0}))+\mathcal{H}^{n-1}(\partial
B_{r}(p_{0})\cap F)\\\ &=P(F)-P(F,M\setminus
B_{r}(p_{0}))+\mathcal{H}^{n-1}(\partial B_{r}(p_{0})\cap F).\end{split}$

Since $E$ is an isoperimetric set we estimate

$\begin{split}P(E)&\leq P(\widetilde{F})=P(F)-P(E,M\setminus
B_{r}(p_{0}))+\mathcal{H}^{n-1}(\partial B_{r}(p_{0})\cap E)\\\ &\leq
P(E)+C\varepsilon-A(r)+|V^{\prime}(r)|,\end{split}$

that is $A(r)\leq|V^{\prime}(r)|+CV(r)$. Hence we see that (B.1) holds for
almost every $r>R$ such that $V(r)<\varepsilon_{0}$, and the proof is
completed. ∎

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