# Volume growth and positive scalar curvature

Guodong Wei, Guoyi Xu, Shuai Zhang  Guodong Wei
School of Mathematics (Zhuhai)
Sun Yat-sen University, Zhuhai, Guangdong
P. R. China, 519082<EMAIL_ADDRESS>Guoyi Xu
Department of Mathematical Sciences
Tsinghua University, Beijing
P. R. China, 100084<EMAIL_ADDRESS>Shuai Zhang
Department of Mathematical Sciences
Tsinghua University, Beijing
P. R. China, 100084<EMAIL_ADDRESS>
###### Abstract.

For three dimensional complete, non-compact Riemannian manifolds with non-
negative Ricci curvature and uniformly positive scalar curvature, we obtain
the sharp linear volume growth ratio and the corresponding rigidity.
Mathematics Subject Classification: 53C21, 53C23, 53C24

G. Wei is partially supported by NSFC 12101619 and 12141106. G. Xu was
partially supported by NSFC 12141103.

section[0em] *[1pc]. subsection[1.5em] *[1pc].

###### Contents

  1. 1 Introduction
  2. 2 The $\mu$-bubble surfaces and diameter estimate
  3. 3 The separating $\mu$-bubble surfaces
  4. 4 The height estimate of almost cylindrical region
  5. 5 The volume comparison of hypersurfaces
  6. 6 The sharp linear volume growth
  7. A The existence of smooth functions for $\mu$-bubbles

##  1\. Introduction

From the celebrated Bishop-Gromov Volume Comparison Theorem, it is well-known
that for any complete non-compact Riemannian manifold $(M^{n},g)$ with Ricci
curvature $Rc\geq 0$, the $n$-dim asymptotic volume growth ratio of $M^{n}$,
denoted as

$\displaystyle\mathrm{V}_{M^{n},n}\vcentcolon=\lim_{r\rightarrow\infty}\frac{V(B_{p}(r))}{\omega_{n}r^{n}},$

is well-defined and is independent of the choice of $p$; where $B_{p}(r)$ is
the open geodesic ball centered at $p\in M^{n}$ with radius $r>0$,
$V(B_{p}(r))$ is the volume of $B_{p}(r)$, and $\omega_{n}$ is the volume of
unit ball in $\mathbb{R}^{n}$. Furthermore $\mathrm{V}_{M^{n},n}\in[0,1]$, and
$\mathrm{V}_{M^{n},n}=1$ if and only if the manifold $M^{n}$ is isometric to
$\mathbb{R}^{n}$ by the rigidity part of the Bishop-Gromov’s volume comparison
Theorem.

Furthermore, uniformly positive scalar curvature $R$ possibly controls the
volume growth better. Gromov [Gromov-large-mflds, (5’)] made the following
conjecture:

######  Conjecture 1.1 (Volume conjecture).

There is a universal constant $C(n)>0$, such that if $(M^{n},g)$ has $Rc\geq
0$ and the scalar curvature $R\geq 2$, then

$\displaystyle\sup_{p\in M^{n}\atop r>0}\frac{V(B_{p}(r))}{r^{n-2}}\leq C(n).$

######  Remark 1.2.

Note Gromov [Gromov-large-mflds, 2.B.] also sketched a proof of the following
result: If $(M^{n},g)$ has the sectional curvature $K\geq 0$ and the scalar
curvature $R\geq 1$, then there exists some $C(n)>0$ such that $\displaystyle
V(B_{p}(r))\leq C(n)\cdot r^{n-2}$.

There is an asymptotic version of Gromov’s volume conjecture as follows:

######  Conjecture 1.3 (Volume growth conjecture).

There is a universal constant $C(n)>0$, such that if $(M^{n},g)$ has $Rc\geq
0$ and $R\geq 2$, then

$\displaystyle\varlimsup_{r\rightarrow\infty}\frac{V(B_{p}(r))}{r^{n-2}}\leq
C(n),\quad\quad\quad\forall p\in M^{n}.$

Following Gromov’s strategy and combining Cheeger-Colding’s theory of Ricci
limit spaces from [CC-I], Anderson-Cheeger’s compactness theorem in [AC], and
Gromov’s spherical Lipschitz bound theorem in [Gromov-MI], Zhu [Zhu] succeeded
in proving: If $(M^{n},g)$ has $Rc\geq 0,R\geq 2$ and $\displaystyle\inf_{q\in
M^{n}}V(B_{q}(1))\geq\varepsilon>0$, then
$\displaystyle\varlimsup_{r\rightarrow\infty}\frac{V(B_{p}(r))}{r^{n-2}}<\infty$
for any $p\in M^{n}$.

Using the property of some suitable positive harmonic function on $(M^{3},g)$,
Conjecture 1.3 for $n=3$ is firstly proved by Munteanu-Wang [MW-II] as
follows:

######  Theorem 1.4.

There is a universal constant $C>0$, such that if $(M^{3},g)$ has $Rc\geq 0$
and $R\geq 2$, then
$\displaystyle\varlimsup_{r\rightarrow\infty}\frac{V(B_{p}(r))}{r}\leq C$ for
any $p\in M^{3}$.

Later, Chodosh-Li-Stryker [CLS] gave another proof of the volume growth
conjecture for $n=3$ by using Cheeger-Colding’s almost splitting theorem [CC-
Ann] and $\mu$-bubble’s diameter estimate.

Motivated to get a generalization of the Cohn-Vossen’s inequality in higher
dimension, Yau has posed the following question [Yau, Problem 9]: For any
complete Riemannian manifold $(M^{n},g)$ with $Rc\geq 0$, and for any $p\in
M^{n}$, is it true that

$\displaystyle\lim\limits_{r\rightarrow\infty}\frac{\int_{B_{p}(r)}R}{r^{n-2}}<\infty?$

Note that a positive answer to Yau’s question implies Conjecture 1.3.

Shi and Yau [SY] verified Yau’s conjecture on Kähler manifolds with additional
assumption. If there is $p\in(M^{n},g)$ such that the corresponding
exponential map $\exp_{p}:T_{p}M^{n}\rightarrow M^{n}$ is a diffeomorphism, we
call $(M^{n},g)$ is a Riemannian manifold with a pole. For $3$-dim Riemannian
manifold $M^{3}$ with a pole and $Rc\geq 0$, the third named author [Xu-IOCII]
shows that

$\displaystyle\lim_{r\rightarrow\infty}\frac{\int_{B_{p}(r)}R}{r}=8\pi(1-\mathrm{V}_{M^{3},3}).$

This formula shows that general $3$-dim complete, noncompact manifolds with
$Rc\geq 0$ possibly has some sharp asymptotic relationship between scalar
curvature and volume growth. In this paper, we address this sharp asymptotic
relationship reflected in Conjecture 1.3.

Similar to the concept of asymptotic volume ratio, for complete non-compact
Riemannian manifolds $(M^{n},g)$ with $n\geq 3$, we define the $(n-2)$-dim
asymptotic volume growth ratio of $M^{n}$ as follows:

$\displaystyle\mathrm{V}_{M^{n},n-2}\vcentcolon=\varlimsup\limits_{r\rightarrow\infty}\frac{V(B_{p}(r))}{\omega_{n-2}r^{n-2}},\quad\quad\quad\forall
p\in M^{n}.$

It is easy to see that $\mathrm{V}_{M^{n},n-2}$ is well-defined and is
independent of the choice of $p$.

In this paper, we prove the sharp version of Conjecture 1.3 for $n=3$.

######  Theorem 1.5.

For any complete non-compact Riemannian manifold $(M^{3},g)$ with $Rc\geq 0$
and $R\geq 2$,

  1. (1)

. if $(M^{3},g)$ has only one end, then $\displaystyle
0<\mathrm{V}_{M^{3},1}\leq 2\pi$,

  2. (2)

. if $(M^{3},g)$ has at least two ends, then $(M^{3},g)$ is isometric to a
cylinder $S\times\mathbb{R}$ with $S$ being a closed surface of sectional
curvature at least one, and $\displaystyle 0<\mathrm{V}_{M^{3},1}\leq 4\pi$.

  3. (3)

. moreover $\mathrm{V}_{M^{3},1}=4\pi$ if and only if $(M^{3},g)$ is isometric
to the cylinder $\mathbb{S}^{2}\times\mathbb{R}$, where $\mathbb{S}^{2}$ is
the unit round sphere in $\mathbb{R}^{3}$.

######  Remark 1.6.

Meanwhile, for any $0<b<1$, we construct distinct noncompact manifolds
$(M_{i}^{3},g_{i})$ with $Rc\geq 0$ and $R\geq 2$, where $i=1,2$, such that
$\displaystyle\mathrm{V}_{M_{i}^{3},1}=4\pi b$. See Remark 6.2 and Remark 6.3
for details.

The proof of Theorem 1.5 follows the strategy of [CLS] closely.

From Cheeger-Colding’s almost splitting theorem, the $L$-neighborhood (where
$L>0$) of some part of a ray (which is $[(k-5)\cdot L,(k+5)\cdot L]$ away from
$p$, where $k>>1$), is $\delta$-Gromov-Hausdorff close to, the product of an
interval with a length space (a cylindrical region); where $\delta>0$ goes to
$0$ as $k\rightarrow\infty$. On the other hand, from theory of $\mu$-bubble
surfaces, we can get the area upper bound of the $\mu$-bubble surfaces in the
above almost cylindrical region in term of $L$.

######  Remark 1.7.

There are two key observations in this paper:

  1. (A)

. The area of $\mu$-bubble surfaces is bounded by the area of unit round
sphere $\mathbb{S}^{2}$, as $L\rightarrow\infty$. This bound is sharp in
asymptotic sense, where asymptotic means $L\rightarrow\infty$.

  2. (B)

. The area of the intersection of geodesic spheres with the neighborhood of
the ray, has an upper bound in term of the area bound of corresponding
$\mu$-bubble surfaces.

When computing $\mathrm{V}_{M^{3},1}$, we take $r=k\cdot L$ for fixed $L$, and
express the upper bound of $\mathrm{V}_{M^{3},1}$ as a limit in terms of
$\delta,L$ with respect to $k\rightarrow\infty$(see (6.4)). Finally we let
$L\rightarrow\infty,\delta\rightarrow 0$ to obtain the sharp linear volume
growth ratio.

In the end of this section, we sketch the proof of Theorem 1.5 in more details
as follows.

We firstly introduce the following notation, which is used in the rest of the
paper.

######  Notation 1.8.

For a complete Riemannian manifold $(M^{n},g)$ and $L>0$, if $(M^{n},g)$ has
only one end, the unique unbounded connected component of
$M^{n}-\overline{B_{p}(kL)}$ is denote by $E_{k}(p,L)$ . And when the context
is clear (or $p,L$ are fixed), we use $E_{k}$ to denote $E_{k}(p,L)$ for
simplicity. Furthermore, we define
$N_{k}\vcentcolon=E_{k}-\overline{E_{k+1}}$.

  1. (1)

. Cutting the whole manifold into a sequence of cylindrical regions
$\\{N_{k}\\}_{k\in\mathbb{Z}^{+}}$ with respect to the scale $L>0$, reduce the
asymptotic upper bound of geodesic ball’s volume to the upper bound of the
volume of $N_{k}$ (see Figure 1).

  2. (2)

. To bound the volume of $N_{k}$, using the co-area formula, we are reduced to
get the upper bound of the ‘height’ of $N_{k}$ (which is the oscillation of
the distance between points in $N_{k}$ and $p$), and the upper bound of the
area of the intersection of geodesic sphere with $N_{k}$ (this is done in
Section 6).

