# Topology of 3-manifolds with uniformly positive scalar curvature

Jian Wang  Stony brook University, Department of Mathematics,100 Nicolls Road,
NY-11794 Stony Brook, USA<EMAIL_ADDRESS>
###### Abstract.

In this article, we classify (non-compact) $3$-manifolds with uniformly
positive scalar curvature. Precisely, we show that an orientable 3-manifold
has a complete metric with uniformly positive scalar curvature if and only if
it is homeomorphic to an (possibly) infinite connected sum of spherical
$3$-manifolds and some copies of $\mathbb{S}^{2}\times\mathbb{S}^{1}$.
Further, we study a $3$-manifold with mean convex boundary and with uniformly
positive scalar curvature. If the boundary is a disjoint union of closed
surfaces, then the manifold is an (possibly) infinite connected sum of
spherical $3$-manifolds, some copies of $\mathbb{S}^{1}\times\mathbb{S}^{2}$
and some handlebodies.

##  1\. Introduction

If $(M,g)$ is a Riemannian manifold, the scalar curvature is the average of
all sectional curvatures. A metric $g$ on $M$ is said to have _uniformly
positive scalar curvature_ if the scalar curvature is greater than a positive
constant.

We should be concerned here with the topology of (non-compact) 3-manifolds
with uniformly positive scalar curvature. A well-known and fundamental
question one may ask is as follows: (See Problem 27 in [Yau82])

_How to classify $3$-manifolds admitting complete Riemannian metrics of
(uniformly) positive scalar curvature up to diffeomorphisms._

This question was intensively studied in Schoen-Yau’s works [SY79a, SY79b,
SY82], as well as in Gromov-Lawson’s [GL80, GL83] and various other authors.

The major contribution to the compact case is made by Perelman [Per02a,
Per02b, Per03] who developed the so-called Ricci flow with surgery. One
beautiful result of his study is a proof that a closed $3$-manifold with
positive scalar curvature is a connected sum of quotients of $\mathbb{S}^{3}$
(called spherical $3$-manifolds) and some copies of
$\mathbb{S}^{1}\times\mathbb{S}^{2}$. Recently, Bessières, Besson and Maillot
generalized his proof to (non-compact) $3$-manifolds with uniformly positive
scalar curvature and with bounded geometry (see [BBM11]).

However, it was still unknown how to classify open $3$-manifolds with
(uniformly) positive scalar curvature. The topological structure of open
$3$-manifolds is much complicated. Relevant to the present article is the fact
that Kneser’s theorem [Kne29] (i.e. the prime decomposition) fails to be
generalized to open $3$-manifolds (see [Sco77] and [Mai08]).

Let us define what we mean by the _infinite connected sum_.

Let $\mathscr{F}$ be a family of $3$-manifolds. A $3$-manifold $M$ is an
_infinite connected sum_ of members in $\mathscr{F}$, if there is a locally
finite graph $G$ satisfying that

  * (1)

each vertex $v$ of $G$ is associated with a copy $M_{v}$ of some manifold in
$\mathscr{F}$;

  * (2)

for each vertex $v$, $Y_{v}$ is defined as $M_{v}$ with $d_{v}$ punctures,
where $d_{v}$ is the degree of $v$;

  * (3)

$M$ is diffeomorphic to the manifold obtained by the following operation (see
Figure 1): for each edge $e$ in $G$, call $v,v^{\prime}$ vertices connected by
$e$; choose two $2$-spheres $S\subset\partial Y_{v}$,
$S^{\prime}\subset\partial Y_{v^{\prime}}$, and glue $Y_{v}$ and
$Y_{v^{\prime}}$ together along $S$ and $S^{\prime}$ using an orientation-
reversing diffeomorphism. (See Definition 2.9)

$v_{1}$
---
$v_{3}$
---
$v_{2}$
---
$v_{4}$
---
$v_{5}$
---
$v_{6}$
---
$v_{9}$
---
$v_{8}$
---
$v_{7}$
---
$M_{v_{1}}$
---
$M_{v_{4}}$
---
$M_{v_{5}}$
---
$M_{v_{7}}$
---
$M_{v_{6}}$
---
$M_{v_{3}}$
---
$M_{v_{2}}$
---
$M_{v_{9}}$
---
$M_{v_{8}}$
---

Figure 1.

For example, $\mathbb{R}^{1}\times\mathbb{S}^{2}$ and $\mathbb{R}^{3}$ are
both the infinite connected sums of $3$-spheres (see Section 2.2). Remark that
$\mathbb{S}^{1}\times\mathbb{S}^{2}$ is a connected sum of $\mathbb{S}^{3}$
with itself.

When $G$ is a finite tree, it is the usual definition of the connected sum.
When $G$ is a finite graph, we could transform the graph into a tree at the
expense of adding some factors $\mathbb{S}^{1}\times\mathbb{S}^{2}$ (see
Figure 3). Generally, for a fixed $3$-manifold $M$, the graph $G$ is not
uniquely associated with $M$.

It is well-known from [GL80] and [SY79b] that any infinite connected sum of
spherical $3$-manifolds and some copies of
$\mathbb{S}^{1}\times\mathbb{S}^{2}$ admits a complete metric with (uniformly)
positive scalar curvature. Conversely, we show that:

######  Theorem 1.1.

Any complete orientable $3$-manifold of uniformly positive scalar curvature is
homeomorphic to an infinite connected sum of spherical $3$-manifolds and some
copies of $\mathbb{S}^{1}\times\mathbb{S}^{2}$.

It was also proved by Gromov in [Gro19].

Further, we study $3$-manifolds with mean convex boundary. Such compact
manifolds were intensively studied in [CL19]. We prove the following theorem.

######  Theorem 1.2.

Let $(M,g)$ be an orientable $3$-manifold with mean convex boundaries, and
with uniformly positive scalar curvature. If its boundary is a disjoint union
of closed surfaces, then $M$ is homeomorphic to an infinite connected sum of
spherical $3$-manifolds, handlebodies and some copies of
$\mathbb{S}^{2}\times\mathbb{S}^{1}$.

When $M$ is compact, it was proved by Ricci flow in [CL19].

###  1.1. Prime decomposition and $\mu$-bubbles

The $\mu$-bubble is the stationary for the prescribed-mean-curvature
functional. It was first introduced to scalar curvature problems by Gromov in
[Section $5\frac{5}{6}$, [Gro96]].

Recently, it was demonstrated that as minimal surfaces, $\mu$-bubbles could
also give topological informations on manifolds. This fact appeared in a
series of Gromov’s works (see [Gro18, Gro19, Gro20]) as well as in Chodosh-
Li’s (see [CL20]), and various other authors (see [Ric20] and [Zhu20b,
Zhu20a]).

In this article, we use $\mu$-bubbles to show the existence of the prime
decomposition for complete $3$-manifolds with uniformly positive scalar
curvature.

Let $(M,g)$ be a complete orientable $3$-manifold with scalar curvature
$\kappa\geq 1$. There are infinitely many stable $\mu$-surfaces which are
topological boundaries of $\mu$-bubbles (see Corollary 3.10). The uniform
positivity of scalar curvature shows that each component of a stable
$\mu$-surface is a $2$-sphere (see Proposition 3.9).

These $2$-spheres from $\mu$-bubbles cuts $M$ into some components. The
closure of each component is a compact $3$-manifold whose boundary consists of
some $2$-spheres. The prime decomposition theorem for a closed $3$-manifold
shows that each component comes from some closed prime $3$-manifolds. Thus, we
have that a prime decomposition theorem for $M$. It also appeared in the
recent version of Gromov’s note (see [Gro19]).

######  Theorem 1.3.

Any complete orientable $3$-manifold with uniformly positive scalar curvature
can be written as an (possibly) infinite connected sum of some closed prime
$3$-manifolds.

Recently, Chang, Weinberger and Yu [CWY10] use K-theory to prove a similar
decomposition under the assumption that the fundamental group is finitely
generated. Moreover, the Ricci flow also could give such a decomposition under
a geometric condition (see [BBM11]).

###  1.2. Idea of the proof of Theorem 1.1

Let $(M,g)$ be a complete (non-compact) orientable $3$-manifold with scalar
curvature $\kappa\geq 1$. By Theorem 1.3, $M$ has a prime decomposition (i.e.
$M$ is homeomorphic to an infinite connected sum of some closed prime
$3$-manifold).

In the following, we explain that each prime factor in the decomposition is
homeomorphic to a spherical $3$-manifold or
$\mathbb{S}^{1}\times\mathbb{S}^{2}$.

Let $\hat{M}_{l}$ be a closed prime $3$-manifold which appears in the
decomposition of $M$. There is a compact $3$-manifold $M_{l}\subset M$
homeomorphic to $\hat{M}_{l}$ with an open $3$-ball removed.

Remark that from Theorem 1.3, there is a non-compact $3$-manifold
$M^{\prime}_{l}$ with $M\cong\hat{M}_{l}\\#M^{\prime}_{l}$

We assume that $\hat{M}_{l}$ is not homeomorphic to one of $\mathbb{S}^{3}$,
$\mathbb{S}^{2}\times\mathbb{S}^{1}$ and $\mathbb{R}P^{3}$. It is irreducible
and $\partial M_{l}$ is not homotopically trivial in $M$.

Consider the isotopy class of $\partial M_{l}$ and an area-minimizing sequence
in the class. If the area-minimizing sequence converges, there is a component
$M^{\prime}_{l}\subset M$ of the complement of the limit, which is
homeomorphic to $\hat{M}_{l}$ with finitely many punctures (see Proposition
4.11). In addition, $(M^{\prime}_{l},g|_{M^{\prime}_{l}})$ has mean convex
boundary.

The uniform positivity of scalar curvature implies that
$\pi_{1}(M^{\prime}_{l})$ is finite (see Lemma 6.3). By the Pioncaré
conjecture, $M^{\prime}_{l}$ is a spherical $3$-manifold with punctures.
Namely, $\hat{M}_{l}$ is a spherical $3$-manifold.

Generally, the area-minimizing sequence of the isotopy class of $\partial
M_{l}$ may go to infinity, since $M$ is non-compact. It yields a new kind of
boundary condition, _mixed boundary_ (see Definition 5.1).