Figure 1. Picture for (1) and (2)

  3. (3)

. The asymptotic equivalence between the ‘height’ of $N_{k}$ and the scale
$L$, is showed in Section 4. And this asymptotic equivalence relies on the
uniform diameter bound of suitable subset of geodesic sphere, which is the
intersection of geodesic sphere with $N_{k}$. Note the uniform diameter bound
(denoted as $c$ in most of the rest argument) is independent of the large
scale $L$ (see Figure 2).

    1. (3.a)

. To prove the uniform diameter bound of the intersection of geodesic sphere
with $N_{k}$, we use Cheeger-Colding’s almost splitting theorem to reduce it
to the uniform diameter bound of corresponding, separating $\mu$-bubble
surfaces.

    2. (3. b)

. The uniformly positive scalar curvature assumption and the second variation
formula for $\mu$-bubble surface, together a conformal version of Bonnet-
Myers’ type estimate (obtained in Section 2), are used to get such diameter
bound of the above $\mu$-bubble surfaces in Section 3.

Figure 2. Picture for (3)

  4. (4)

. The upper bound of the area of the intersection of geodesic sphere with
$N_{k}$ is obtained from the following two results:

    1. (4.a)

The volume comparison for suitable part of geodesic spheres and corresponding
separating $\mu$-bubble surfaces, is provided in Section 5. This volume
comparison result is in the similar spirit of Bishop-Gromov’s volume
comparison Theorem (also see [HK]), and holds for general dimensions (see
Figure 3).

Figure 3. Picture for (4.a)

    2. (4.b)

Based on some results about separating $\mu$-bubble surfaces in [CLS] and
[CLS-0], the area bound of separating $\mu$-bubble surfaces is obtained in
Section 3.

######  Remark 1.9.

Another key point is: all estimates (including some ‘height’ estimates of
cylindrical region) related to area or volume obtained in this paper are
asymptotically sharp with respect to $L\rightarrow\infty$.

There are some smooth functions required in the construction of $\mu$-bubbles
and corresponding $\mu$-bubble hypersurfaces, and we put the existence result
of those functions into Appendix A.

For readers’ convenience, we provide a diagram as follows, which is the
structure of the proof of Theorem 1.5 (see Figure 4).

Theorem 6.1 (Theorem 1.5)Proposition 3.3Proposition 3.2Lemma 4.2Lemma 4.5Lemma
4.6Lemma 5.1Lemma 5.2Lemma 5.3Proposition 2.2Lemma 2.3Lemma 2.4Lemma A.1Lemma
A.2Lemma A.3 Figure 4. Structure of the proof of Theorem 1.5

##  2\. The $\mu$-bubble surfaces and diameter estimate

The soap bubble was firstly proposed by Schoen to Gromov [Gromov-Macro,
Section $5\frac{5}{6}$]. We recall its definition as follows.

######  Definition 2.1.

Assume $N$ is a smooth Riemannian manifold with boundary $\partial
N=\partial_{-}N\cup\partial_{+}N$, where
$\partial_{-}N\cap\partial_{+}N=\emptyset$ and $\partial_{\pm}N\neq\emptyset$.
Fix a smooth function $u>0$ on $N$ and a smooth function $h$ on $\mathring{N}$
with $h\rightarrow\pm\infty$ on $\partial_{\pm}N$, where $\mathring{N}$ is the
interior of $N$. Choose a Caccioppoli set $\mathcal{C}\subseteq N$ such that

(2.1) $\displaystyle\partial\mathcal{C}-\partial_{+}N\ \text{is
smooth},\quad\quad\partial\mathcal{C}-\partial_{+}N\subseteq\mathring{N}\quad\quad\text{and}\quad\quad\partial_{+}N\subseteq\partial\mathcal{C}.$

Define $\mathscr{C}(N,\mathcal{C})$ as the set of all Caccioppoli sets
$\Omega\subseteq N$ with $\Omega\triangle\mathcal{C}\subseteq\mathring{N}$.

We define $\mathcal{A}:\mathscr{C}(N,\mathcal{C})\rightarrow\mathbb{R}$ as
follows:

$\displaystyle\mathcal{A}(W)=\int_{\partial^{*}W-\partial
N}ud\mathcal{H}^{n-1}-\int_{N}(\chi_{W}-\chi_{\mathcal{C}})\cdot h\cdot
ud\mathcal{H}^{n},\quad\quad\quad\forall W\in\mathscr{C}(N,\mathcal{C}),$

where $\partial^{*}W$ is the reduced boundary of $W$.

If
$\displaystyle\mathcal{A}(\Omega)=\inf_{W\in\mathscr{C}(N,\mathcal{C})}\mathcal{A}(W)$
for some $\Omega\in\mathscr{C}(N,\mathcal{C})$, we call $\Omega$ as a
$\mu$-bubble with respect to $(h,u,\mathcal{C})$ in
$(N,\partial_{-}N,\partial_{+}N)$ (See Figure 5).

For any $\mu$-bubble $\Omega$ with respect to $(h,u,\mathcal{C})$ in
$(N,\partial_{-}N,\partial_{+}N)$, we call any connected component of
$\partial\Omega-\partial N$ as a $\mu$-bubble hypersurface with respect to
$(h,u,\mathcal{C})$ in $(N,\partial_{-}N,\partial_{+}N)$. When
$\partial\Omega-\partial N$ is two dimensional, we also call its connected
components as $\mu$-bubble surfaces.

Figure 5. The $\mu$-bubble and $\mu$-bubble surface

The following existence result of $\mu$-bubble is [CL, Proposition $12$] (also
see [Zhu-bubble, Proposition $2.1$])

######  Proposition 2.2.

There is a $\mu$-bubble $\Omega$ with respect to $(h,u,\mathcal{C})$ in
$(N,\partial_{-}N,\partial_{+}N)$, and any $\mu$-bubble hypersurface
corresponding to $\Omega$ is a smooth hypersurface.

∎

We use the second variation formula for $\mu$-bubble hypersurface $\Sigma$
repeatedly (see [CL, Lemma $13$ and $14$]), the statement is as follows.

######  Lemma 2.3.

If $\Sigma$ is a $\mu$-bubble hypersurface with respect to $(h,u,\mathcal{C})$
in $(N,\partial_{-}N,\partial_{+}N)$, where $N$ is $n$-dim Riemannian manifold
with $n\geq 3$; then for any $\psi\in C^{\infty}(\Sigma)$, we have

$\displaystyle\int_{\Sigma}|\nabla_{\Sigma}\psi|^{2}u-\frac{1}{2}(R_{N}-R_{\Sigma}+|\mathring{A}_{\Sigma}|^{2})\cdot\psi^{2}u+(\Delta_{N}u-\Delta_{\Sigma}u)\psi^{2}$
(2.2)
$\displaystyle-\frac{1}{2}u^{-1}\langle\nabla_{N}u,\vec{n}\rangle^{2}\psi^{2}-\frac{1}{2}\\{h^{2}+2\langle\nabla_{N}h,\vec{n}\rangle\\}\psi^{2}u\geq
0;$

where $\mathring{A}_{\Sigma}$ is the traceless part of second fundamental form
of $\Sigma\subseteq N$, $\vec{n}$ is the unit normal vector of
$\Sigma\subseteq N$, and $R_{N},R_{\Sigma}$ are the intrinsic scalar curvature
of $N$ and $\Sigma$ respectively.

∎

For later height estimates of cylindrical region, we need the diameter bound
of $\mu$-bubble surfaces. Similar diameter bound is given in [CL, Lemma $16$].
Our method follows closely [SY-CMP] (also see [SY-PSC]).

######  Lemma 2.4.

For compact $(\Sigma^{2},g)$, if there is a smooth function $u>0$ such that

$\displaystyle-\Delta u\geq(K-K_{\Sigma})u,$

for some constant $K>0$, then
$\mathrm{Diam}(\Sigma)\leq\frac{2\pi}{\sqrt{3K}}$.

######  Remark 2.5.

The sharp upper bound of $\mathrm{Diam}(\Sigma)$ in the above lemma is
unknown. Comparing the well-known Bonnet-Myers Theorem, we can view Lemma 2.4
as a type of conformal Bonnet-Myers Theorem for surfaces.

Proof: Step (1). Assume $\mathrm{Diam}(\Sigma)=d_{g}(p,q)$, where
$p,q\in\Sigma$. Define $I[\gamma]$ by

$\displaystyle I[\gamma]=\int_{\gamma}uds,$

where $ds$ is arclength along $\gamma$ and $\gamma$ is any curves from $p$ to
$q$. Let $\gamma_{0}$ be one minimizer of $I[\cdot]$.

As in the proof of [SY-CMP, Proposition $1$] (also see [LZ, Proposition
$2.2$]), let $V(s)=f(s)\cdot v(s)$ be the variation vector field along
$\gamma(s)$, where $v(s)$ is the unit normal vector field along $\gamma$. The
first variation formula yields

(2.3) $\displaystyle u(\gamma_{0}(s))\nabla_{\partial s}\partial s-(\nabla
u)^{\perp}=0,\quad\text{along}\ \ \gamma_{0}.$

The non-negativity of the second variation of $I$ at $\gamma_{0}$ gives

(2.4) $\displaystyle
0\leq\delta^{2}I[\gamma_{0}]=\int_{\gamma_{0}}[D^{2}u(v,v)-2u\cdot H^{2}]\cdot
f^{2}+u\cdot(|f^{\prime}|^{2}-K_{\Sigma}\cdot f^{2})ds,$

where $H$ is the mean curvature of $\gamma\subseteq\Sigma$.

Note $\displaystyle\Delta_{\Sigma}u=D^{2}u(v,v)+D^{2}u(\partial s,\partial
s)$, combining (2.3), we get

$\displaystyle\Delta_{\Sigma}u$ $\displaystyle=D^{2}u(v,v)+D^{2}u(\partial
s,\partial s)=D^{2}u(v,v)+u^{\prime\prime}-\nabla_{\nabla_{\partial s}\partial
s}u$ (2.5) $\displaystyle=D^{2}u(v,v)+u^{\prime\prime}-H^{2}\cdot u,$

where $u^{\prime}\vcentcolon=\frac{d}{ds}u(\gamma_{0}(s))$.