An orientable $3$-manifold $(X,g)$ has _mixed boundary_ if it satisfies that

  * •

the boundary $\partial X$ consists of $2$-spheres, $\\{S_{j}\\}_{j\in
J^{\prime}\amalg J^{\prime\prime}}$;

  * •

$S_{j}$ is minimal for $g$, where $j\in J^{\prime}$;

  * •

there are disjoint punctured $3$-spheres $\\{\Omega_{l}\\}_{l}$ with
$\amalg_{j\in J^{\prime\prime}}S_{j}\subset\amalg_{l}\partial\Omega_{l}$;

  * •

$\hat{S}_{l}:=\partial\Omega_{l}\cap\text{Int}~{}~{}X$ is a $2$-sphere and
$d(\hat{S}_{l},\partial\Omega_{l}\cap\amalg_{j\in
J^{\prime\prime}}S_{j})>4\pi$.

In this article, we show that there is an open $3$-manifold
$M^{\prime}_{l}\subset M$ so that (1) $M^{\prime}_{l}$ is homeomorphic to
$\hat{M}_{l}$ with finitely many punctures; (2) the closure of
$(M^{\prime}_{l},g|_{M^{\prime}_{l}})$ is a compact $3$-manifold with mixed
boundary (see Theorem 5.3).

We use the uniform positivity of scalar curvature to classify such manifolds.

######  Theorem 1.4.

Let $(M^{3},g)$ be a compact $3$-manifold with scalar curvature $\kappa\geq 1$
and with mixed boundary. If $\pi_{2}(M)$ is generated by the spheres in
$\partial M$, then $\text{Int}~{}M$ is homeomorphic to a spherical
$3$-manifold with finitely many punctures.

When $M$ is non-compact, it is homeomorphic to $\mathbb{R}^{3}$ with (at most)
countably many disjoint $3$-ball removed. The philosophy of the proof follows
from the proof for manifolds with mean convex boundary.

We use Theorem 1.4 to show that $M^{\prime}_{l}$ is a spherical $3$-manifold
with finitely many punctures. Namely, $\hat{M}_{l}$ is a spherical
$3$-manifold.

The paper is organized as follows. In Section 2, we recall related background
and introduce the infinite connected sum and boundary connected sum. In
Sections 3, our focus is on $\mu$-bubbles and the prime decomposition. In
Section 4 , we treat minimal surfaces and related problems. In Sections 5 and
6, we introduce the mixed boundary condition and use it to complete the proof
of Theorem 1.1. In Section 7, we use a geometric version of loop lemma to
prove Theorem 1.2.

We would like to thank L. Bessière, G. Besson and S. Maillot for introducing
this question to me and many helpful discussions. We are grateful to M.
Gromov, B. Lawson and B. Hanke for their interests. I am also thankful to
Jintian Zhu for his helpful discussions and suggestions. This work was
partially supported by SPP2026, “Geometry at infinity”.

##  2\. 3-manifolds and infinite connected sum

In this section, we begin with several topological properties of open
$3$-manifolds. Then, we introduce the so-called _infinite connected sum_ and
give several examples.

###  2.1. Topology Basis

The Poincaré conjecture (see [MT07], [BBB+10] and [CZ06]) tells that a compact
contractible 3-manifold is homeomorphic to the unit ball,
$\mathbb{B}^{3}\subset\mathbb{R}^{3}$. One useful corollary is as follows:

######  Proposition 2.1.

(See [MT07, BBB+10, CZ06] and Proposition 3.10 of [Hat00] on Page 50) Let $M$
be a $3$-manifold (possibly with boundary, not necessarily compact or
orientable) and $\Sigma\subset M$ an embedded 2-sphere. The sphere $\Sigma$ is
homotopically trivial in $M$ if and only if it bounds a $3$-ball in $M$.

If any embedded sphere in $M$ bounds a $3$-ball in $M$, such a $3$-manifold is
called irreducible. In this article, we will repeatedly use a family of
irreducible $3$-manifolds with boundaries, called handlebodies.

######  Definition 2.2.

(See Page 46 of [Rol03]) A closed _handlebody_ is any space obtained from the
closed $3$-ball $D^{3}$ ($0$-handle) by attaching $g$ distinct copies of
$D^{2}\times[-1,1]$ (1-handle) with the homeomorphisms identifying the $2g$
discs $D^{2}\times\\{\pm 1\\}$ to $2g$ disjoint $2$ disks on $\partial D^{3}$,
all to be done in such way that the resulting $3$-manifold is orientable. The
integer $g$ is called the genus of the handlebody.

######  Definition 2.3.

(See Page 39 of [Rol03])Let $M_{1}$ and $M_{2}$ be two $n$-manifolds with
boundaries. A _boundary connected sum_ $M_{1}\\#_{\partial}M_{2}$ is
constructed by identifying standard balls $B^{n-1}_{i}\subset\partial M_{i}$.

######  Remark 2.4.

  * •

A handlebody of genus $g$ is homeomorphic to a boundary connected sum of $g$
solid tori (see Page 46 of [Rol03]).

  * •

A boundary connected sum of two handlebodies is a handlebody.

  * •

A boundary connected sum of a handlebody with itself is a handlebody.

###  2.2. Open $3$-manifolds

We now consider an open manifold $M$ which is homotopically equivalent to
$\mathbb{S}^{1}$.

######  Lemma 2.5.

Let $M$ be an open $3$-manifold which is homotopically equivalent to
$\mathbb{S}^{1}$. Then any closed embedded surface $\Sigma\subset M$ cuts $M$
into an compact part and a non-compact part.

We will prove it in Appendix A.

######  Corollary 2.6.

Let $M$ be assumed in Lemma 2.5. Then there is an exhaustion $\\{M_{i}\\}_{i}$
of $M$ so that for each $i$, $M_{i}$ is compact and $\partial M_{i}$ is a
connected closed surface.

###### Proof.

It is sufficient to show that any compact $K\subset M$ is a subset of some
compact set $M^{\prime}\subset M$, where $\partial M^{\prime}$ is a connected
closed surface.

We may assume that $K$ is connected. (If not, we find a compact and connected
set containing $K$ to replace it.) The boundary $\partial K$ has finitely many
components, $\\{\Sigma_{i}\\}_{i\in I}$. We use Lemma 2.5 to find a compact
manifold $M_{\Sigma_{i}}\subset M$ with $\partial M_{\Sigma_{i}}=\Sigma_{i}$
for each $i$.

We can conclude that there is $K$ is contained in $M_{\Sigma_{i}}$ for some
$i$. If not, consider the set $\hat{K}:=K\cup_{i\in
I}(\cup_{\Sigma_{i}}M_{\Sigma_{i}})\subset M$. It is a closed manifold which
is in contradiction with the fact that $M$ is non-compact and connected.

The set $M_{\Sigma_{i}}$ contains $K$ and the boundary is a closed and
connected surface. It is the required candidate in the previous statement. ∎

######  Lemma 2.7.

Let $M$ be assumed as in Lemma 2.5 and $\gamma$ a closed curve generating
$\pi_{1}(M)$. Assume that a closed surface $\Sigma\subset M$ bounds a compact
manifold $M_{\Sigma}\subset M$. If $\gamma\subset M_{\Sigma}$, there is a
closed curve $\sigma\subset\Sigma$ so that it is contractible in $M$ but non-
contractible in $M\setminus\gamma$.

We will prove it in Appendix A. We use Corollary 2.6 and Lemma 2.7 to get that

######  Corollary 2.8.

Let $M$ and $\gamma$ be assumed as in Lemma 2.7. For any compact set $K$
containing $\gamma$, there is a closed curve $\sigma\subset M\setminus K$ so
that it is contractible in $M$ but non-contractible in $M\setminus\gamma$.

###  2.3. Infinite connected sum

We introduce the notion of an infinite connected sum.

$v_{1}$
---
$v_{3}$
---
$v_{2}$
---
$v_{4}$
---
$v_{5}$
---
$v_{6}$
---
$v_{9}$
---
$v_{8}$
---
$v_{7}$
---
$M_{v_{1}}$
---
$M_{v_{4}}$
---
$M_{v_{5}}$
---
$M_{v_{7}}$
---
$M_{v_{6}}$
---
$M_{v_{3}}$
---
$M_{v_{2}}$
---
$M_{v_{9}}$
---
$M_{v_{8}}$
---

Figure 2.

######  Definition 2.9.

Let $\mathscr{F}$ be a family of $3$-manifolds. A $3$-manifold $M$ is an
_infinite connected sum_ of members in $\mathscr{F}$, if there is a locally
finite graph $G$ satisfying that

$\bullet$ each vertex $v$ of $G$ is associated with a copy $M_{v}$ of some
manifold in $\mathscr{F}$;

$\bullet$ for each vertex $v$, $Y_{v}$ is defined as $M_{v}$ with $d_{v}$
punctures, where $d_{v}$ is the degree of $v$;

$\bullet$ $M$ is diffeomorphic to the manifold obtained by the following
operation:

For each edge $e$ in $G$, call $v,v^{\prime}$ vertices connected by $e$;
choose two $2$-spheres $S\subset\partial Y_{v}$ , $S^{\prime}\subset\partial
Y_{v^{\prime}}$, and glue $Y_{v}$ and $Y_{v^{\prime}}$ together along $S$ and
$S^{\prime}$ using an orientation-reversing diffeomorphism (see Figure 2).

When $G$ is a finite tree, it is the usual definition of the connected sum.
When $G$ is a finite graph, we could transform the graph into a tree at the
expense of adding some factor $\mathbb{S}^{1}\times\mathbb{S}^{2}$ (see Figure
3).

######  Remark.

For a fixed $3$-manifold $M$, the graph $G$ is not uniquely associated with
$M$. For example, the manifold $M_{1}$ in Figure 3 is homeomorphic to $M_{2}$
obtained from the graph $G_{2}$, where $M_{v_{0}}$ (described in Figure 3) is
homeomorphic to $\mathbb{S}^{1}\times\mathbb{S}^{2}$.

$v$
---
$M_{v}$
---
$\rightsquigarrow$
---
$G_{1}$
---
$v_{1}$
---
$v_{4}$
---
$v_{3}$
---
$v_{2}$
---
$\rightsquigarrow$
---
$M_{v_{1}}$
---
$M_{v_{2}}$
---
$M_{v_{4}}$
---
$M_{v_{3}}$
---
$M_{v_{0}}$
---
$M_{v_{1}}$
---
$M_{v_{4}}$
---
$M_{v_{3}}$
---
$M_{v_{2}}$
---
$v_{2}$
---
$v_{3}$
---
$v_{4}$
---
$v_{1}$
---
$v_{0}$
---
$\rightsquigarrow$
---
$G_{0}$
---
$M_{0}$
---
$M_{1}$
---
$G_{2}$
---
Figure 3.