Plugging (2.5) into (2.4), using the equation of $u$ in the assumption, we
obtain

$\displaystyle 0$
$\displaystyle\leq\int_{\gamma_{0}}[\Delta_{\Sigma}u-u^{\prime\prime}(s)-u\cdot
H^{2}]\cdot f^{2}+u\cdot(|f^{\prime}|^{2}-K_{\Sigma}\cdot f^{2})ds$
$\displaystyle\leq\int_{\gamma_{0}}-u^{\prime\prime}(s)\cdot
f^{2}+u\cdot(|f^{\prime}|^{2}-Kf^{2})ds.$

Assume $l=\mathrm{length}(\gamma_{0})$, then integration by parts yields

(2.6)
$\displaystyle\int_{0}^{l}u(f^{\prime})^{2}+2fu^{\prime}f^{\prime}-Kuf^{2}ds\geq
0,\quad\quad\forall f\in C_{0}^{\infty}[0,l].$

Step (2). Put $f(s)=u^{-\frac{1}{2}}(\gamma_{0}(s))\sin(\frac{\pi}{l}s)$ into
(2.6), we get

$\displaystyle\quad\int_{0}^{l}-\frac{3}{4}[(\ln
u)^{\prime}]^{2}\sin^{2}(\frac{\pi}{l}s)+\frac{\pi}{l}(\ln
u)^{\prime}\sin(\frac{\pi}{l}s)\cdot\cos(\frac{\pi}{l}s)ds$ (2.7)
$\displaystyle\geq\int_{0}^{l}\left(K\sin^{2}(\frac{\pi}{l}s)-(\frac{\pi}{l})^{2}\cos^{2}(\frac{\pi}{l}s)\right)ds.$

By Cauchy-Schwarz inequality, we have

(2.8) $\displaystyle-\frac{3}{4}[(\ln
u)^{\prime}]^{2}\sin^{2}(\frac{\pi}{l}s)+\frac{\pi}{l}(\ln
u)^{\prime}\sin(\frac{\pi}{l}s)\cdot\cos(\frac{\pi}{l}s)\leq\frac{1}{3}\cdot(\frac{\pi}{l})^{2}\cos^{2}(\frac{\pi}{l}s).$

Plugging (2.8) into (2.7), we obtain

$\displaystyle\frac{4}{3}\cdot(\frac{\pi}{l})^{2}\int_{0}^{l}\cos^{2}(\frac{\pi}{l}s)ds\geq
K\int_{0}^{l}\sin^{2}(\frac{\pi}{l}s)ds.$

This implies $l\leq\frac{2\pi}{\sqrt{3K}}$. ∎

##  3\. The separating $\mu$-bubble surfaces

######  Definition 3.1.

If a smooth compact hypersurface $\Sigma_{k}\subseteq
E_{k}-\overline{E_{k+1}}$ satisfies the following property: “for any
continuous curve $\gamma:[0,1]\rightarrow\overline{E_{k}}-E_{k+1}$ with
$\gamma(0)\in\partial E_{k},\gamma(1)\in\partial E_{k+1}$, we have
$\gamma\cap\Sigma_{k}\neq\emptyset$”, then we say that $\Sigma_{k}$ separates
$\partial E_{k}$ from $\partial E_{k+1}$.

In the applications of $\mu$-bubble surface, we require that it satisfies the
above separating property. The following proposition is important to obtain
such separating property.

######  Proposition 3.2.

If $(M^{n},g)$ is a complete Riemannian manifold with only one end and is
simply connected, then there is $k_{0}\in\mathbb{Z}^{+}$, such that
$\overline{E_{k}}-E_{k+1}$ and $\partial E_{k}$ are both connected for $k\geq
k_{0}$.

Proof: See [CLS-0, Proposition $3.2$]. ∎

Now we present the main result of this section, which provides us the
separating $\mu$-bubble surfaces with upper bound of area and diameter in
cylindrical region $E_{k}-\overline{E_{k+1}}$ when $k$ is big enough. One
crucial thing is that the upper bound of the area is asymptotically sharp with
respect to the large scale $L\rightarrow\infty$.

######  Proposition 3.3.

Assume $(M^{3},g)$ is a complete non-compact Riemannian manifold with $R\geq
2$ and only one end, also it is simply connected and $p\in M^{3}$. For any
$\varepsilon>0$ and $L>\sqrt{2}\pi(1+\varepsilon)$, there is
$k_{0}=k_{0}(\varepsilon,L)\in\mathbb{Z}^{+}$; such that for any $k\geq
k_{0}$, one can find a smooth compact surface $\Sigma_{k}\subseteq
E_{k}-\overline{E_{k+1}}$, which satisfies

  1. (1)

. $\Sigma_{k}$ separates $\partial E_{k}$ from $\partial E_{k+1}$;

  2. (2)

. $\displaystyle
V(\Sigma_{k})\leq\frac{4\pi}{1-\frac{2\pi^{2}(1+\varepsilon)^{2}}{L^{2}}}$.

  3. (3)

.
$\mathrm{Diam}(\Sigma_{k})\leq\frac{2\pi}{\sqrt{3(1-\frac{2\pi^{2}(1+\varepsilon)^{2}}{L^{2}})}}$.

######  Remark 3.4.

The separating property of suitable $\mu$-bubble surface $\Sigma_{k}$ (the
property (1) above) was firstly observed in the argument of [CLS-0, Lemma
$5.4$], which is used for our later argument of Lemma 5.3.

Proof: Step (1). It is obvious that $E_{k+1}\subseteq E_{k}$. Recall
$N_{k}=E_{k}-\overline{E_{k+1}}$, and $\partial_{+}N_{k}=\partial
E_{k},\partial_{-}N_{k}=\partial E_{k+1}$. Then $\partial
N_{k}=\partial_{-}N_{k}\sqcup\partial_{+}N_{k}$. From Lemma A.2, we can choose
a smooth function $f\in C^{\infty}(M^{3})$ with

$\displaystyle f\big{|}_{\partial E_{k}}\leq kL,\quad f\big{|}_{\partial
E_{k+1}}>(k+\frac{1}{2})L,\quad\text{and}\ (k+\frac{1}{2})L\ \text{is a
regular value of}\ f.$

Define $\mathcal{C}_{k}=\\{x\in M^{3}:f(x)<(k+\frac{1}{2})L\\}\cap N_{k}$.
Then $\mathcal{C}_{k}$ is a Caccioppoli set satisfying (2.1).

From Lemma A.3 there is a function
$\rho_{k}(x):\overline{N_{k}}\to[-L/2,L/2]$, which is continuous on
$\overline{N_{k}}$ and smooth on $N_{k}$. Moreover
$\rho_{k}(N_{k})\subset(-L/2,L/2)$, and

$\displaystyle\rho_{k}|_{\partial E_{k}}=-L/2,\quad\rho_{k}|_{\partial
E_{k+1}}=L/2,\quad|\nabla_{M}\rho_{k}|<(1+\varepsilon)^{2}.$

Now choose
$h_{k}(x)=-\frac{2\pi(1+\varepsilon)}{L}\tan(\frac{\pi}{L}\rho_{k}(x))$ on
$N_{k}$, from Proposition 2.2, there exists one $\mu$-bubble $\Omega_{k}$ with
respect to $(h_{k},1,\mathcal{C}_{k})$ in
$(N_{k},\partial_{-}N_{k},\partial_{+}N_{k})$.

The choice of $h_{k}$ implies

(3.1)
$\displaystyle\frac{2\pi^{2}(1+\varepsilon)^{2}}{L^{2}}+\frac{1}{2}h_{k}^{2}-|\nabla_{M}h_{k}|\geq
0.$

Step (2). From Proposition 3.2 and the assumption, there is
$k_{0}\in\mathbb{Z}^{+}$, such that the sets
$\overline{E_{k}}-E_{k+1},\partial E_{k}$ are connected for any $k\geq k_{0}$.
In the rest argument, we assume $k\geq k_{0}$.

Then there is a simple curve $\gamma:[0,1]\to\overline{E_{k}}-E_{k+1}$, with
$\gamma(0)=q_{1}\in\partial E_{k}$, $\gamma(1)=q_{2}\in\partial E_{k+1}$ and
$\gamma((0,1))\cap(\partial E_{k}\cup\partial E_{k+1})=\varnothing$.
Furthermore, form the density of transverse intersections, we may assume
$\gamma$ intersects $\Omega_{k}$ and $\partial\Omega_{k}$ transversely.

Let $I_{2}(\gamma,\partial\Omega_{k})$ be the mod 2 intersection number of
$\gamma$ and $\partial\Omega_{k}$. From the transversality,
$\gamma\cap\Omega_{k}$ is an 1-dimensional manifold, which turns out to be a
union of closed intervals. So
$\displaystyle\\#\partial\Big{(}\gamma\cap\Omega_{k}\Big{)}=0\mod 2$. And then

$\displaystyle
I_{2}(\gamma,\partial\Omega_{k})=\\#\gamma\cap\big{(}\partial\Omega_{k}\big{)}=\\#\partial\Big{(}\gamma\cap\Omega_{k}\Big{)}=0\mod
2.$

From the definition of $\Omega_{k}$, we know $\partial
E_{k}\subset\Omega_{k}$. Let $\displaystyle\partial\Omega_{k}-\partial
E_{k}=\bigcup_{i}\Sigma_{k}^{i}$ be the disjoint union of connected
components. Since $\gamma\cap(\partial E_{k})=\\{q_{1}\\}$, we get

$\displaystyle\sum_{i}I_{2}(\gamma,\Sigma_{k}^{i})=\\#\gamma\cap\big{(}\partial\Omega_{k}\big{)}-\\#\gamma\cap(\partial
E_{k})=1\mod 2.$

So there is $i_{0}$ such that

$\displaystyle I_{2}(\gamma,\Sigma_{k}^{i_{0}})=1.$

From the definition of $\Omega_{k}$, we know $\Omega_{k}\cap\partial
E_{k+1}=\varnothing$. Thus $\Sigma_{k}^{i_{0}}\cap\partial
E_{k+1}=\varnothing$. And we also have $\Sigma_{k}^{i_{0}}\cap\partial
E_{k}=\varnothing$ from the definition.

Step (3). We use $\Sigma_{k}$ to denote $\Sigma_{k}^{i_{0}}$, then
$\Sigma_{k}$ satisfies the conclusion (1). If not, then there is a simple
curve $\check{\gamma}:[0,1]\to\overline{E_{k}}-E_{k+1}$, such that
$\check{\gamma}(0)=q_{3}\in\partial E_{k+1}$,
$\check{\gamma}(1)=q_{4}\in\partial E_{k}$ and
$\check{\gamma}((0,1))\cap(\partial E_{k}\cup\partial E_{k+1})=\varnothing$.
Moreover

$\displaystyle\check{\gamma}([0,1])\cap\Sigma_{k}=\varnothing.$

Since $\partial E_{k}$ and $\partial E_{k+1}$ are connected, there are two
simple curves $\gamma_{1}:[0,1]\to\partial E_{k}$ and
$\gamma_{2}:[0,1]\to\partial E_{k+1}$, such that $\gamma_{1}(0)=q_{4}$,
$\gamma_{1}(1)=q_{1}$, $\gamma_{2}(0)=q_{2}$, $\gamma_{2}(1)=q_{3}$. Let
$\tilde{\gamma}=\gamma*\gamma_{2}*\check{\gamma}*\gamma_{1}$ (see Figure 6),
then

(3.2) $\displaystyle
I_{2}(\tilde{\gamma},\Sigma_{k})=I_{2}(\gamma,\Sigma_{k})+I_{2}(\gamma_{2},\Sigma_{k})+I_{2}(\check{\gamma},\Sigma_{k})+I_{2}(\gamma_{1},\Sigma_{k})=1\mod
2.$

On the other hand, from the assumption that $M$ is simply connected,
$\tilde{\gamma}$ is homotopic equivalent to a constant curve at $q_{1}$. The
mod 2 intersection number is invariant under the homotopic equivalence. So

(3.3) $\displaystyle
I_{2}(\tilde{\gamma},\Sigma_{k})=I_{2}(q_{1},\Sigma_{k})=0\mod 2,$

which is a contradiction.

Figure 6. the closed curve $\tilde{\gamma}$

Step (4). Now from Lemma 2.3, for any $\psi\in C^{\infty}(\Sigma_{k})$, we
have

$\displaystyle\int_{\Sigma_{k}}|\nabla_{\Sigma_{k}}\psi|^{2}-\frac{1}{2}(R_{M}-R_{\Sigma_{k}}+|\mathring{A}_{\Sigma_{k}}|^{2})\cdot\psi^{2}-\frac{1}{2}\\{h_{k}^{2}+2\langle\nabla_{M}h_{k},\vec{n}\rangle\\}\psi^{2}\geq
0.$

Combining the assumption $R_{M}\geq 2$, we obtain

(3.4)
$\displaystyle\int_{\Sigma_{k}}|\nabla_{\Sigma_{k}}\psi|^{2}+(\frac{1}{2}R_{\Sigma_{k}}-1-\frac{1}{2}|\mathring{A}_{\Sigma_{k}}|^{2})\cdot\psi^{2}-\frac{1}{2}\\{h_{k}^{2}+2\langle\nabla_{M}h_{k},\vec{n}\rangle\\}\psi^{2}\geq
0.$

By (3.4) and (3.1), there is

(3.5)
$\displaystyle\int_{\Sigma_{k}}|\nabla_{\Sigma_{k}}\psi|^{2}+(\frac{1}{2}R_{\Sigma_{k}}-1+\frac{2\pi^{2}(1+\varepsilon)^{2}}{L^{2}})\cdot\psi^{2}$
$\displaystyle\geq\int_{\Sigma_{k}}\\{\frac{2\pi^{2}(1+\varepsilon)^{2}}{L^{2}}+\frac{1}{2}h_{k}^{2}+\langle\nabla_{M}h_{k},\vec{n}\rangle\\}\psi^{2}\geq
0.$

Let $\psi=1$ in the above inequality, we get

$\displaystyle\int_{\Sigma_{k}}(1-\frac{2\pi^{2}(1+\varepsilon)^{2}}{L^{2}})\leq\frac{1}{2}\int_{\Sigma_{k}}R_{\Sigma_{k}}.$

Note $\Sigma_{k}$ is a connected surface without boundary, by Gauss-Bonnet
Theorem,

$\displaystyle\frac{1}{2}\int_{\Sigma_{k}}R_{\Sigma_{k}}=2\pi\cdot\chi(\Sigma_{k})\leq
4\pi.$

This implies $\displaystyle
V(\Sigma_{k})\leq\frac{4\pi}{1-\frac{2\pi^{2}(1+\varepsilon)^{2}}{L^{2}}}$.

Step (5). Define
$\displaystyle\mathcal{L}_{k}(f)\vcentcolon=-\Delta_{\Sigma_{k}}f+(\frac{1}{2}R_{\Sigma_{k}}-1+\frac{2\pi^{2}(1+\varepsilon)^{2}}{L^{2}})\cdot
f$, which is a self-dual operator. From (3.5), we know that $\mathcal{L}_{k}$
is a non-negative operator.

Hence from the spectrum theory of self-dual operators, there is a smooth
function $\psi_{k}>0$ defined on $\Sigma_{k}$ such that

$\displaystyle-\Delta_{\Sigma_{k}}\psi_{k}+(\frac{1}{2}R_{\Sigma_{k}}-1+\frac{2\pi^{2}(1+\varepsilon)^{2}}{L^{2}})\cdot\psi_{k}\geq
0.$

From Lemma 2.4, we get
$\displaystyle\mathrm{Diam}(\Sigma_{k})\leq\frac{2\pi}{\sqrt{3(1-\frac{2\pi^{2}(1+\varepsilon)^{2}}{L^{2}})}}$.
∎

##  4\. The height estimate of almost cylindrical region

The following lemma is closely related to [CLS, Proposition $A.4$]; and our
assumption is a little bit different from there, we include its proof here for
completeness.

We firstly recall the definition of length space (in some literature, the
length space defined below is also called complete length space).

######  Definition 4.1.

Let $(Y,d)$ be a metric space. For a path $\gamma:[a,b]\to Y$, let

$\displaystyle\mathrm{length}(\gamma)=\sup\Big{\\{}\sum_{i=0}^{n-1}d(\gamma(t_{i}),\gamma(t_{i+1})):a=t_{0}<t_{1}<\cdots<t_{n}=b\Big{\\}}.$

For $x,y\in Y$, define $\displaystyle
d^{\prime}(x,y)=\inf\\{\mathrm{length}(\gamma):\gamma\ \text{ is a path from}\
x\ \text{to}\ y\\}$.

If for any $x,y\in Y$, there is a path $\gamma_{0}$ from $x$ to $y$, such that
$\mathrm{length}(\gamma_{0})=d^{\prime}(x,y)=d(x,y)<\infty$, then we call
$(Y,d)$ a length space.

######  Lemma 4.2.

Suppose $(Y,d)$ is a length space. If $\displaystyle
0<r<\min\\{\frac{1}{2}\mathrm{Diam}(Y,d),L\\}$, then
$B_{(y_{0},0)}(L)-B_{(y_{0},0)}(r)$ is path connected, where
$B_{(y_{0},0)}(r)$ is the open ball in $(Y\times\mathbb{R};d\times
d_{\mathbb{R}})$ centered at $(y_{0},0)\in Y\times\mathbb{R}$.

######  Remark 4.3.

If $(Y,d)$ is merely a path connected metric space , the conclusion in Lemma
4.2 does not hold.111The counterexample here is due to Jianqiao Shang in
Université Paris-Saclay.

Let $Y=\mathbb{R}^{2}-(0,1)\times(-1,1)$ with the standard Euclidean metric,
$y_{0}=(0,0)$ and $L=1.01$, then $B_{y_{0}}(L)$ is disconnected. Hence
$B_{(y_{0},0)}(L)-B_{(y_{0},0)}(r)$ is disconnected (see Figure 7).

Figure 7. Counterexample when $(Y,d)$ is a path connected metric space

###### Proof.

We first show

$\displaystyle
X_{+}\vcentcolon=(B_{(y_{0},0)}(L)-B_{(y_{0},0)}(r))\cap(Y\times\overline{\mathbb{R}^{+}})$

is path connected. For any $(y_{1},t_{1}),(y_{2},t_{2})\in X_{+}$, let
$\eta(s)=(y(s),t(s))$ be a continuous curve from $(y_{1},t_{1})$ to
$(y_{2},t_{2})$ in $B_{(y_{0},0)}(L)$, where we use the fact that $Y$ is a
length space. Replacing $t(s)$ by $\max\\{t(s),0\\}$, we may assume $t(s)\geq
0$.

Let

$\displaystyle
T(s)=\left\\{\begin{array}[]{ll}\sqrt{r^{2}-d(y(s),y_{0})^{2}},&(y(s),t(s))\in
B_{(y_{0},0)}(r)\\\ t(s),&\text{otherwise}\end{array}\right..$

Then $T(s)\geq 0$ and $r^{2}\leq d(y(s),y_{0})^{2}+T(s)^{2}<L^{2}$. So
$(y(s),T(s))$ is a continuous curve in $X_{+}$ from $(y_{1},t_{1})$ to
$(y_{2},t_{2})$. The same argument shows

$\displaystyle
X_{-}=(B_{(y_{0},0)}(L)-B_{(y_{0},0)}(r))\cap(Y\times{\bf\mathbb{R}}_{\leq
0})$

is also path connected.

For any $(y_{1},t_{1})\in X_{+}$, $(y_{2},t_{2})\in X_{-}$, since
$r<\frac{1}{2}\mathrm{Diam}(Y,d)$, there is a point $y^{\prime}\in Y$ such
that $r<d(y^{\prime},y_{0})<L$. Since $X_{+}$, $X_{-}$ are path connected,
there is a path $\eta_{1}$ from $(y_{1},t_{1})$ to $(y^{\prime},0)$, and a
path $\eta_{2}$ from $(y_{2},t_{2})$ to $(y^{\prime},0)$. Then
$\eta_{1}*\eta_{2}$ is a path from $(y_{1},t_{1})$ to $(y_{2},t_{2})$. Hence
$B_{(y_{0},0)}(L)-B_{(y_{0},0)}(r)=X_{+}\cup X_{-}$ is path connected. ∎

######  Definition 4.4.

Assume $(X,d)$ is a metric space, for any set $W\subseteq X$ and $\tau>0$, we
define the $\tau$-neighborhood of $W$ as follows:

$\displaystyle U_{\tau}(W)\vcentcolon=\\{x\in X:d(x,W)<\tau\\}.$

######  Lemma 4.5.

Assume $(M^{3},g)$ has only one end with $Rc\geq 0$ and $R\geq 2$. We also
assume it is simply connected. For any $p\in M^{3}$ and a unit speed geodesic
ray $\gamma$ starting from $p$, any
$\varepsilon\in(0,1),\delta\in(0,\frac{1}{100})$ and $L>8\pi$, there is
$k_{0}=k_{0}(\delta,\varepsilon,L)\in\mathbb{Z}^{+}$ such that

$\displaystyle\sup_{x,y\in\big{(}\partial B_{p}(r)\cap B_{\gamma(k\cdot
L)}(5L)\big{)}}d_{g}(x,y)\leq\frac{4\pi}{\sqrt{3(1-\frac{2\pi^{2}(1+\varepsilon)^{2}}{L^{2}})}}+9\delta\cdot
L,\quad\quad\quad\forall k\geq k_{0},r>0.$

Proof: Step (1). In the rest of the argument, we assume $\displaystyle
c=\frac{2\pi}{\sqrt{3(1-\frac{2\pi^{2}(1+\varepsilon)^{2}}{L^{2}})}}$.

By Cheeger-Colding’s almost splitting theorem [CC-Ann, Theorem $6.62$], for
any $\delta\in(0,\frac{1}{100})$, there is
$k_{1}=k_{1}(\delta)\in\mathbb{Z}^{+}$ such that for all $L>0$, the following
holds:

$\displaystyle
d_{GH}\Big{(}B_{\gamma(kL)}(5L)\subseteq(M^{3},g),B_{(y_{0},0)}(5L)\subseteq(Y\times\mathbb{R},d\times
d_{\mathbb{R}})\Big{)}\leq\delta\cdot L,\quad\quad\quad\forall k\geq k_{1},$

where $(Y,d)$ is a length space.

Furthermore, the map $f:B_{\gamma(kL)}(5L)\rightarrow B_{(y_{0},0)}(5L)$
defined as follows is a $(\delta\cdot L)$-Gromov-Hausdorff approximation:

$\displaystyle f(x)=(\varphi(x),d(p,x)-kL),$

where $\varphi:B_{\gamma(kL)}(5L)\rightarrow Y$ is a suitable map satisfying
$\varphi(\gamma(t))=y_{0}$ for any $t\in(kL-5L,kL+5L)$.

For $\varepsilon\in(0,1),L>\sqrt{2}\pi(1+\varepsilon)$, from Proposition 3.3;
there is $k_{2}=k_{2}(\varepsilon,L)\in\mathbb{Z}^{+}$, and for each $k\geq
k_{2}$, we can choose $\Sigma_{k}\in N_{k}$ as in Proposition 3.3.

We define
$k_{0}\vcentcolon=k_{0}(\delta,\varepsilon,L)=k_{1}(\delta)+k_{2}(\varepsilon,L)$,
in the rest argument we always assume $k\geq k_{0}$.

Now we define

$\displaystyle t_{k}\vcentcolon=\min\\{t>0:\gamma(t)\in\Sigma_{k}\\}.$

From the separating property of $\Sigma_{k}$, we know that

(4.1) $\displaystyle(E_{k-1}-\overline{E_{k+2}})-\Sigma_{k}=A_{1}\bigsqcup
A_{2},$

where $A_{i}$ is one of the two connected components of
$(E_{k-1}-\overline{E_{k+2}})-\Sigma_{k}$ and $i=1,2$ (see Figure 8).