For example, if $M_{v}$ (in Figure 3) is homeomorphic to $\mathbb{S}^{3}$,
then $M_{0}$ is homeomorphic to $\mathbb{S}^{2}\times\mathbb{S}^{1}$ (see
Figure 3). If $M_{v}$ is homeomorphic to $\mathbb{S}^{3}/\Gamma$, where
$\Gamma\subset O(4)$ is a finite group, $M_{0}$ is homeomorphic to
$\mathbb{S}^{2}\times\mathbb{S}^{1}\\#\mathbb{S}^{3}/\Gamma$ (see Figure 3).

Consider a non-compact example as follows. If each $M_{v_{i}}$ (in Figure 4)
is homeomorphic to $\mathbb{S}^{3}$ , $M$ is homeomorphic to $\mathbb{R}^{3}$.
That is to say, $\mathbb{R}^{3}$ is the infinite connected sum of some copies
of $\mathbb{S}^{3}$.

$\dots$
---
$\dots$
---
$v_{2}$
---
$v_{3}$
---
$v_{4}$
---
$v_{1}$
---
$v_{0}$
---
$\rightsquigarrow$
---
$v_{5}$
---
Figure 4.

##  3\. $\mu$-bubble and Prime decomposition

Classically, an orientable closed $3$-manifold can be written as a connected
sum of some prime $3$-manifolds. Our focus is on the prime decomposition for
complete $3$-manifolds with uniformly positive scalar curvature.

Our approach is to find a family of $2$-spheres which cut the complete
$3$-manifold into some compact $3$-manifolds. Then, we apply the prime
decomposition for closed $3$-manifolds to each component and get a required
decomposition for the complete $3$-manifold. The decomposition is not unique.

The existence of such a family of $2$-spheres comes from $\mu$-bubble theory.

###  3.1. $\mu$-bubbles

In this section, we recall the general existence and the stability results for
$\mu$-bubbles. Consider a connected orientable compact manifold $(M^{3},g)$
with boundary. Assume that $\partial M:=\partial_{-}M\cup\partial_{+}M$ is a
choice of labeling the components of $\partial M$ so that neither of the sets,
$\partial_{\pm}M$, are empty.

Fix a smooth function $h$ on $\text{Int}~{}M$. Choose a Caccioppoli set
$\Omega_{0}$ with smooth reduced boundary
$\partial^{*}\Omega_{0}:=\partial\Omega\cap\text{Int}~{}M$ and
$\partial_{+}M\subset\Omega_{0}$. Consider the following functional

$\mathcal{A}(\Omega)=\mathcal{H}^{2}(\partial^{*}\Omega)-\int_{M}(\chi_{\Omega}-\chi_{\Omega_{0}})h~{}d\mathcal{H}^{3},$
(3.1)

where $\Omega$ is any Caccioppoli set of $\text{Int}~{}M$ with reduced
boundary $\partial^{*}\Omega:=\partial\Omega\cap\text{Int}~{}~{}M$

$\Omega\Delta\Omega_{0}\Subset\text{Int}~{}M$

and $\chi_{\Omega}$ is the characteristic function of region $\Omega$.

######  Remark.

If $h$ vanishes on $\text{Int}~{}~{}M$ and $\Omega$ is a critical point of
$\mathcal{A}$, then $\partial^{*}\Omega$ is minimal.

The existence and regularity of a minimizer of $\mathcal{A}$ among all
Caccioppoli sets is claimed by Gromov (see Section 5.1 of [Gro19] on Page 102)
and rigorously carried out by Zhu (see Proposition 2.1 of [Zhu20b] on Page 4).

######  Proposition 3.1.

(See Proposition 2.1 of [Zhu20b] on Page 4) If the function $h$ satisfies

$\lim_{x\rightarrow\partial_{-}M}h(x)=-\infty~{}~{}\text{and}~{}~{}\lim_{x\rightarrow\partial_{+}M}h(x)=+\infty,$
(3.2)

then there exists a smooth minimizer $\Omega$ for $\mathcal{A}$ such that
$\Omega\Delta\Omega_{0}$ is compactly contained in $\text{Int}~{}M$.

###### Proof.

Define $\mathcal{C}:=\\{$Capccipoli sets $\Omega$ in $\text{Int}~{}~{}M$ such
that $\Omega\Delta\Omega_{0}\Subset\text{Int}~{}~{}M\\}$ and

$I:=\inf\\{\mathcal{A}(\Omega)~{}|~{}\Omega\in\mathcal{C}\\}.$

We first prove $I>-\infty$ and then show that a minimizing sequence converges
to the required candidate.

We now show that $I>-\infty$. For $\tau>0$, denote $\Omega^{\tau}_{\pm}$ the
distance-$\tau$ neighborhood of $\partial_{\pm}M$. Choose $\tau$ small enough,
$\Omega^{\tau}_{\pm}$ has a foliation $\\{S^{\rho}_{\pm}\\}_{\rho\in[0,\tau]}$
by smooth equidistant hypersurfaces to $\partial_{\pm}M$. Denote $\eta$ the
unit normal vector field of $\\{S^{\rho}_{\pm}\\}$ defined in this
$\Omega^{\tau}_{\pm}$, pointing out $M$ along $\partial_{\pm}M$. Let $\tau>0$
be sufficiently small so that

$h>\text{div}(\eta)~{}\text{in}~{}\Omega^{\tau}_{+}$ (3.3)
$h<-\text{div}(\eta)~{}\text{in}~{}\Omega^{\tau}_{-}.$ (3.4)

We compute that

$\begin{split}&\mathcal{A}(\Omega\cup\Omega^{\tau}_{+})-\mathcal{A}(\Omega)\\\
&=\int_{\partial^{*}(\Omega\cup\Omega^{\tau}_{+})}d\mathcal{H}^{2}-\int_{\partial^{*}\Omega}d\mathcal{H}^{2}-\int_{\Omega^{\tau}_{+}\setminus\Omega}h~{}d\mathcal{H}^{3}\\\
&=\int_{\partial^{*}\Omega^{\tau}_{+}\setminus\Omega}d\mathcal{H}^{2}-\int_{\partial^{*}\Omega\cap\Omega^{\tau}_{+}}d\mathcal{H}^{2}-\int_{\Omega^{\tau}_{+}\setminus\Omega}h~{}d\mathcal{H}^{3}\\\
&<\int_{\partial^{*}\Omega^{\tau}_{+}\setminus\Omega}d\mathcal{H}^{2}-\int_{\partial^{*}\Omega\cap\Omega^{\tau}_{+}}d\mathcal{H}^{2}-\int_{\Omega^{\tau}_{+}\setminus\Omega}\text{div}(\eta)~{}d\mathcal{H}^{3}\end{split}$
(3.5)

We have that $\partial(\Omega^{\tau}_{+}\setminus\Omega)$ consists of two
parts, $\partial^{*}\Omega^{\tau}_{+}\setminus\Omega$ and
$\partial^{*}\Omega\cap\Omega^{\tau}_{+}$. It follows from the fact that
$\Omega^{\tau}_{+}\setminus\Omega\subset\text{Int}~{}~{}M$. Moreover,

$\begin{split}\int_{\Omega^{\tau}_{+}\setminus\Omega}\text{div}(\eta)&=\int_{\partial^{*}\Omega^{\tau}_{+}\setminus\Omega}\eta\cdot\nu~{}d\mathcal{H}^{2}-\int_{\partial*\Omega\cap\Omega^{\tau}_{+}}\eta\cdot\nu~{}d\mathcal{H}^{2}\\\
&\geq\int_{\partial^{*}\Omega^{\tau}_{+}\setminus\Omega}d\mathcal{H}^{2}-\int_{\partial^{*}\Omega\cap\Omega^{\tau}_{+}}d\mathcal{H}^{2},\end{split}$
(3.6)

where $\nu$ is the unit normal vector and $\eta\cdot\nu=1$ on
$\partial^{*}\Omega^{\tau}_{+}\setminus\Omega$.

Thus, $\mathcal{A}(\Omega\cup\Omega^{\tau}_{+})<\mathcal{A}(\Omega)$. By an
analogous calculation,
$\mathcal{A}(\Omega\setminus\Omega^{\tau}_{-})<\mathcal{A}(\Omega)$. It
follows that for any $\Omega\in\mathcal{C}$,

$\mathcal{A}(\Omega)>\mathcal{A}((\Omega\cup\Omega^{\tau}_{+})\setminus\Omega^{\tau}_{-})\geq-C\mathcal{H}^{3}(M),$
(3.7)

where $|h|\leq C$ on $M\setminus(\Omega^{\tau}_{+}\cup\Omega^{\tau}_{-})$.
Thus, $I>-\infty$.

Take a sequence $\\{\Omega_{k}\\}\subset\mathcal{C}$ with
$\mathcal{A}(\Omega_{k})\rightarrow I$ as $k\rightarrow\infty$. From the
above, we may assume that

$\Omega_{k}\Delta\Omega_{0}\subset
M\setminus(\Omega^{\tau}_{+}\cup\Omega^{\tau}_{-}).$ (3.8)

We have that for $k$ large enough, $\mathcal{A}(\Omega_{k})\leq I+1$ and
$|h|\leq C$ on $M\setminus(\Omega^{\tau}_{+}\cup\Omega^{\tau}_{-})$. It
follows that

$\mathcal{H}^{2}(\partial^{*}\Omega_{k})\leq I+1+C\mathcal{H}^{3}(M)$

By BV-compactness, taking a subsequence, the sets $\Omega_{k}$ converges to a
Caccioppoli set $\hat{\Omega}$. It follows that $\hat{\Omega}$ is a minimizer
of $\mathcal{A}$ and thus has smooth boundary from the standard regularity
theory [Tam84]. ∎

###  3.2. First and Second variations

######  Lemma 3.2.

If $\Omega_{t}$ is a smooth 1-parameter family of regions with
$\Omega_{t}|_{t=0}=\Omega$ and normal speed $\phi$ at $t=0$, then

$\frac{d}{dt}\mathcal{A}(\Omega_{t})\Big{|}_{t=0}=\int_{\Sigma_{0}}(H-h)\phi~{}d\mathcal{H}^{2}$
(3.9)

where $\Sigma_{0}:=\partial\Omega\cap\text{Int}~{}~{}M$ and $H$ is the mean
curvature of $\Sigma_{0}$. In particular, a critical point $\Omega$ of
$\mathcal{A}$ satisfies

$H=h$ (3.10)

along $\Sigma_{0}$.