Figure 8. Picture for (4.1)

Step (2). Next we show

(4.2) $\displaystyle\mathrm{Diam}(Y,d)\leq 2(c+4\delta\cdot L).$

By contradiction, assume (4.2) does not hold. Note $\delta<\frac{1}{100}$ and
$L>8\pi$, then we further have $\displaystyle c+4\delta\cdot
L<\min\\{\frac{1}{2}\mathrm{Diam}(Y,d),L\\}$. By Lemma 4.2, we obtain that
$B_{z_{1}}(L)-B_{z_{1}}(c+4\delta L)$ is connected, where
$z_{1}\vcentcolon=(y_{0},t_{k}-kL)\in Y\times\mathbb{R}$.

From Proposition 3.3, we know that $\mathrm{Diam}(\Sigma_{k})\leq c$. Define
$\displaystyle S_{k}\vcentcolon=U_{\delta L}\Big{(}f\big{[}U_{2\delta
L}(\Sigma_{k})\big{]}\Big{)}$, then we have

(4.3) $\displaystyle S_{k}\subseteq B_{z_{1}}(c+4\delta L).$

Now we define
$\displaystyle\Lambda_{i}\vcentcolon=B_{z_{1}}(L)\cap\Big{[}U_{\delta\cdot
L}(f(A_{i}))-S_{k}\Big{]}$ for $i=1,2$. Note

$\displaystyle B_{z_{1}}(L)$ $\displaystyle\subseteq U_{\delta\cdot
L}\Big{(}f(E_{k-1}-\overline{E}_{k+2})\Big{)}\subseteq U_{\delta\cdot
L}\Big{(}f(A_{1})\Big{)}\bigcup U_{\delta\cdot L}\Big{(}f(A_{2})\Big{)}\bigcup
U_{\delta\cdot L}\Big{(}f(\Sigma_{k})\Big{)}.$

Combining the above, we obtain (see Figure 9)

(4.4) $\displaystyle B_{z_{1}}(L)=\Lambda_{1}\cup S_{k}\cup\Lambda_{2}.$

Figure 9. Picture for (4.4)

From (4.4) and (4.3), we have

(4.5) $\displaystyle\Big{[}B_{z_{1}}(L)-B_{z_{1}}(c+4\delta
L)\Big{]}\subseteq\Lambda_{1}\cup\Lambda_{2}.$

Now for any $q_{i}\in\Lambda_{i}-S_{k}$, there are $\tilde{q}_{i}\in
A_{i}-U_{2\delta L}(\Sigma_{k})$ such that
$d(q_{i},f(\tilde{q}_{i}))<\delta\cdot L$. Therefore

$\displaystyle d(q_{1},q_{2})$ $\displaystyle\geq
d(f(\tilde{q}_{1}),f(\tilde{q}_{2}))-2\delta L\geq
d(\tilde{q}_{1},\tilde{q}_{2})-3\delta L$ (4.6) $\displaystyle\geq
d(\tilde{q}_{1},\Sigma_{k})+d(\tilde{q}_{2},\Sigma_{k})-3\delta L\geq\delta
L>0.$

By (4.3) and (4.6), we get that

(4.7) $\displaystyle
d\Big{(}\Lambda_{1}\cap\Big{[}B_{z_{1}}(L)-B_{z_{1}}(c+4\delta
L)\Big{]},\Lambda_{2}\cap\Big{[}B_{z_{1}}(L)-B_{z_{1}}(c+4\delta
L)\Big{]}\Big{)}\geq\delta L.$

Now from (4.5), we get that

$\displaystyle\Big{\\{}\Lambda_{1}\cap\Big{[}B_{z_{1}}(L)-B_{z_{1}}(c+4\delta
L)\Big{]}\Big{\\}}\cup\Big{\\{}\Lambda_{2}\cap\Big{[}B_{z_{1}}(L)-B_{z_{1}}(c+4\delta
L)\Big{]}\Big{\\}}$ $\displaystyle=\Big{[}B_{z_{1}}(L)-B_{z_{1}}(c+4\delta
L)\Big{]}.$

Combining (4.7), we get that $\Big{[}B_{z_{1}}(L)-B_{z_{1}}(c+4\delta
L)\Big{]}$ is not connected, which leads to a contradiction. We complete the
proof of (4.2).

By the definition of the $(\delta\cdot L)$-Gromov-Hausdorff approximation map
$f$ and (4.2), for any $x,y\in\big{(}\partial B_{p}(r)\cap B_{\gamma(k\cdot
L)}(5L)\big{)}$, we get

$\displaystyle d_{g}(x,y)$ $\displaystyle\leq d(f(x),f(y))+\delta\cdot
L=d((\varphi(x),r-kL),(\varphi(y),r-kL))+\delta\cdot L$
$\displaystyle=d_{Y}(\varphi(x),\varphi(y))+\delta\cdot L\leq 2c+9\delta\cdot
L.$

∎

The main result of this section is Lemma 4.6, which is an improvement of [CLS,
Claim $3$, page $5$]. In fact, this improvement is asymptotically sharp, which
will be used in the proof of Theorem 6.1.

######  Lemma 4.6.

Assume $(M^{3},g)$ has only one end and $Rc\geq 0,R\geq 2$, furthermore assume
it is simply connected. For any
$\varepsilon\in(0,1),\delta\in(0,\frac{1}{100}),L>8\pi$, there is
$k_{0}=k_{0}(\varepsilon,\delta,L)\in\mathbb{Z}^{+}$; such that (see Figure
10)

(4.8) $\displaystyle N_{k}\subseteq\Big{[}B_{p}((k+1)L+2c+10\delta\cdot
L)-\overline{B_{p}(kL)}\Big{]},\quad\quad\quad\forall k\geq k_{0},$

where $\displaystyle
c=\frac{2\pi}{\sqrt{3(1-\frac{2\pi^{2}(1+\varepsilon)^{2}}{L^{2}})}}$.

Figure 10. Picture for (4.8)

Proof: Step (1). Choose $k_{0}$ from Lemma 4.5.

By contradiction. Otherwise, there is $x^{\prime}\in
N_{k}-B_{p}((k+1)L+2c+10\delta\cdot L)$, where $L>8\pi$ and $k\geq k_{0}$ is
some integer. Let $\tilde{\gamma}$ be the geodesic segment
$\overline{px^{\prime}}$, and define
$x^{\prime\prime}=\tilde{\gamma}\cap\partial B_{p}((k+1)L+2c+10\delta\cdot
L)$.

Choose a ray $\gamma$ with $\gamma(0)=p$ and
$\gamma\big{|}_{(kL,\infty)}\subseteq E_{k}$. By the choice of $L$, we have
$c\leq\frac{1}{4}L$.

By Proposition 3.3, we can find one $\mu$-bubble surface
$\Sigma_{k-1}\subseteq N_{k-1}$ with $\mathrm{Diam}(\Sigma_{k-1})\leq c$,
which separates $\partial E_{k-1}$ from $\partial E_{k}$. So we can take
$x_{k-1}\in(\tilde{\gamma}\cap\Sigma_{k-1}),y_{k-1}\in(\gamma\cap\Sigma_{k-1})$.
Then we have (see Figure 11.)

$\displaystyle d(x^{\prime\prime},\gamma(kL))$ $\displaystyle\leq
d(x^{\prime\prime},x_{k-1})+d(x_{k-1},y_{k-1})+d(y_{k-1},\gamma(kL))\leq(2L+2c+10\delta
L)+c+L$ (4.9) $\displaystyle\leq 5L.$

Therefore $x^{\prime\prime}\in B_{\gamma(kL)}(5L)$.

Figure 11. Picture for (4.9)

Step (2). Note $x^{\prime\prime},\gamma((k+1+10\delta)L+2c)\in\partial
B_{p}((k+1+10\delta)L+2c)\cap B_{\gamma(kL)}(5L)$, from Lemma 4.5, we have

(4.10) $\displaystyle d(x^{\prime\prime},\gamma((k+1+10\delta)L+2c))\leq
2c+9\delta\cdot L.$

On the other hand, because $x^{\prime\prime}\in\overline{N_{k}}$, the set
$\partial B_{p}((k+1)L)$ separates $x^{\prime\prime}$ from $\gamma$. From
(4.10), we have

$\displaystyle 2c+9\delta\cdot L\geq d(x^{\prime\prime},\gamma)\geq
d(x^{\prime\prime},\partial B_{p}((k+1)L))=2c+10\delta\cdot L.$

This is the contradiction. ∎

##  5\. The volume comparison of hypersurfaces

For any $q\in M^{n}$, we can write $q=(r,\theta)$ in terms of polar normal
coordinates at $p$, where $r=d(p,q)$. And it is well known that we can write
the volume element of $(M^{n},g)$ as

$\displaystyle d\mu_{g}=J(t,\theta)dtd\theta,$

where $d\theta$ is the area element of the unit $(n-1)$-sphere
$\mathbb{S}^{n-1}$. The area element of geodesic sphere $\partial B_{p}(t)$ is
given by $J(t,\theta)d\theta$.

######  Lemma 5.1.

Let $X,Y$ be two positive semi-definite $(n\times n)$-matrices, then

$\displaystyle\det(X+Y)\geq\det(X)+\det(Y).$

###### Proof.

The inequality is trivial when $\det(X)+\det(Y)=0$. Without loss of
generality, we assume $\det(X)>0$ in the following proof.

$\displaystyle\det(X+Y)=$
$\displaystyle\det(\sqrt{X}(I+\sqrt{X}^{-1}Y\sqrt{X}^{-1})\sqrt{X})=\det(X)\det(I+\sqrt{X}^{-1}Y\sqrt{X}^{-1}).$

Let $\lambda_{i}\geq 0$ be the eigenvalues of $\sqrt{X}^{-1}Y\sqrt{X}^{-1}$,
then

$\displaystyle\det(I+\sqrt{X}^{-1}Y\sqrt{X}^{-1})=$
$\displaystyle\prod_{i=1}^{n}(1+\lambda_{i})\geq 1+\prod_{i=1}^{n}\lambda_{i}$
$\displaystyle=$ $\displaystyle
1+\det(\sqrt{X}^{-1}Y\sqrt{X}^{-1})=1+\det(Y)\det(X)^{-1}.$

We complete the proof. ∎

For $p\in M$, we define $\mathrm{Cut}(p)$ as the set of all cut points of $p$.

######  Lemma 5.2.

If $(M^{n},g)$ is a complete Riemannian manifold and
$r:\Omega\rightarrow\mathbb{R}^{+}$ is a smooth function, where
$\Omega\subseteq\mathbb{S}_{p}M$ is an open set. Let
$\varphi(\theta)=\exp_{p}(r(\theta)\theta)$; then

$\displaystyle J_{\varphi}(\theta)\geq
J(r(\theta),\theta),\quad\quad\quad\forall\theta\in\Omega;$

where $J_{\varphi}(\theta)$ is the Jacobian determinant of $\varphi$.

###### Proof:.

Assume the local normal coordinates of $\mathbb{S}_{p}M$ are
$\theta^{1},\theta^{2},\cdots,\theta^{n-1}$, then

$\displaystyle
J_{\varphi}(\theta)=\sqrt{\det\Bigg{[}\Bigg{(}\Big{<}\frac{\partial\varphi(\theta)}{\partial\theta^{i}},\frac{\partial\varphi(\theta)}{\partial\theta^{j}}\Big{>}_{q}\Bigg{)}_{1\leq
i,j\leq n-1}\Bigg{]}}.$

where $\left<\cdot,\cdot\right>_{q}$ is the inner product with respect to the
metric on $T_{q}M$ and $q=\varphi(\theta)$.