###### Proof.

The reduced boundary $\Sigma_{t}:=\partial\Omega_{t}\cap\text{Int}~{}~{}M$
gives a variation $F:(-\epsilon,\epsilon)\times\Sigma_{0}\rightarrow M$ with
$F(t,\Sigma_{0})=\Sigma_{t}$. The variational field is $F_{t}(0,x)$ and its
normal component
$F^{N}_{t}(0,x)=\phi\overset{\rightarrow}{\mathbf{n}}_{\Sigma_{0}}(x)$.

The first variation of area functional gives that

$\begin{split}\frac{d}{dt}\int_{\Sigma_{t}}d\mathcal{H}^{2}\Big{|}_{t=0}&=\int_{\Sigma_{0}}H\phi~{}d\mathcal{H}^{2}+\int_{\Sigma_{0}}\text{div}_{\Sigma_{0}}(F^{T}_{t}(0,x))~{}d\mathcal{H}^{2},\\\
&=\int_{\Sigma_{0}}H\phi~{}d\mathcal{H}^{2},\end{split}$ (3.11)

where $\text{div}_{\Sigma_{0}}$ is the divergence operation of the induced
metric on $\Sigma_{0}$ and $F^{T}_{t}(0,x)$ is the tangential component of
$F_{t}(0,x)$. The second equation comes from Stokes’ theorem.

A computation gives that

$\frac{d}{dt}\int_{\Omega_{t}}h~{}d\mathcal{H}^{2}\Big{|}_{t=0}=\int_{\Sigma_{0}}h~{}d\mathcal{H}^{2}.$
(3.12)

Thus, we have that
$\frac{d}{dt}\mathcal{A}(\Omega_{t})\Big{|}_{t=0}=\int_{\Sigma_{0}}(H-h)\phi~{}d\mathcal{H}^{2}$∎

######  Lemma 3.3.

Let $\Omega$ be a minimizer of $\mathcal{A}$ with the reduced boundary
$\partial^{*}\Omega:=\partial\Omega\cap\text{Int}~{}~{}M$. Then, for a smooth
function $\phi$ on the reduced boundary $\Sigma_{0}:=\partial^{*}\Omega$, one
has that

$\int_{\Sigma_{0}}|\nabla_{\Sigma_{0}}\phi|^{2}-(\text{Ric}_{M}(\overset{\rightarrow}{\mathbf{n}}_{\Sigma_{0}})+|A|^{2})\phi^{2}~{}d\mathcal{H}^{2}-\int_{\Sigma_{0}}\langle\nabla_{M}h,\overset{\rightarrow}{\mathbf{n}}_{\Sigma_{0}}\rangle\phi^{2}d\mathcal{H}^{2}\geq
0$ (3.13)

where $|A|^{2}$ is the squared norm of the second fundamental form of
$\Sigma_{0}$.

###### Proof.

Let $\Omega_{t}$ be a smooth 1-parameter family of regions with
$\Omega_{t}\Big{|}_{t=0}=\Omega$ and
$F:(-\epsilon,\epsilon)\times\Sigma_{0}\rightarrow M$ be a variation of
$\Sigma_{0}$, with $F(t,\Sigma_{0})=\Sigma_{t}$, where
$\Sigma_{t}:=\partial\Omega_{t}\cap\text{Int}~{}~{}M$ is the reduced boundary
$\partial^{*}\Omega_{t}$ of $\Omega_{t}$.

Assume that $F$ is a normal variation on $\Sigma_{0}$. We have that

$F_{t}(0,x)=\phi\overset{\rightarrow}{\mathbf{n}}_{\Sigma_{0}},$

where $\phi$ is a smooth function of $\Sigma_{0}$. The second variational
formula of the area functional shows that

$\begin{split}\frac{d^{2}}{dt^{2}}\int_{\Sigma_{t}}d\mathcal{H}^{2}\Big{|}_{t=0}&=\int_{\Sigma_{0}}|\nabla_{\Sigma_{0}}\phi|^{2}-(\text{Ric}(\overset{\rightarrow}{\mathbf{n}}_{\Sigma_{0}})+|A|^{2})\phi^{2}+H^{2}\phi^{2}d\mathcal{H}^{2}+\int_{\Sigma_{0}}\text{div}_{\Sigma_{0}}(F_{tt}(0,x))d\mathcal{H}^{2},\\\
&=\int_{\Sigma_{0}}|\nabla_{\Sigma_{0}}\phi|^{2}-(\text{Ric}(\overset{\rightarrow}{\mathbf{n}}_{\Sigma_{0}})+|A|^{2})\phi^{2}+H^{2}\phi^{2}d\mathcal{H}^{2}.\end{split}$
(3.14)

where $H$ is the mean curvature of $\Sigma_{0}$. The last equation follows
from Stokes’ theorem. Notice that $F_{tt}(0,x)$ belongs to $T_{x}\Sigma_{0}$,
where $x\in\Sigma_{0}$.

Differentiating Equation (3.12), we have that

$\frac{d^{2}}{dt^{2}}\int_{\Omega_{t}}h~{}d\mathcal{H}^{2}\Big{|}_{t=0}=\int_{\Sigma_{0}}\langle\nabla_{M}h,\overset{\rightarrow}{\mathbf{n}}_{\Sigma_{0}}\rangle\phi^{2}+hH\phi^{2}~{}d\mathcal{H}^{2}.$
(3.15)

Since $\Omega$ is a minimizer of $\mathcal{A}$, we use Lemma 3.2 to have that
the mean curvature $H$ is equal to $h|_{\Sigma}$ and
$\frac{d^{2}}{dt^{2}}\mathcal{A}(\Omega_{t})|_{t=0}\geq 0$. Thus, we can
conclude that

$\begin{split}\frac{d^{2}}{dt^{2}}\mathcal{A}(\Omega_{t})\big{|}_{t=0}&=\int_{\Sigma_{0}}|\nabla_{\Sigma_{0}}\phi|^{2}-(\text{Ric}(\overset{\rightarrow}{\mathbf{n}}_{\Sigma_{0}})+|A|^{2})\phi^{2}-\langle\nabla_{M}h,\overset{\rightarrow}{\mathbf{n}}_{\Sigma_{0}}\rangle\phi^{2}~{}d\mathcal{H}^{2}\\\
&\geq 0.\end{split}$ (3.16)

∎

######  Remark 3.4.

The proof of Lemma in [SY79b] shows that

$\text{Ric}_{M}(\overset{\rightarrow}{\mathbf{n}}_{\Sigma_{0}})=\kappa_{M}-K_{\Sigma_{0}}+\frac{H^{2}}{2}-\frac{1}{2}|A|^{2},$
(3.17)

where $\kappa_{M}$ is the scalar curvature of $(M,g)$ and $K_{\Sigma_{0}}$ is
the sectional curvature of the induced metric on $\Sigma_{0}$. The inequality
(3.13) can be written as follows:

$\int_{\Sigma_{0}}|\nabla_{\Sigma_{0}}\phi|^{2}+K_{\Sigma_{0}}\phi^{2}-\frac{1}{2}|A|^{2}\phi^{2}-(\kappa_{M}+\frac{h^{2}}{2}+\langle\nabla_{M}h,\overset{\rightarrow}{\mathbf{n}}_{\Sigma_{0}}\rangle)\phi^{2}~{}d\mathcal{H}^{2}\geq
0$ (3.18)

###  3.3. Free boundary $\mu$-bubble

Let $(M^{3},g)$ be a Riemannian $3$-manifold with boundary. Assume that
$\partial M=\partial_{+}M\amalg\partial_{-}M\amalg\partial_{0}M$, where
$\partial_{\pm}M$, $\partial_{0}M$ are smooth submanifolds of $M$.

Fix a function $h\in C^{\infty}(M\setminus(\partial_{+}M\cup\partial_{-}M))$
with $h\rightarrow\pm\infty$ on $\partial_{\pm}M$. We assume that
$\partial_{0}M$ is minimal for $g$ .That is to say,

$H_{\partial_{0}M}=0.$ (3.19)

It prevents tangential contact between free boundary $\mu$-bubble and
$\partial_{0}M$ in the usual way that that free boundary minimal surface does
not make tangential contact with mean convex components of the boundary.

Let $\Omega_{0}$ be a Caccioppoli set with the smooth reduced boundary
$\partial^{*}\Omega_{0}:=\partial\Omega_{0}\cap(\text{Int}~{}M\cup\partial_{0}M)$.
Consider the functional with free boundary (as in Section 3.1)

$\mathcal{A}(\Omega)=\mathcal{H}^{2}(\partial^{*}\Omega)-\int_{M}(\chi_{\Omega}-\chi_{\Omega_{0}})h~{}d\mathcal{H}^{3},$
(3.20)

for all Caccioppoli sets $\Omega$ in $M$ with $\Omega\Delta\Omega_{0}\Subset
M\setminus(\partial_{+}M\cup\partial_{-}M)$. By similar arguments to
Proposition 3.1, we can conclude that

######  Proposition 3.5.

There exists $\Omega$ with the smooth reduced boundary
$\partial^{*}\Omega:=\partial\Omega\cap(\text{Int}~{}M\cup\partial_{0}M)$
minimizing $\mathcal{A}$ among such regions.

In the rest, we study the variational formulas for $\mu$-bubbles with free
boundary. The following notions will be used:

  * •

We let
$(\Sigma,\partial\Sigma)\subset(\text{Int}~{}~{}M\cup\partial_{0}M,\partial_{0}M)$
denote an embedded surface intersecting $\partial_{0}M$ transversally;

  * •

the vector $\overset{\rightarrow}{\alpha}$ is the unit tangential vector of
$\partial\Sigma:=\Sigma\cap\partial_{0}M$;

  * •

the unit vector $\overset{\rightarrow}{\beta}$ is the outward normal vector of
$\partial\Sigma$ in $\Sigma$;

  * •

the unit vector $\overset{\rightarrow}{\gamma}$ on $\partial\Sigma$ is
orthogonal to $\overset{\rightarrow}{\alpha}$ in
$T(\partial_{0}M)|_{\partial\Sigma}$.

######  Lemma 3.6.

Let $M$ and $\mathcal{A}$ be defined as above and $\Omega$ a critical point of
$\mathcal{A}$. If $\Sigma$ is a component of the reduced boundary
$\partial^{*}\Omega:=\partial\Omega\cap(\text{Int}~{}~{}M\cup\partial_{0}M)$,
then one has that the mean curvature of $\Sigma$

$H=h|_{\Sigma},$

and $\Sigma$ meets $\partial_{0}M$ orthogonally.