Note

$\displaystyle\frac{\partial\varphi(\theta)}{\partial\theta^{i}}=$
$\displaystyle\mathrm{d}\exp_{p}\big{(}r(\theta)\theta\big{)}\left(\frac{\partial\big{(}r(\theta)\theta\big{)}}{\partial\theta^{i}}\right)=\mathrm{d}\exp_{p}\big{(}r(\theta)\theta\big{)}\bigg{(}\frac{\partial
r(\theta)}{\partial\theta^{i}}\theta+r(\theta)\frac{\partial\theta}{\partial\theta^{i}}\bigg{)}$
$\displaystyle=$ $\displaystyle\frac{\partial
r(\theta)}{\partial\theta^{i}}\mathrm{d}\exp_{p}\big{(}r(\theta)\theta\big{)}(\theta)+r(\theta)\mathrm{d}\exp_{p}\big{(}r(\theta)\theta\big{)}\bigg{(}\frac{\partial\theta}{\partial\theta^{i}}\bigg{)}.$

For simplicity, let

$\displaystyle r_{i}=\frac{\partial r(\theta)}{\partial\theta^{i}},\quad\quad
A=\mathrm{d}\exp_{p}\big{(}r(\theta)\theta\big{)}(\theta),\quad\quad
B_{i}=r(\theta)\mathrm{d}\exp_{p}\big{(}r(\theta)\theta\big{)}\bigg{(}\frac{\partial\theta}{\partial\theta^{i}}\bigg{)}.$

We have
$\displaystyle\frac{\partial\varphi(\theta)}{\partial\theta^{i}}=r_{i}A+B_{i}$.

From the definition, we know $\left<\theta,\theta\right>_{p}=1$. So

$\displaystyle\left<\theta,\frac{\partial\theta}{\partial\theta^{i}}\right>_{p}=0,\quad
i=1,2,\cdots,n-1.$

From Gauss lemma,

$\displaystyle\left<A,B_{i}\right>_{q}=$ $\displaystyle
r(\theta)\left<\mathrm{d}\exp_{p}\big{(}r(\theta)\theta\big{)}(\theta),\mathrm{d}\exp_{p}(r(\theta)\theta)\left(\frac{\partial\theta}{\partial\theta^{i}}\right)\right>_{q}=r(\theta)\left<\theta,\frac{\partial\theta}{\partial\theta^{i}}\right>_{p}=0.$

From the above we get

$\displaystyle
J_{\varphi}(\theta)^{2}=\det\Big{[}\Big{(}r_{i}r_{j}|A|_{q}^{2}+\left<B_{i},B_{j}\right>_{q}\Big{)}_{1\leq
i,j\leq n-1}\Big{]}.$

For fixed $t>0$, define
$\displaystyle\psi_{t}(\theta)=\exp_{p}(t\theta):\mathbb{S}_{p}(M)\rightarrow
M^{n}$, then

$\displaystyle\left.\frac{\partial\psi_{t}(\theta)}{\partial\theta^{i}}\right|_{t=r(\theta)}$
$\displaystyle=\left.\mathrm{d}\exp_{p}(t\theta)\left(t\frac{\partial\theta}{\partial\theta^{i}}\right)\right|_{t=r(\theta)}=r(\theta)\mathrm{d}\exp_{p}\big{(}r(\theta)\theta\big{)}\bigg{(}\frac{\partial\theta}{\partial\theta^{i}}\bigg{)}=B_{i}.$

And the Jacobian of $J_{\psi_{t}}$ is as follows:

$\displaystyle J_{\psi_{t}}(\theta)|_{t=r(\theta)}=$
$\displaystyle\sqrt{\left.\det\Bigg{[}\Bigg{(}\Big{<}\frac{\partial\psi_{t}(\theta)}{\partial\theta^{i}},\frac{\partial\psi_{t}(\theta)}{\partial\theta^{j}}\Big{>}_{q}\Bigg{)}_{1\leq
i,j\leq n-1}\Bigg{]}\right|_{t=r(\theta)}}$ $\displaystyle=$
$\displaystyle\sqrt{\det\Big{[}\Big{(}\left<B_{i},B_{j}\right>_{q}\Big{)}_{1\leq
i,j\leq n-1}\Big{]}}.$

Since $\theta^{1},\theta^{2},\cdots,\theta^{n-1}$ are normal coordinates,

$\displaystyle\left<\frac{\partial\theta}{\partial\theta^{i}},\frac{\partial\theta}{\partial\theta^{j}}\right>=$
$\displaystyle\delta_{ij},\quad 1\leq i,j\leq n-1.$

Combining $\left<\theta,\theta\right>_{p}=1$ and
$\left<\theta,\frac{\partial\theta}{\partial\theta^{i}}\right>_{p}=0$, we know
that
$\theta,\frac{\partial\theta}{\partial\theta^{1}},\frac{\partial\theta}{\partial\theta^{2}},\cdots,\frac{\partial\theta}{\partial\theta^{n-1}}$
form an orthonormal basis of $T_{p}M$. So $J(t,\theta)=J_{\psi_{t}}(\theta)$.
From Lemma 5.1, we have

$\displaystyle J_{\varphi}(\theta)^{2}-J(r(\theta),\theta)^{2}$
$\displaystyle=$
$\displaystyle\det\Big{[}\Big{(}r_{i}r_{j}|A|_{q}^{2}+\left<B_{i},B_{j}\right>_{q}\Big{)}_{1\leq
i,j\leq
n-1}\Big{]}-\det\Big{[}\Big{(}\left<B_{i},B_{j}\right>_{q}\Big{)}_{1\leq
i,j\leq n-1}\Big{]}\geq 0.$

The conclusion follows. ∎

######  Lemma 5.3.

If $(M^{n},g)$ has $Rc\geq 0$ and only one end, suppose $\Sigma\subset
N_{k-1}$ is a smooth, compact hypersurface separating $\partial E_{k-1}$ from
$\partial E_{k}$, then

$\displaystyle V\Big{(}(\partial B_{p}(s)\cap
N_{k})-\mathrm{Cut}(p)\Big{)}\leq(\frac{s}{(k-1)L})^{n-1}\cdot
V(\Sigma),\quad\quad\forall s>kL.$

###### Proof:.

Let $\tilde{\Sigma}\vcentcolon=\Sigma-\mathrm{Cut}(p)$, define
$\Pi:\tilde{\Sigma}\to\mathbb{S}_{p}(M)$ as
$\displaystyle\Pi(q)=\frac{\exp_{p}^{-1}(q)}{|\exp_{p}^{-1}(q)|_{g(p)}}$,
where $\mathbb{S}_{p}M$ is the unit tangent space of $M$ at $p$. From the
definition of cut points, the map $\Pi$ is a well-defined smooth map.

From the separating property of $\Sigma$, we know that

$\displaystyle\Theta_{k}\vcentcolon=\frac{1}{s}\exp_{p}^{-1}\big{(}(\partial
B_{p}(s)\cap N_{k})-\mathrm{Cut}(p)\big{)}\subseteq\Pi(\tilde{\Sigma}).$

Applying Sard’s theorem on the map $\Pi:\tilde{\Sigma}\to\mathbb{S}_{p}(M)$,
we obtain that the set of critical values of $\Pi$ (denoted as
$\mathcal{C}(\Pi)$) satisfies $\mathcal{H}^{n-1}(\mathcal{C}(\Pi))=0$, where
$\mathcal{H}^{n-1}$ is $(n-1)$-dim Hausdorff measure on $\mathbb{S}_{p}M$.

Define

(5.1) $\displaystyle\Theta_{k}^{\prime}=\Theta_{k}-\mathcal{C}(\Pi).$

Let $\Sigma^{\prime}\subset(\tilde{\Sigma}\cap\Pi^{-1}(\Theta_{k}))$ be the
regular points of $\Pi$ with image in $\Theta_{k}$. Then $\Sigma^{\prime}$ is
an open subset of ${\bf\mathbb{S}}_{p}M$ from the implicit function theorem,
and

$\displaystyle\Theta_{k}^{\prime}\subset\Pi(\Sigma^{\prime}).$

For any $q_{0}\in\Sigma^{\prime}$, there is a local coordinate chart $(U,h)$
of $\Sigma^{\prime}$ containing $q_{0}$, such that $\Pi|_{U}$ is a
diffeomorphism between $U$ and $\Pi(U)$. And

$\displaystyle\varphi(\theta)\vcentcolon=\exp_{p}(r(\theta)\cdot\theta)=(\Pi\big{|}_{U})^{-1}(\theta),$

is a smooth function defined on $\Pi(U)\subseteq\mathbb{S}_{p}M$.

For any $\theta\in\Theta_{k}^{\prime}$, note $(k-1)L\leq r(\theta)\leq kL<s$.
From Lemma 5.2 and Bishop-Gromov’s volume comparison Theorem, we get

$\displaystyle J_{(\Pi\big{|}_{U})^{-1}}(\theta)$
$\displaystyle=J_{\varphi}(\theta)\geq
J(r(\theta),\theta)\geq\bigg{(}\frac{r(\theta)}{s}\bigg{)}^{n-1}J(s,\theta)\geq\bigg{(}\frac{(k-1)L}{s}\bigg{)}^{n-1}J(s,\theta).$

Finally by the area formula, (5.1) and
$\mathcal{H}^{n-1}(\mathcal{C}(\Pi))=0$, we get

$\displaystyle V(\Sigma)\geq$ $\displaystyle V(\Sigma^{\prime})\geq
V(\Pi^{-1}(\Theta_{k}^{\prime}))\geq\bigg{(}\frac{(k-1)L}{s}\bigg{)}^{n-1}\int_{\Theta_{k}^{\prime}}J(s,\theta)\mathrm{d}\mathcal{H}^{{n-1}}(\theta)$
$\displaystyle=\bigg{(}\frac{(k-1)L}{s}\bigg{)}^{n-1}\int_{\Theta_{k}}J(s,\theta)\mathrm{d}\mathcal{H}^{{n-1}}(\theta)$
$\displaystyle=\bigg{(}\frac{(k-1)L}{s}\bigg{)}^{n-1}V\Big{(}(\partial
B_{p}(s)\cap N_{k})-\mathrm{Cut}(p)\Big{)}.$

∎

##  6\. The sharp linear volume growth

######  Theorem 6.1.

For any complete non-compact Riemannian manifold $(M^{3},g)$ with $Rc\geq 0$
and $R\geq 2$,

  1. (1)

. if $(M^{3},g)$ has only one end, then $\displaystyle
0<\mathrm{V}_{M^{3},1}\leq 2\pi$,

  2. (2)

. if $(M^{3},g)$ has at least two ends, then $(M^{3},g)$ is isometric to a
cylinder $S\times\mathbb{R}$ with $S$ being a closed surface of sectional
curvature at least one, and $\displaystyle 0<\mathrm{V}_{M^{3},1}\leq 4\pi$.

  3. (3)

. moreover $\mathrm{V}_{M^{3},1}=4\pi$ if and only if $(M^{3},g)$ is isometric
to a cylinder $\mathbb{S}^{2}\times\mathbb{R}$.

Proof: Step (1). Note any complete manifold with nonnegative Ricci curvature
has at least linear volume growth (see [Yau-linear]). We know that
$\displaystyle\mathrm{V}_{M^{3},1}>0$ generally.

If $(M^{3},g)$ has more than one ends, then there is at least one geodesic
line in $M^{3}$. From Cheeger-Gromoll’s splitting Theorem, we know that
$M^{3}$ is isometric to $N^{2}\times\mathbb{R}$. Using $R\big{|}_{M}\geq 2$,
we get that $R\big{|}_{N}\geq 2$. From Bonnet-Myers’ Theorem and the Bishop-
Gromov’s volume comparison Theorem, we get that $N^{2}$ is compact and
$V(N^{2})\leq V(\mathbb{S}^{2})=4\pi$. The direct computation yields

(6.1)
$\displaystyle\mathrm{V}_{M^{3},1}=\varlimsup\limits_{r\rightarrow\infty}\frac{V(B_{p}(r))}{2r}\leq\varlimsup\limits_{r\rightarrow\infty}\frac{V(N^{2})\times(2r)}{2r}\leq
4\pi.$

The rigidity part of (6.1) follows from the rigidity part of the Bishop-
Gromov’s volume comparison Theorem for $N^{2}$.