###### Proof.

Let $\Omega_{t}$ be a smooth 1-parameter family of regions with
$\Omega_{t}\Big{|}_{t=0}=\Omega$ and
$F:(-\epsilon,\epsilon)\times\partial^{*}\Omega\rightarrow M$ be a variation
of $\Sigma$, with $F(t,\partial^{*}\Omega)=\partial^{*}\Omega_{t}$, where
$\partial^{*}\Omega_{t}:=\partial\Omega_{t}\cap(\text{Int}~{}~{}M\cup\partial_{0}M)$
is the reduced boundary of $\Omega_{t}$. At
$x\in\partial\Sigma:=\Sigma\cap\partial_{0}M$, $F_{t}(0,x)$ always belongs to
$T_{x}(\partial_{0}M)$.

In the following, we assume that

$\begin{cases}F(t,x)=x\quad\text{for}~{}x\in\partial^{*}\Omega\setminus\Sigma~{}\text{and
any}~{}t;\\\
F^{N}_{t}(0,x)=\phi(x)\overset{\rightarrow}{\mathbf{n}}_{\Sigma}(x)\quad\text{on}\quad
x\in\Sigma,\end{cases}$

where $\phi$ is a smooth function on $\Sigma$ and $F^{T}_{t}(0,x)$ and
$F^{N}_{t}(0,x)$ are the tangential and normal component of $F_{t}(0,x)$
respectively. The first variational formula for area functional gives that

$\begin{split}\frac{d}{dt}\int_{\Sigma_{t}}d\mathcal{H}^{2}\Big{|}_{t=0}&=\int_{\Sigma}H\phi~{}d\mathcal{H}^{2}+\int_{\Sigma}\text{div}_{\Sigma}(F^{T}_{t}(0,x))~{}d\mathcal{H}^{2},\\\
&=\int_{\Sigma}H\phi~{}d\mathcal{H}^{2}+\int_{\partial\Sigma}\langle
F^{T}_{t}(0,x),\overset{\rightarrow}{\beta}\rangle
d\mathcal{H}^{1}\quad\quad\quad\text{By Stokes's theorem}\\\
&=\int_{\Sigma}H\phi~{}d\mathcal{H}^{2}+\int_{\partial\Sigma}\langle
F_{t}(0,x),\overset{\rightarrow}{\beta}\rangle\phi
d\mathcal{H}^{1}\quad\quad\quad\text{since}\langle
F^{N}_{t}(0,x),\overset{\rightarrow}{\beta}\rangle=0.\end{split}$

Equation (3.12) shows that
$\frac{d}{dt}\int_{\Omega_{t}}h~{}d\mathcal{H}^{2}\Big{|}_{t=0}=\int_{\Sigma}h~{}d\mathcal{H}^{2}$.
Thus, we can conclude that

$\frac{d}{dt}\mathcal{A}(\Omega_{t})\Big{|}_{t=0}=\int_{\Sigma}(H-h)\phi~{}d\mathcal{H}^{2}+\int_{\partial\Sigma}\langle
F_{t}(0,x),\overset{\rightarrow}{\beta}\rangle\phi d\mathcal{H}^{1}.$

Since $\Omega$ is a critical point of $\mathcal{A}$, then
$\frac{d}{dt}\mathcal{A}(\Omega_{t})\Big{|}_{t=0}=0$ for any variation $F$.
That is to say,

  * •

$H=h$ on $\Sigma$

  * •

$\langle F_{t}(0,x),\overset{\rightarrow}{\beta}\rangle=0$ on $\partial\Sigma$
for any variation $F$.

Remark that $F_{t}(0,x)$ belongs to $T_{x}(\partial_{0}M)$ at
$x\in\partial\Sigma$. The second fact implies that
$\overset{\rightarrow}{\beta}$ is orthogonal to $\partial_{0}M$. Namely,
$\Sigma$ meets $\partial_{0}M$ orthogonally. ∎

######  Lemma 3.7.

Let $M$ and $\mathcal{A}$ be defined in Section 3.3 and $\Omega$ a minimizer
of $\mathcal{A}$. If $\Sigma$ is a component of $\partial^{*}\Omega$, then for
any $\phi\in C^{1}(\Sigma)$, one has that

$\begin{split}\int_{\Sigma}&(|\nabla_{\Sigma}\phi|^{2}+K_{\Sigma}\phi^{2})~{}d\mathcal{H}^{2}+\int_{\partial\Sigma}k_{g}\phi^{2}~{}d\mathcal{H}^{1}-\frac{1}{2}\int_{\Sigma}|A|^{2}\phi^{2}d\mathcal{H}^{2}\\\
&-\int_{\Sigma}(\frac{h^{2}}{2}+\langle\nabla_{M}h,\overset{\rightarrow}{\mathbf{n}}_{\Sigma}\rangle+\kappa_{M})\phi^{2}~{}d\mathcal{H}^{2}\geq
0,\end{split}$ (3.21)

where $k_{g}$ is the geodesic curvature of $\partial\Sigma$ in $\Sigma$.

###### Proof.

Let $F$ be a variation as in the proof of Lemma 3.6. We may assume

$F_{t}(0,x)=\left\\{\begin{array}[]{cc}0,&\quad\text{on}\quad\partial^{*}\Omega\setminus\Sigma;\\\
\phi(x)\overset{\rightarrow}{\mathbf{n}}_{\Sigma}(x),&\quad\text{on}\quad\Sigma,\end{array}\right.$

where $\phi$ is a smooth function on $\Sigma$. As in the proof of Lemma 3.3,
we have that

$\begin{split}\frac{d^{2}}{dt^{2}}\int_{\Sigma_{t}}d\mathcal{H}^{2}\Big{|}_{t=0}&=\int_{\Sigma}\\{|\nabla_{\Sigma}\phi|^{2}-(\text{Ric}(\overset{\rightarrow}{\mathbf{n}}_{\Sigma})+|A|^{2})\phi^{2}+H^{2}\phi^{2}\\}d\mathcal{H}^{2}+\int_{\Sigma}\text{div}_{\Sigma}(F_{tt}(0,x))d\mathcal{H}^{2},\\\
&=\int_{\Sigma}\\{|\nabla_{\Sigma}\phi|^{2}-(\text{Ric}(\overset{\rightarrow}{\mathbf{n}}_{\Sigma})+|A|^{2})\phi^{2}+H^{2}\phi^{2}\\}d\mathcal{H}^{2}+\int_{\partial\Sigma}\langle
F_{tt}(0,x),\overset{\rightarrow}{\beta}\rangle d\mathcal{H}^{1}\\\
&=\int_{\Sigma}\\{|\nabla_{\Sigma}\phi|^{2}-(\text{Ric}(\overset{\rightarrow}{\mathbf{n}}_{\Sigma})+|A|^{2})\phi^{2}+H^{2}\phi^{2}\\}d\mathcal{H}^{2}+\int_{\partial\Sigma}\langle\nabla_{\overset{\rightarrow}{\mathbf{n}}_{\Sigma}}\overset{\rightarrow}{\mathbf{n}}_{\Sigma},\overset{\rightarrow}{\beta}\rangle\phi^{2}d\mathcal{H}^{1}\end{split}$

Lemma 3.6 shows that $\Sigma$ meets $\partial_{0}M$ orthogonally. Namely,
$\langle\overset{\rightarrow}{\mathbf{n}}_{\Sigma}(x),\overset{\rightarrow}{\mathbf{n}}_{\partial_{0}M}(x)\rangle=0$
on $\partial\Sigma$. Thus, for $x\in\partial\Sigma$, $\vec{\beta}(x)$ is the
unit normal vector $\overset{\rightarrow}{\mathbf{n}}_{\partial_{0}M}(x)$ of
$\partial_{0}M$ and $T_{x}(\partial_{0}M)$ can be spanned by
$\overset{\rightarrow}{\alpha}(x)$ and
$\overset{\rightarrow}{\mathbf{n}}_{\Sigma}(x)$. We have that

$\begin{cases}H_{\partial_{0}M}(x)=\langle\overset{\rightarrow}{\mathbf{n}}_{\Sigma},\nabla_{\overset{\rightarrow}{\mathbf{n}}_{\Sigma}}\overset{\rightarrow}{\beta}\rangle+\langle\overset{\rightarrow}{\alpha},\nabla_{\overset{\rightarrow}{\alpha}}\overset{\rightarrow}{\beta}\rangle=0,\quad
x\in\partial\Sigma.\\\
\langle\overset{\rightarrow}{\alpha},\nabla_{\overset{\rightarrow}{\alpha}}\overset{\rightarrow}{\beta}\rangle=-\langle\overset{\rightarrow}{\beta},\nabla_{\overset{\rightarrow}{\alpha}}\overset{\rightarrow}{\alpha}\rangle\quad\text{and}\quad\langle\overset{\rightarrow}{\mathbf{n}}_{\Sigma},\nabla_{\overset{\rightarrow}{\mathbf{n}}_{\Sigma}}\overset{\rightarrow}{\beta}\rangle\quad=-\langle\overset{\rightarrow}{\beta},\nabla_{\overset{\rightarrow}{\mathbf{n}}_{\Sigma}}\overset{\rightarrow}{\mathbf{n}}_{\Sigma}\rangle,\\\
k_{g}=-\langle\nabla_{\overset{\rightarrow}{\alpha}}\overset{\rightarrow}{\alpha},\overset{\rightarrow}{\beta}\rangle.\end{cases}$

Thus,
$\int_{\partial\Sigma}\langle\nabla_{\overset{\rightarrow}{\mathbf{n}}_{\Sigma}}\overset{\rightarrow}{\mathbf{n}}_{\Sigma},\overset{\rightarrow}{\beta}\rangle\phi^{2}d\mathcal{H}^{1}=\int_{\partial\Sigma}k_{g}\phi^{2}d\mathcal{H}^{1}$.
Equation (3.15) gives that

$\frac{d^{2}}{dt^{2}}\int_{\Omega_{t}}h~{}d\mathcal{H}^{2}\Big{|}_{t=0}=\int_{\Sigma}\langle\nabla_{M}h,\overset{\rightarrow}{\mathbf{n}}_{\Sigma}\rangle\phi^{2}+hH\phi^{2}~{}d\mathcal{H}^{2}.$

Since $\Omega$ is a minimizer of $\mathcal{A}$, then $H=h$ on $\Sigma$ and the
following inequality holds:

$\begin{split}\frac{d^{2}}{dt^{2}}\mathcal{A}(\Omega_{t})\big{|}_{t=0}=&\int_{\Sigma}\\{|\nabla_{\Sigma}\phi|^{2}-(\text{Ric}(\overset{\rightarrow}{\mathbf{n}}_{\Sigma})+|A|^{2})\phi^{2}-\langle\nabla_{M}h,\overset{\rightarrow}{\mathbf{n}}_{\Sigma}\rangle\phi^{2}\\}~{}d\mathcal{H}^{2}\\\
&+\int_{\partial\Sigma}k_{g}\phi^{2}~{}d\mathcal{H}^{1}\\\
&=\int_{\Sigma}(|\nabla_{\Sigma}\phi|^{2}+K_{\Sigma}\phi^{2})~{}d\mathcal{H}^{2}+\int_{\partial\Sigma}k_{g}\phi^{2}~{}d\mathcal{H}^{1}-\int_{\Sigma}\frac{1}{2}|A|^{2}\phi^{2}~{}d\mathcal{H}^{2}\\\
&-\int_{\Sigma}(\kappa_{M}+h^{2}+\langle\nabla_{M}h,\overset{\rightarrow}{\mathbf{n}}_{\Sigma}\rangle)\phi^{2}~{}d\mathcal{H}^{2}\\\
&\geq 0.\end{split}$

∎

###  3.4. Topology of stable $\mu$-surfaces and the Separation Lemma

Let $(M,g)$ be a Riemannian manifold assumed as in Section 3.3. If the
distance $d(\partial_{+}M,\partial_{-}M)>4\pi$, there is a smooth function
$\tau$ on $M$ so that

$\begin{cases}\tau(x)=0\quad
x\in\partial_{-}M,\quad\text{and}\quad\tau(x)=4\pi\quad x\in\partial_{+}M,\\\
|\nabla_{M}\tau|(x)\leq 1\quad x\in M.\end{cases}$

The construction of $\tau$ comes from the distance function
$d(\cdot,\partial_{-}M)$.

######  Remark 3.8.

Let $(M,g)$ and $\rho$ be assumed as above. Consider the function

$h:=\tan(\frac{\tau(x)-2\pi}{4}).$

Thus $h(x)=+\infty$ for $x\in\partial_{+}M$ and $h(x)=-\infty$ for
$x\in\partial_{-}M$.

If the scalar curvature $\kappa\geq 1$, then we have that

$\kappa_{M}+\frac{h^{2}}{2}-|\nabla h|\geq
1+\frac{1}{2}\tan^{2}(\frac{\tau-2\pi}{4})-\frac{1}{4}|\nabla\rho|\cos^{-2}(\frac{\tau-2\pi}{4})\\\
\geq 1/2>0$

######  Corollary 3.9.

(1) Let $(M,g)$ and $\mathcal{A}$ be assumed in Sections 3.1 and 3.2 with
$d(\partial_{+}M,\partial_{-}M)>4\pi$ and $h$ chosen as above. If
$\kappa_{M}\geq 1$ and $\Sigma$ is a component of the reduced boundary
$\partial^{*}\Omega$, then $\Sigma$ is a $2$-sphere, where $\Omega$ is a
minimizer of $\mathcal{A}$.

(2) Let $(M,g)$ and $\mathcal{A}$ be assumed in Section 3.3 with
$d(\partial_{+}M,\partial_{-}M)>4\pi$ and $h$ chosen as above. If
$\kappa_{M}\geq 1$ and $\Sigma$ is a component of $\partial^{*}\Omega$, then
$\Sigma$ is a disc or a $2$-sphere, where $\Omega$ is a minimizer of the free
boundary functional $\mathcal{A}$.

###### Proof.

(1) Since the scalar curvature $\kappa_{M}\geq 1$, then the choice of $h$
gives that (see Remark 3.8):

$\kappa_{M}+\frac{h^{2}}{2}-|\nabla h|>0$

Lemma 3.3 and Remark 3.4 (see Equation (3.18)) show that for any smooth
function $\phi$ on $\Sigma$,

$\int_{\Sigma}|\nabla_{\Sigma}\phi|^{2}+K_{\Sigma}\phi^{2}-\frac{1}{2}|A|^{2}\phi^{2}-(\kappa_{M}+\frac{h^{2}}{2}+\langle\nabla_{M}h,\overset{\rightarrow}{\mathbf{n}}_{\Sigma}\rangle)\phi^{2}~{}d\mathcal{H}^{2}\geq
0$

We now choose $\phi=1$ and get the following:

$\chi(\Sigma)=\int_{\Sigma}K_{\Sigma}~{}d\mathcal{H}^{2}\geq\int_{\Sigma}(\kappa_{M}+\frac{h^{2}}{2}+\langle\nabla_{M}h,\overset{\rightarrow}{\mathbf{n}}_{\Sigma}\rangle)~{}d\mathcal{H}^{2}+\frac{1}{2}\int_{\Sigma}|A|^{2}~{}d\mathcal{H}^{2}>0.$

Namely, $\Sigma$ is a $2$-sphere.

(2) As mentioned above, we also have that $\kappa_{M}+\frac{h^{2}}{2}-|\nabla
h|>0$. If $\Sigma$ is closed (without boundary), the above stability
inequality also holds for $\Sigma$. By a similar argument, we have that
$\Sigma$ is a $2$-sphere.

It remains to show the case when $\Sigma$ has non-empty boundary. We use
Equation (3.21) to have that for a smooth function $\phi$ on $\Sigma$,

$\begin{split}\int_{\Sigma}&|\nabla_{\Sigma}\phi|^{2}+K_{\Sigma}\phi^{2}~{}d\mathcal{H}^{2}+\int_{\partial\Sigma}k_{g}\phi^{2}~{}d\mathcal{H}^{1}-\frac{1}{2}\int_{\Sigma}|A|^{2}\phi^{2}d\mathcal{H}^{2}\\\
&-\int_{\Sigma}(\frac{h^{2}}{2}+\langle\nabla_{M}h,\overset{\rightarrow}{\mathbf{n}}_{\Sigma}\rangle+\kappa_{M})\phi^{2}~{}d\mathcal{H}^{2}\geq
0.\end{split}$

We choose $\phi=1$ and get that

$\chi(\Sigma)=\int_{\Sigma}K_{\Sigma}~{}d\mathcal{H}^{2}+\int_{\partial\Sigma}k_{g}~{}d\mathcal{H}^{1}\geq\int_{\Sigma}(\kappa_{M}+\frac{h^{2}}{2}+\langle\nabla_{M}h,\overset{\rightarrow}{\mathbf{n}}_{\Sigma}\rangle)~{}d\mathcal{H}^{2}+\frac{1}{2}\int_{\Sigma}|A|^{2}~{}d\mathcal{H}^{2}>0.$

Thus, we can conclude that $\Sigma$ is a disc. ∎

We use Corollary 3.9 to have the following separation lemma.

######  Lemma 3.10.

(The separation lemma) Let $(M,g)$ be a compact and orientable $3$-manifold
with boundary $\partial M$. Assume that $\partial M$ satisfies the following

  * •

$\partial M=\partial_{+}M\amalg\partial_{-}M\amalg\partial_{0}M$;

  * •

$d(\partial_{-}M,\partial_{+}M)>4\pi$;

  * •

$\partial_{0}M$ is minimal (i.e. its mean curvature vanishes) and a disjoint
union of $2$-spheres.

Then, if the scalar curvature $\kappa_{M}\geq 1$, there is a finite collection
$\\{S_{i}\\}_{i}$ of embedded $2$-spheres in $\text{Int}~{}~{}M$ satisfying
the following:

  * •

$\\{S_{i}\\}_{i}$ are disjoint;

  * •

$M\setminus\amalg_{i}S_{i}$ has no component intersecting both $\partial_{-}M$
and $\partial_{+}M$.

###### Proof.

As above, there is a smooth function $h$ on
$\text{Int}~{}~{}M\cup\partial_{0}M$ so that (see Remark 3.8)

$\begin{cases}\lim\limits_{x\rightarrow\partial_{-}M}h(x)=-\infty\quad\text{and}\quad\lim\limits_{x\rightarrow\partial_{+}M}h(x)=+\infty,\\\
\kappa_{M}+\frac{h^{2}}{2}-|\nabla h|>0\end{cases}$

By Proposition 3.5, there exists $\Omega$ with
$\partial^{*}\Omega\subset\text{Int}~{}~{}M^{\prime}$ minimizing the free
boundary functional

$\mathcal{A}(\Omega)=\mathcal{H}^{2}(\partial^{*}\Omega)-\int_{M}(\chi_{\Omega}-\chi_{\Omega_{0}})h~{}d\mathcal{H}^{3},$

where $\Omega_{0}:=\tau^{-1}([0,2\pi])$ and $\tau$ is a smooth function
defined above and
$\Omega\Delta\Omega_{0}\Subset\text{Int}~{}~{}M^{\prime}\cup\partial_{0}M^{\prime}$.
The reduced boundary
$\partial^{*}\Omega\subset\text{Int}~{}~{}M\cup\partial_{0}M$ has finitely
many components, $\\{\Sigma_{i}\\}$. The space $M\setminus\amalg\Sigma_{i}$
has no component intersecting both $\partial_{+}M$ and $\partial_{-}M$.

Corollary 3.9 shows that each $\Sigma_{i}$ is a disc or a $2$-sphere. The
collection $\\{S^{\prime}_{i}\\}$ is constructed as follows:

  * •

If $\Sigma_{i}$ is a sphere, we define $S^{\prime}_{i}$ as $\Sigma_{i}$.

  * •

If $\Sigma_{i}$ is a disc with $\partial\Sigma_{i}\subset\partial_{0}M$, we
let $S^{\prime}_{i}$ denote $\Sigma_{i}\cup_{\partial\Sigma_{i}}D_{i}$, where
$D_{i}\subset\partial_{0}M$ is an embedded disc with $\partial\Sigma_{i}$.

The existence of $D_{i}$ follows from the fact that $\partial_{0}M$ is a
disjoint union of $2$-spheres.

Remark that since $D_{i}$ may be a subset of $D_{i^{\prime}}$ for some
$i^{\prime}$, $S^{\prime}_{i}$ may intersect $S^{\prime}_{i}$. If necessary,
we deform $\\{S^{\prime}_{i}\\}_{i}$ to get a disjoint collection
$\\{S_{i}\\}_{i}$ in $\text{Int}~{}~{}M$.