Step (2). In the rest argument, we assume that $(M^{3},g)$ has only one end.
We firstly consider the case $\pi_{1}(M^{3})=0$.

For any $\varepsilon\in(0,1),\delta\in(0,\frac{1}{100})$, choose $L>8\pi$.
Choose $k_{0}=k_{0}(\varepsilon,\delta,L)\in\mathbb{Z}^{+}$, such that for
$k\geq k_{0}$, the conclusions in Proposition 3.3 and Lemma 4.6 both hold.

In the rest argument, unless otherwise mentioned, all $k\geq k_{0}$.

Now from Proposition 3.3, there is a compact smooth surface
$\Sigma_{k}\subseteq N_{k}$ separating $\partial E_{k}$ from $\partial
E_{k+1}$, and

(6.2) $\displaystyle
V(\Sigma_{k})\leq\frac{4\pi}{1-\frac{2\pi^{2}(1+\varepsilon)^{2}}{L^{2}}}.$

From Lemma 5.3 and (6.2), for any $r\in(kL,(k+1+10\delta)\cdot L+2c)$, we have

(6.3) $\displaystyle V\Big{(}(\partial B_{p}(r)\cap
N_{k})-\mathrm{Cut}(p)\Big{)}\leq\left(\frac{r}{(k-1)L}\right)^{2}\cdot
V(\Sigma_{k-1})\leq\left(\frac{r}{(k-1)L}\right)^{2}\cdot\frac{4\pi}{1-\frac{2\pi^{2}(1+\varepsilon)^{2}}{L^{2}}},$

where $\displaystyle
c=\frac{2\pi}{\sqrt{3(1-\frac{2\pi^{2}(1+\varepsilon)^{2}}{L^{2}})}}$.

Note $V(\mathrm{Cut}(p))=0$, where $V$ is the measure with respect to metric
$g$. Now using (6.3) and Lemma 4.6, we get

$\displaystyle V(N_{k})$
$\displaystyle=V(N_{k}-\mathrm{Cut}(p))\leq\int_{kL}^{(k+1+10\delta)\cdot
L+2c}V\Big{(}(\partial B_{p}(r)\cap N_{k})-\mathrm{Cut}(p)\Big{)}dr$
$\displaystyle\leq\Big{(}\frac{k^{2}+(1+10\delta+\frac{2c}{L})k+3^{-1}(1+10\delta+\frac{2c}{L})^{2}}{(k-1)^{2}}\Big{)}\cdot\frac{4\pi\cdot[(1+10\delta)L+2c]}{1-\frac{2\pi^{2}(1+\varepsilon)^{2}}{L^{2}}}.$

Therefore for $r\in(kL,(k+1)L]$, using $c\leq\frac{1}{4}L$, there is

$\displaystyle V(B_{p}(r))$ $\displaystyle\leq
V(M-E_{k_{0}})+\sum_{j=k_{0}}^{k}V(N_{j})$ $\displaystyle\leq
C+\frac{4\pi\cdot[(1+10\delta)L+2c]}{1-\frac{2\pi^{2}(1+\varepsilon)^{2}}{L^{2}}}\cdot\sum_{j=k_{0}}^{k}\Big{(}\frac{j^{2}+(1+10\delta+\frac{2c}{L})j+3^{-1}(1+10\delta+4^{-1})^{2}}{(j-1)^{2}}\Big{)}$
$\displaystyle\leq
C+\frac{4\pi\cdot[(1+10\delta)L+2c]}{1-\frac{2\pi^{2}(1+\varepsilon)^{2}}{L^{2}}}\cdot\sum_{j=k_{0}}^{k}\Big{(}\frac{j^{2}+(1+10\delta+\frac{2c}{L})j}{(j-1)^{2}}\Big{)},$

where $C>0$ is some universal constant independent of $k$, and $C$ may depend
on $L$ and change from line to line.

Direct computation yields

(6.4)
$\displaystyle\varlimsup_{r\rightarrow\infty}\frac{V(B(r))}{r}\leq\varlimsup_{k\rightarrow\infty}\frac{C+\frac{4\pi\cdot[(1+10\delta)L+2c]}{1-\frac{2\pi^{2}(1+\varepsilon)^{2}}{L^{2}}}\cdot\sum_{j=k_{0}}^{k}\Big{(}\frac{j^{2}+(1+10\delta+\frac{2c}{L})j}{(j-1)^{2}}\Big{)}}{kL}\leq\frac{4\pi\cdot(1+10\delta+\frac{2c}{L})}{1-\frac{2\pi^{2}(1+\varepsilon)^{2}}{L^{2}}}.$

Note $\displaystyle\lim_{L\rightarrow\infty}\frac{c}{L}=0$, let
$L\rightarrow\infty,\delta\rightarrow 0$ in (6.4), we get
$\displaystyle\mathrm{V}_{M^{3},1}\leq 2\pi$.

Step (3). If the fundamental group $\pi_{1}(M^{3})\neq\\{1\\}$, assume
$(\tilde{M}^{3},\tilde{g})$ is the universal cover of $(M^{3},g)$. Because the
number of elements in $\pi_{1}(M^{3})$ satisfies $\\#\pi_{1}(M^{3})\geq 2$, we
get
$\displaystyle\mathrm{V}_{M^{3},1}\leq\frac{1}{2}\mathrm{V}_{\tilde{M}^{3},1}$.

Note $(\tilde{M}^{3},\tilde{g})$ is simply connected with $Rc\geq 0,R\geq 2$.
From Step (1) and Step (2), we get that
$\displaystyle\mathrm{V}_{\tilde{M}^{3},1}\leq 4\pi$. Therefore we obtain that
$\displaystyle\mathrm{V}_{M^{3},1}\leq 2\pi$ in this case. ∎

In the end of this article, we construct examples to show that only
$\displaystyle\mathrm{V}_{M^{3},1}=4\pi$ case yields the rigidity.

######  Remark 6.2.

When $(M^{3},g)$ has more than one ends, the model space is as in Figure 12.

Let $(M^{3},g)={\bf\mathbb{S}}^{2}(r_{0})\times\mathbb{R}$, where $0<r_{0}<1$;
then we have $V_{M^{3},1}=4\pi r_{0}^{2}\in(0,4\pi)$. On the other hand, let
$E(a)$ be the ellipsoid

$\displaystyle
E(a)=\\{(x,y,z)\in{\bf\mathbb{R}}^{3}:\frac{x^{2}}{a^{2}}+y^{2}+z^{2}=r_{0}^{2}\\},$

where $1<a\leq r_{0}^{-1}$. Let $\displaystyle
b=\frac{V(E(a))}{V({\bf\mathbb{S}}^{2}(r_{0}))}>1$ and
$N(a)=\frac{1}{\sqrt{b}}E(a)$. Then $Rc(N(a))\geq 0$,
$R(N(a))\geq\frac{2b}{a^{2}r_{0}^{2}}\geq 2$, and $V(N(a))=4\pi r_{0}^{2}$. We
also have $V_{M^{3},1}=4\pi r_{0}^{2}$ where $(M^{3},g)=N(a)\times\mathbb{R}$.

Hence there is no rigidity for the case: $V_{M^{3},1}\in(0,4\pi)$ and
$(M^{3},g)$ has more than one ends.

Figure 12. The model space with two ends

######  Remark 6.3.

When $(M^{3},g)$ has only one end, the model space is as in Figure 13.

We construct examples as follows, which show that there is no any rigidity
result for any $\mathrm{V}_{M^{3},1}\in(0,2\pi]$.

Let $(M,g)=({\bf\mathbb{R}}^{3},dr^{2}+f(r)^{2}d\mathbb{S}^{2})$, and $\theta$
is coordinate on $\mathbb{S}^{2}$. Simple computations show that

$\displaystyle Rc(\partial r,\partial
r)=-2\frac{f^{\prime\prime}}{f},\quad\quad\quad
Rc(\partial\theta,\partial\theta)=\frac{1-(f^{\prime})^{2}}{f^{2}}-\frac{f^{\prime\prime}}{f},$
$\displaystyle
R=2\frac{1-(f^{\prime})^{2}}{f^{2}}-4\frac{f^{\prime\prime}}{f}.$

We want to construct function
$f:[0,\infty)\rightarrow\overline{\mathbb{R}^{+}}$ satisfies

(6.5) $Rc\geq 0,\quad\quad\quad R\geq 2,$

and

(6.6) $\displaystyle f^{(\text{even})}(0)=0,\quad\quad\quad f^{\prime}(0)=1.$

Direct computation yields

$\displaystyle\mathrm{V}_{M^{3},1}=\varlimsup_{r\rightarrow\infty}\frac{V(B_{p}(r))}{2r}=\varlimsup_{r\rightarrow\infty}\frac{1}{2r}\int_{0}^{r}f^{2}(s)\cdot
4\pi ds=2\pi\varlimsup_{r\rightarrow\infty}\frac{\int_{0}^{r}f^{2}(s)ds}{r}.$

Now, fix $0<b\leq 1$.

  1. Example $(1)$

. Let $f(r)=f_{1}(r)\vcentcolon=\frac{2b}{\pi}\arctan(\frac{\pi}{2b}r)$. Note
$f_{1}$ is a smooth odd function on ${\bf\mathbb{R}}$, so
$f_{1}^{(\text{even})}(0)=0$. Furthermore,

$f_{1}^{\prime}(r)=\frac{1}{1+(\frac{\pi r}{2b})^{2}},\quad
f_{1}^{\prime\prime}(r)=-\frac{\pi^{2}r}{2b^{2}(1+(\frac{\pi
r}{2b})^{2})^{2}}\leq 0\quad\text{when }r\geq 0.$

Direct computation yields

$1-f_{1}^{\prime}(r)^{2}-f_{1}(r)^{2}\geq 0.$

Hence, $f_{1}$ satisfies (6.5) and (6.6).

  2. Example $(2)$

. Let

$f(r)=f_{2}(r)\vcentcolon=b\tanh(\frac{r}{b}).$

Then

$\displaystyle
f_{2}^{\prime}(r)=\frac{4}{\left(e^{-r/b}+e^{r/b}\right)^{2}},\quad
f_{2}^{\prime\prime}(r)=\frac{8\left(e^{-r/b}-e^{r/b}\right)}{b\left(e^{-r/b}+e^{r/b}\right)^{3}}\leq
0\quad\text{when }r\geq 0.$

Furthermore,

$\displaystyle
1-f_{2}^{\prime}(r)^{2}-f_{2}(r)^{2}=\tanh^{2}(\frac{r}{b})\left(1-b^{2}+\frac{4}{\left(e^{-r/b}+e^{r/b}\right)^{2}}\right)\geq
0.$

Hence $f_{2}$ satisfies (6.5) and (6.6).

Now $f_{1}$ and $f_{2}$ are two different smooth functions satisfying (6.5),
and (6.6). Moreover,

$\displaystyle\varlimsup_{r\rightarrow\infty}\frac{\int_{0}^{r}f_{i}^{2}(s)ds}{r}=b,\quad
i=1,2.$

Therefore, for noncompact Riemannian manifold $(M^{3},g)$ with one end,
$\mathrm{V}_{M^{3},1}(M^{3},g)=2\pi b,\ (0<b\leq 1)$ does not imply rigidity.