We also have that $M\setminus\amalg_{i}S_{i}$ has no component intersecting
both $\partial_{+}M$ and $\partial_{-}M$. The collection $\\{S_{i}\\}_{i}$ is
the required candidate in our assertion. ∎

###  3.5. Prime decomposition

We now use $\mu$-bubbles to complete the proof for Theorem 1.3.

###### Proof.

Let $(M,g)$ be a complete orientable $3$-manifold with scalar curvature
$\kappa\geq 1$. Fix a point $x\in M$ and $\rho(\bullet):=d(\bullet,x)$. For
our connivence, we may assume that $\rho^{-1}(8\pi k)$ is a smooth surface for
each $k\in\mathbf{Z}_{+}$. We define that $k\in\mathbf{Z}_{+}$

$\begin{cases}Y_{k}:=\rho^{-1}([8\pi k,8\pi(k+1)])\\\
\partial_{+}Y_{k}:=\rho^{-1}(8\pi k)\\\
\partial_{-}Y_{k}:=\rho^{-1}(8\pi(k+1)).\end{cases}$

One has that $\partial Y_{k}=\partial_{-}Y_{k}\amalg\partial_{+}Y_{k}$ and
$d(\partial_{-}Y_{k},\partial_{+}Y_{k})>4\pi$ for $k\in\mathbf{Z}_{+}$.

Corollary 3.10 gives a finite collection $\\{S_{i}\\}_{i\in I_{k}}$ of
disjoint $2$-spheres so that $Y_{k}\setminus\amalg_{i\in I_{k}}S_{i}$ has no
component intersecting both $\partial_{+}Y_{k}$ and $\partial_{-}Y_{k}$. We
have that

  * •

The family, $\\{S_{i}\\}_{i\in I}$, is locally finite, where
$I:=\amalg_{k\in\mathbb{Z}_{+}}I_{k}$ (i.e. a compact set intersects (at most)
finitely many elements in $\\{S_{i}\\}_{i\in I}$);

  * •

$\\{S_{i}\\}_{i\in I}$ cut $M$ into some connected $3$-manifolds,
$\\{X_{l}\\}$;

  * •

the boundary of $X_{l}$ consists of $2$-spheres

We can conclude that each $X_{l}$ is contained in $\rho^{-1}([8\pi
k,8\pi(k+2)])$ for some $k\in\mathbb{Z}$. If not, one of components of
$Y_{k}\cap X_{l}$ is a component of $Y_{k}\setminus\amalg_{i\in I_{k}}S_{i}$
intersecting both $\partial_{-}Y_{k}$ and $\partial_{+}Y_{k}$ for some $k$ and
$l$, which leads to a contradiction with the above fact.

Let $\hat{X}_{l}$ be a closed $3$-manifold obtained from $X_{l}$ by gluing
some $3$-balls along $\partial X_{l}$. The prime decomposition (see Theorem
1.5 of [Hat00] on Page 5) gives a finite collection $\\{S_{i}\\}_{i\in
I^{\prime}_{l}}$ of disjoint $2$-spheres so that each component of
$\hat{X}_{l}\setminus\amalg_{i\in I^{\prime}_{l}}S_{i}$ is homeomorphic to a
closed prime $3$-manifold with finitely many punctures.

We may assume that each $S_{i}$ is contained in $\text{Int}~{}~{}X_{l}$ for
each $i\in I^{\prime}_{l}$. For our convenience, we write $\\{S_{e}\\}_{e\in
E}$, $\\{M_{v}\\}_{v\in V}$ for $\\{S_{i}\\}_{i\in
I}\amalg_{l}\\{S_{i}\\}_{i\in I^{\prime}_{l}}$ and the components of
$M\setminus\amalg_{e\in E}S_{e}$. Each $M_{v}$ is homeomorphic to a closed
prime $3$-manifold with finitely many punctures.

There is a locally finite graph $G:=(V,E)$ constructed as follows:

  * •

Two distinct vertices, $v,v^{\prime}\in V$ are connected by the edge $e\in E$
if both $\partial M_{v}$ and $\partial M_{v^{\prime}}$ contain the $2$-sphere
$S_{e}$;

  * •

the edge $e\in E$ connects the vertex $v$ with itself, if $\partial M_{v}$ has
two components coming from $S_{e}$.

Thus, $M$ is an infinite connected sum of some closed prime $3$-manifolds. ∎

We use free boundary functional to generalize the result to the manifold with
boundary.

######  Corollary 3.11.

Let $(M,g)$ be an orientable $3$-manifold with scalar curvature $\kappa\geq
1$. If $\partial M$ is a disjoint union of some $2$-spheres and minimal for
$g$, then $M$ is an infinite connected sum of some closed prime $3$-manifolds
and $3$-balls.

Based on the work of Bär and Hanke [BH20], Carlotto and Li [CL19], it can be
generalized to mean convex boundary. See Theorem 7.7

##  4\. Minimal surfaces and their existence

In this section, we talk about minimal surfaces theory and the existence
theory of minimal $2$-spheres and discs.

###  4.1. Minimal surfaces and a metric inequality

The geometry of minimal surface is constrained by the scalar curvature (see
[SY83] and [GL83]). We begin with a metric inequality:

######  Proposition 4.1.

(See Theorem 2 of [SY83] on Page 576 and Theorem 1 of [Ros06] on Page 228) Let
$(M^{3},g)$ be a 3-manifold and $(\Sigma,\partial\Sigma)\subset(M,\partial M)$
a stable minimal surface. If the scalar curvature $\kappa\geq 1$, then one has
that, for any point $x\in\Sigma$,

$d(x,\partial\Sigma)\leq\frac{2\pi}{\sqrt{3}}.$

where $d$ is the distance function on $(M,g)$.

######  Corollary 4.2.

Let $(M,g)$ be a $3$-manifold with boundary and
$(\Sigma,\partial\Sigma)\subset(M,S_{1}\amalg S_{2})$ a connected stable
minimal surface with $\partial\Sigma\cap S_{j}\neq\emptyset$ for $j=1,2$,
where $S_{1}\amalg S_{2}\subset\partial M$. If the scalar curvature
$\kappa\geq 1$, then $d(S_{1},S_{2})\leq\frac{4\pi}{\sqrt{3}}$.

###  4.2. Embedded minimal disc and mean convex boundary

In the following, we will repeatedly use the geometric version of loop lemma.

######  Theorem 4.3.

(See [MY82] and Theorem 6.28 of [CM11] on Page 224) Let $(X^{3},g)$ be a
compact Riemannian 3-manifold whose boundary is mean convex and $\gamma$ a
simple closed curve in $\partial X$ which is null-homotopic in $X$. Then,
$\gamma$ bounds an area-minimizing disc and any such least area disc is
embedded.

The existence of minimal surfaces always requires a geometric condition, mean
convex boundary. To overcome it, we introduce a metric deformation. For a
compact manifold $(X^{3},g)$ with non-empty boundary, we find a new metric
$g_{\epsilon}$ for $\epsilon\geq 0$ so that

  * •

$g_{\epsilon}$ is equal to $g$ on $X\setminus B(\partial X,\epsilon)$;

  * •

the mean curvature of $\partial X$ is non-negative for $g_{\epsilon}$,

where $B(\partial X,\epsilon)$ is the tubular neighborhood of $\partial X$
with radius $\epsilon$.

The metric $g_{\epsilon}$ is constructed as below:

_Let $h(t)$ be a positive smooth function on $\mathbb{R}$ so that $h(t)=1$,
for any $t\in\mathbb{R}\setminus[-\epsilon/2,\epsilon/2]$. Consider the
function $f(x):=h(d(x,\partial X))$ and the metric $g_{\epsilon}:=f^{2}g$.
Under $(X,g_{\epsilon})$, the mean curvature $\hat{H}(x)$ of $\partial X$ is_

$\hat{H}(x)=h^{-1}(0)(H(x)+2h^{\prime}(0)h^{-1}(0))$

_Choosing a function $h$ with $h(0)=2$ and $h^{\prime}(0)>2\max_{x\in\partial
X}|H(x)|+2$, one gets the metric $g_{\epsilon}$ which is the required
candidate in the assertion._

###  4.3. Isotopy classes and Minimal surfaces

We talk about how to use the isotopy class of $2$-spheres to find a minimal
surface.

######  Definition 4.4.

Let $\mathscr{C}_{0}$ be the collection of compact embedded surfaces such that
each component is a $2$-sphere. Given $\Sigma\in\mathscr{C}_{0}$, the _isotopy
class_ $\mathscr{I}^{M}(\Sigma)$ of $\Sigma$ in $M$ is the collection of all
$\Sigma^{\prime}\in\mathscr{C}_{0}$ such that $\Sigma^{\prime}$ is isotopic to
$\Sigma$ via a smooth isotopy $\phi_{\bullet}:[0,1]\times M\rightarrow M$,
where $\phi_{0}=\textbf{1}_{M}$ and each $\phi_{t}$ is a diffeomorphism of
$M$.

When $M$ is non-compact, we also require that for an isotopy $\phi_{\bullet}$,
there is a compact set $K\subset M$ such that $\phi_{t}|_{M\setminus
K}=\textbf{1}_{M\setminus K}$ for each $t$.

######  Theorem 4.5.

(See Theorem 1 and Theorem 1’ in [MSY82]) Let $(M,g)$ be a compact orientable
$3$-manifold with mean convex boundary and $\Sigma\subset M$ an embedded
$2$-sphere. Consider an area-minimizing sequence
$\\{\Sigma_{i}\\}^{\infty}_{i=1}$ in $\mathscr{I}^{M}(\Sigma)$. Then, if
$\Sigma$ is not contractible in $M$, then after passing to a subsequence,
there are positive integers, $R$, $n_{1}$,$\cdots$, $n_{R}$ and pairwise
disjoint stable minimal surfaces $\Sigma^{(1)}$, $\cdots$, $\Sigma^{(R)}$ such
that

$\Sigma_{k}\rightarrow
n_{1}\Sigma^{(1)}+n_{2}\Sigma^{(2)}+\cdots+n_{R}\Sigma^{(R)}$

in the sense of varifolds. Moreover, $g(\Sigma^{(i)})=0$ for $1\leq i\leq R$.

Remark that from [HS88], we have that $\inf_{i}\text{Area}(\Sigma_{i})>0$.

######  Remark 4.6.

We have that

  * •

Each $\Sigma^{(i)}$ is $\mathbb{S}^{2}$ or 1-sided $\mathbb{R}P^{2}$. If
$n_{i}$ is odd, then $\Sigma^{(i)}$ is $\mathbb{S}^{2}$. However, it may be
1-sided $\mathbb{R}P^{2}$ when $n_{i}$ is even. See Pages 624-625 in [MSY82].