Figure 13. The model space with one end

## Acknowledgments

We are indebted to Otis Chodosh for insightful comments. We are grateful to
Jianqiao Shang for carefully reading the earlier version of the paper and
pointing out some gaps. We particularly thank Jiaping Wang for his suggestion
on the paper.

##  Appendix A The existence of smooth functions for $\mu$-bubbles

######  Lemma A.1.

For any closed subset $K\subsetneq(M^{3},g)$, there is $f\in
C^{\infty}(M^{3})$ such that

$\displaystyle f|_{K}=0,\quad f|_{K^{c}}>0.$

###### Proof.

Let $\\{B_{p_{i}}(r_{i})\\}_{i=1}^{\infty}$ be a countable open cover of
$K^{c}$ consisting of open balls, such that $\rho_{i}(x):=d(p_{i},x)^{2}$ is
smooth on each $B_{p_{i}}(r_{i})$. Let

$\displaystyle
f_{i}(x)=e^{-\frac{1}{r_{i}^{2}-\rho_{i}(x)}}1_{B_{p_{i}}(r_{i})}$

be a smooth function which is positive on $B_{p_{i}}(r_{i})$ and vanishes
outside $B_{p_{i}}(r_{i})$. For each $i$, define

$\displaystyle M_{i}=1+\sup\left\\{\left|D^{\alpha}f_{i}(x)\right|:x\in
M,\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})\in\mathbb{N}^{3},|\alpha|=\alpha_{1}+\alpha_{2}+\alpha_{3}\leq
i\right\\}.$

We have $M_{i}<\infty$, since all the derivatives of $f_{i}$ vanishes outside
a compact set.

Let

$\displaystyle f=\sum_{i=1}^{\infty}\frac{f_{i}}{2^{i}M_{i}}.$

Then for each multi-index $\alpha$,
$|D^{\alpha}\frac{f_{i}}{2^{i}M_{i}}|\leq\frac{1}{2^{i}}$ when
$i\geq|\alpha|$. So the series
$\sum_{i=1}^{\infty}D^{\alpha}\frac{f_{i}}{2^{i}M_{i}}$ converges uniformly,
which implies the smoothness of $f$. From the definition of $f$, we also have

$\displaystyle f|_{K}=0,\quad f|_{K^{c}}>0.$

∎

######  Lemma A.2.

For any $p\in(M^{3},g)$ and $k\in\mathbb{Z}^{+},L>0$, there is $f\in
C^{\infty}(M^{3})$ with

$\displaystyle f\big{|}_{\partial E_{k}}\leq kL,\quad f\big{|}_{\partial
E_{k+1}}>(k+\frac{1}{2})L,\quad\text{and}\ (k+\frac{1}{2})L\ \text{is a
regular value of}\ f.$

###### Proof.

From Lemma A.1 there exists a smooth function $g$ on $M$ such that
$g|_{\partial E_{k}}=0$ and $g|_{(\partial E_{k})^{c}}>0$. Because $\partial
E_{k+1}\subset\partial B_{p}((k+1)L)$ is compact, let $\displaystyle
c:=\inf_{x\in\partial E_{k+1}}g(x)>0$. From Sard’s theorem, $g$ has a regular
value $c^{\prime}\in(0,c)$. Then $f(x)=\frac{(k+1/2)L}{c^{\prime}}g(x)$ is the
desired smooth function. ∎

######  Lemma A.3.

For any $\varepsilon>0$, there is a continuous function
$\rho_{k}(x):\overline{N_{k}}\to[-L/2,L/2]$, where
$N_{k}=E_{k}-\overline{E_{k+1}}$, such that $\rho_{k}$ is smooth on $N_{k}$,
$\rho_{k}(N_{k})\subset(-L/2,L/2)$ and

$\displaystyle\rho_{k}|_{\partial E_{k}}=-L/2,\quad\rho_{k}|_{\partial
E_{k+1}}=L/2,\quad|\nabla_{M}\rho_{k}|<(1+\varepsilon)^{2}.$

###### Proof.

Let $\delta>0$ such that $\frac{1}{1-2\delta}<1+\frac{\varepsilon}{2}$, and
define $\tilde{\rho}_{k}$ on
$\overline{N_{k}}\cup\Big{(}\overline{B_{p}((k+1+\delta)L)}-B_{p}((k-\delta)L)\Big{)}$
by

$\displaystyle\tilde{\rho}_{k}(x)=\left\\{\begin{array}[]{ll}-\frac{L}{2},&x\in\overline{B_{p}((k+\delta)L)}\\\
\frac{L}{2},&x\in B_{p}((k+1-\delta)L)^{c}\\\
\frac{1}{1-2\delta}\big{(}d(p,x)-(k+\frac{1}{2})L\big{)},&x\in\overline{B_{p}((k+1-\delta)L)}-B_{p}((k+\delta)L)\end{array}\right..$

$\tilde{\rho}_{k}$ is a Lipschitz function with Lipschitz constant
$\frac{1}{1-2\delta}$. Let $B_{0}^{3}(1)$ be the Euclidean ball in
${\bf\mathbb{R}}^{3}$, $\eta(v)=Ce^{-\frac{1}{1-|v|^{2}}}1_{B_{0}^{3}(1)}$ be
a smooth function on ${\bf\mathbb{R}}^{3}$ and
$\int_{{\bf\mathbb{R}}^{3}}\eta(v)\mathrm{d}v=1$. Let $\alpha>0$ be
sufficiently small such that $\exp_{x}$ is a diffeomorphism on $\\{v\in
T_{x}M:|v|<3\alpha\\}$ for all
$x\in\overline{B_{p}((k+1+\delta)L)}-B_{p}((k-\delta)L)$. Then we define

$\displaystyle\rho_{k,\alpha}(x)=\frac{1}{\alpha^{3}}\int_{T_{x}M}\eta\Big{(}\frac{v}{\alpha}\Big{)}\tilde{\rho}_{k}(\exp_{x}(v))\mathrm{d}v.$

The integration is taken with respect to the Lebesgue measure on
$T_{x}M={\bf\mathbb{R}}^{3}$

We claim that $\rho_{k,\alpha}(x)$ is smooth. Fix $x\in M$. For any smooth
curve $\gamma:(-\alpha,\alpha)\to M$ with $\gamma(0)=x$, $\gamma$ can be
written as $\gamma(t)=\exp_{x}(\tilde{\gamma}(t))$, where $\tilde{\gamma}$ is
a smooth curve in $T_{x}M$. Let
$\varphi_{t}(v)=\mathrm{d}\exp_{x}(\tilde{\gamma}(t))(v)$ be the linear
isomorphism between $T_{x}M$ and $T_{\gamma(t)}M$. Then

$\displaystyle\rho_{k,\alpha}(\gamma(t))=$
$\displaystyle\frac{1}{\alpha^{3}}\int_{T_{\gamma(t)}M}\eta\Big{(}\frac{v}{\alpha}\Big{)}\tilde{\rho}_{k}(\exp_{\gamma(t)}(v))\mathrm{d}v$
$\displaystyle=$
$\displaystyle\frac{1}{\alpha^{3}}\int_{T_{x}M}\eta\Big{(}\frac{\varphi_{t}(v)}{\alpha}\Big{)}\tilde{\rho}_{k}(\exp_{\gamma(t)}(\varphi_{t}(v)))J_{\varphi_{t}}(v)\mathrm{d}v.$

Since $\exp_{x}(\tilde{\gamma}(t)+sv)$ is smooth about $(s,t)$ in a small
neighborhood of $(0,0)$,
$\varphi_{t}(v)=\mathrm{d}\exp_{x}(\tilde{\gamma}(t))(v)=\frac{\mathrm{d}}{\mathrm{d}s}|_{s=0}\exp_{x}(\tilde{\gamma}(t)+sv)$
is also smooth about $t$ in a small neighborhood of $0$. So
$\eta\Big{(}\frac{\varphi_{t}(v)}{\alpha}\Big{)}$,
$\exp_{\gamma(t)}(\varphi_{t}(v))$ and $J_{\varphi_{t}}(v)$ are smooth
functions of $t$ near $0$. Then $\rho_{k,\alpha}(\gamma(t))$ is smooth about
$t$ near $0$. Hence $\rho_{k,\alpha}$ is smooth at $x$.

We further assume
$\exp_{x}(B_{0}^{3}(\alpha))\subset\overline{B_{p}((k+1+\delta)L)}-B_{p}((k+1-\delta)L)$
for all $x\in\partial B_{p}((k+1)L)$, and
$\exp_{x}(B_{0}^{3}(\alpha))\subset\overline{B_{p}((k+\delta)L)}-B_{p}((k-\delta)L)$
for all $x\in\partial B_{p}(kL)$ by taking $\alpha$ small. When $x\in\partial
B_{p}((k+1)L)$,

$\displaystyle\rho_{k,\alpha}(x)=\frac{1}{\alpha^{3}}\int_{T_{x}M}\eta\Big{(}\frac{v}{\alpha}\Big{)}\frac{L}{2}\mathrm{d}v=\frac{L}{2}.$

When $x\in\partial B_{p}(kL)$,

$\displaystyle\rho_{k,\alpha}(x)=\frac{1}{\alpha^{3}}\int_{T_{x}M}\eta\Big{(}\frac{v}{\alpha}\Big{)}\Big{(}-\frac{L}{2}\Big{)}\mathrm{d}v=-\frac{L}{2}.$

Moreover $-L/2\leq\rho_{k,\alpha}(x)\leq L/2$ on $N_{k}$.

Since $\rho_{k,\alpha}(x)$ converges to $\tilde{\rho}_{k}$ uniformly on
$\overline{B_{p}((k+1+\delta)L)}-B_{p}((k-\delta)L)$, there is $\alpha_{0}$
such that the Lipschitz constants of $\rho_{k,\alpha_{0}}$ and
$\tilde{\rho}_{k}$ satisfy
$\mathrm{Lip}(\rho_{k,\alpha_{0}})<\mathrm{Lip}(\tilde{\rho}_{k})+\frac{\varepsilon}{2}<1+\varepsilon$.

Let $U=\\{x\in N_{k}:\rho_{k,\alpha_{0}}(x)<0\\}$, $V=\\{x\in
N_{k}:\rho_{k,\alpha_{0}}(x)>0\\}$ be two open sets. From Lemma A.1 there
exists smooth bump functions $f$, $g$ on $M$ such that

$\displaystyle f|_{U}>0,\quad f|_{U^{c}}=0,\quad g|_{V}>0,\quad g|_{V^{c}}=0.$

And we may assume $f<L/4$, $g<L/4$, $|\nabla_{M}f|<\varepsilon/2$,
$|\nabla_{M}g|<\varepsilon/2$ after a scaling.

Let

$\displaystyle\rho_{k}=\rho_{k,\alpha_{0}}|_{\overline{N_{k}}}+f|_{\overline{N_{k}}}-g|_{\overline{N_{k}}}$

then $\rho_{k}$ is smooth, $\rho_{k}|_{\partial E_{k}}=-\frac{L}{2}$,
$\rho_{k}|_{\partial E_{k+1}}=\frac{L}{2}$, $\rho_{k}(N_{k})\subset(-L/2,L/2)$
and

$\displaystyle|\nabla_{M}\rho_{k}|\leq\mathrm{Lip}(\rho_{k,\alpha_{0}})+|\nabla_{M}f|+|\nabla_{M}g|<(1+\varepsilon)^{2}.$

∎

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