  * •

If $\Sigma^{(i)}$ is $\mathbb{S}^{2}$, then $\partial
B(\Sigma^{(i)},\epsilon)$ has two components and each component is a
$2$-sphere isotopic to $\Sigma^{(i)}$ for $\epsilon$ sufficiently small; if
$\Sigma^{(i)}$ is 1-sided $\mathbb{R}P^{2}$ then $\partial
B(\Sigma^{(i)},\epsilon)$ is a smooth $2$-sphere.

  * •

If $\Sigma^{(i)}$ is 1-sided $\mathbb{R}P^{2}$, the boundary
$\tilde{\Sigma}^{(i)}$ of the closure of
$(M\setminus\Sigma^{(i)},g|_{M\setminus\Sigma^{(i)}})$, coming from
$\Sigma^{(i)}$, is a stable minimal $2$-sphere.

  * •

The boundaries of
$(M\setminus\coprod\limits^{R}_{i=1}\Sigma^{(i)},g|_{M\setminus\coprod\limits^{R}_{i=1}\Sigma^{(i)}})$,
coming from $\amalg^{R}_{i=1}\Sigma^{(i)}$, are a disjoint union of stable
minimal $2$-spheres.

###  4.4.  $\gamma$-Reduction

In this part, we study the minimal surfaces (introduced by Theorem 4.5) and
its relationship with $\mathscr{I}^{M}(\Sigma)$. Their relationship is
clarified by the so-called $\gamma$-reduction. See the general definition in
[MSY82].

######  Definition 4.7.

(See Pages 628-629 in [MSY82])Given $\Sigma_{1},\Sigma_{2}\in\mathscr{C}_{0}$,
we say $\Sigma_{2}$ is a _$\gamma$ -reduction_ of $\Sigma_{1}$, if the
following conditions are satisfied:

  * (1)

$\Sigma_{1}\setminus\Sigma_{2}$ has clousure $A$ diffeomorphic to the closed
annulus;

  * (2)

$\Sigma_{2}\setminus\Sigma_{1}$ has closure consisting of two disc, $D_{1}$
and $D_{2}$;

  * (3)

$\partial A=\partial D_{1}\cup\partial D_{2}$ and
$\text{Area}(A)+\text{Area}(D_{1})+\text{Area}(D_{2})\leq\gamma$;

  * (4)

the sphere $A\cup D_{1}\cup D_{2}$ bounds a $3$-ball $Y$ with
$\text{Int}~{}~{}Y\cap(\Sigma_{1}\cup\Sigma_{2})=\emptyset$;

  * (5)

The set $\Sigma^{*}_{1}\setminus A$ is not connected and each component has
area $\geq\delta^{2}/2$,

where $\delta$ and $\gamma$ only depend on $(M,g)$ and $\Sigma^{*}_{1}$ is the
component of $\Sigma_{1}$ containing $A$.

$\Sigma^{*}_{1}$$\Sigma^{*}_{1}$$A$$Y$$A$$\Sigma_{2}$$\Sigma_{2}$${D_{1}}$$D_{2}$
Figure 5.

Remark that since $\Sigma^{*}_{1}$ is a sphere, each component of
$\Sigma^{*}_{1}\setminus A$ is simply-connected.

######  Remark 4.8.

(See Remark 3.27 of [MSY82] on Page 635) Let $\Sigma_{k}$ and $\Sigma^{(i)}$
be as in Theorem 4.5. For $k$ large enough and $1\leq i\leq R$, we define that

$S^{(i)}_{k}=\left\\{\begin{array}[]{cc}\amalg^{m_{i}}_{l=1}\partial
B(\Sigma^{(i)},{l}/{k}),&\text{if}~{}n_{i}=2m_{i}~{}\text{is
even}\\\\[8.53581pt] \Sigma^{(i)}\cup(\amalg^{m_{i}}_{l=1}\partial
B(\Sigma^{(i)},{l}/{k})),&\text{if}~{}n_{i}=2m_{i}+1~{}\text{is
odd}.\end{array}\right.$

For $k$ large enough, there is a sequence of $\gamma$-reductions from
$\Sigma_{k}$. After doing a sequence of $\gamma$-reductions, we get a finite
collection $\tilde{\Sigma}_{k}$ of disjoint $2$-spheres, which is isotopic to
$S_{k}:=\amalg^{R}_{i=0}S^{(i)}_{k}$ in $M$, where for $k$ sufficiently large,
each sphere in $S^{(0)}_{k}$ is homotopically trivial in $M$.

###  4.5. Irreducible manifolds and $\gamma$-reduction

In the following, we let $(M,g)$, $\Sigma$, $\\{\Sigma_{k}\\}$ and
$\Sigma^{(i)}$ be assumed in Theorem 4.5. We now consider the case when
$\Sigma$ (assumed as in Theorem 4.5) bounds a compact manifold $M_{0}\subset
M$, homeomorphic to an irreducible $3$-manifold with a puncture.

######  Definition 4.9.

Let $X$ be a compact $3$-manifold whose boundary is a union of $2$-spheres.
Its boundary filling $\hat{X}$ is a closed 3-manifold obtained from $X$ by
filling the boundary with some 3-balls.

######  Lemma 4.10.

Let $\Sigma_{1}$ and $\Sigma_{2}$ be two finite collections of disjoint
$2$-spheres in a compact $3$-manifold $(M,g)$ (possibly with boundary), where
$\Sigma_{2}$ is a $\gamma$-reduction of $\Sigma_{1}$. If $\Sigma_{1}$
satisfies that

  * •

$\Sigma_{1}$ bounds a compact $3$-manifold $M_{1}\subset M$;

  * •

the boundary filling $\hat{M}_{1}$ of $M_{1}$ is irreducible and
$\pi_{1}(\hat{M}_{1})\neq\\{1\\}$,

then there is a compact component of $M\setminus\Sigma_{2}$ whose boundary
filling is homeomorphic to $\hat{M}_{1}$.

###### Proof.

Let $A$, $Y$, $D_{1}$, $D_{2}$ and $\Sigma^{*}_{1}$ be as in Definition 4.7.
We assume that $\Sigma^{i}_{2}$ is the component of $\Sigma_{2}$ containing
$D_{i}$ for $i=1,2$. Thus,
$\Sigma_{1}\setminus\Sigma^{*}_{1}=\Sigma_{2}\setminus\\{\Sigma^{i}_{2}\\}_{i=1,2}$.

Since $\text{Int}~{}Y\cap\Sigma_{1}=\emptyset$, we have that that
$\text{Int}~{}Y\subset M_{1}$ or $\text{Int}~{}Y\cap M_{1}=\emptyset$. In the
following, we will discuss two different cases:

Case I: If $\text{Int}~{}Y\subset M_{1}$, then $\Sigma_{2}$ is a subset of
$\overline{M}_{1}$. The set $\Sigma_{2}\subset\overline{M}_{1}$ can be
considered as a subset of $\hat{M}_{1}$. The spheres $\Sigma_{2}$ gives a
prime decomposition for $\hat{M}_{1}$. Since $\hat{M}_{1}$ is irreducible and
$\pi_{1}(\hat{M}_{1})\neq\\{1\\}$, we use the uniqueness of the prime
decomposition to find a unique component $M_{2}\subset M_{1}$ of
$\hat{M}_{1}\setminus\Sigma_{2}$ with $\pi_{1}(M_{2})\neq\\{1\\}$. Its
boundary filling is homeomorphic to $\hat{M}_{1}$.

Case II: If $\text{Int}~{}Y\cap M_{1}=\emptyset$,
$M_{2}:=M_{1}\cup_{A}\text{Int}~{}Y$ is a component of $M\setminus\Sigma_{2}$
(becasuse
$\Sigma_{1}\setminus\Sigma^{*}_{1}=\Sigma_{2}\setminus\\{\Sigma^{i}_{2}\\}_{i=1,2}$).

Consider the manifold
${M}^{\prime}_{2}:=M_{2}\bigcup_{i=1,2}(\cup_{\Sigma^{i}_{2}}B_{i})$, where
$B_{i}$ is a $3$-ball. The set $B:=B_{1}\cup_{D_{1}}Y\cup_{D_{2}}B_{2}$ is a
$3$-ball with boundary $\Sigma^{*}_{1}$. Thus, ${M}^{\prime}_{2}$ is
homeomorphic to $M_{1}\cup_{\Sigma^{*}_{1}}B$. The sets $M^{\prime}_{2}$ and
$\Sigma_{2}$ can be viewed as a subset of $\hat{M}_{1}$. We use the argument
in last paragraph to find a required component in $M\setminus\Sigma_{2}$. ∎

We now use Lemma 4.10 to study the spheres in Theorem 4.5.

######  Proposition 4.11.

Let $(M,g)$, $\Sigma$, $\\{\Sigma_{k}\\}$ and $\Sigma^{(i)}$ be as in Theorem
4.5. If $\Sigma$ satisfies that

  * •

the sphere $\Sigma$ bounds a compact $3$-manifold $M_{0}\subset M$;

  * •

the boundary filling $\hat{M}_{0}$ of $M_{0}$ is irreducible and is not
$\mathbb{R}P^{3}$.

then there is a component $M^{\prime}$ of
$M\setminus\amalg^{R}_{i=1}\Sigma^{(i)}$ whose boundary filling is
homeomorphic to $\hat{M}_{0}$.

Since $\Sigma$ is not homotopically trivial in $M$, then ${M_{0}}$ is not a
$3$-ball. Namely, $\hat{M}_{0}$ is not homeomorphic to $\mathbb{S}^{3}$. Each
$\Sigma_{k}\in\mathscr{I}^{M}(\Sigma)$ gives a diffeomorphism $\phi_{k}$ of
$M$ with $\phi_{k}(\Sigma)=\Sigma_{k}$. Thus, it also bounds
$\phi_{k}(M_{0})$, diffeomorphic to $\hat{M}_{0}$ with a puncture.

###### Proof.

Let $\tilde{\Sigma}_{k}$, $m_{i}$ and $S_{k}$ be as in Remark 4.8. For $k$
large enough, each sphere in $S^{(0)}_{k}$ is homotopically trivial in $M$.
For simplifying our proof, we may assume that $S^{(0)}_{k}$ is empty.

Since $\tilde{\Sigma}_{k}$ is obtained by a sequence of $\gamma$-reductions
from $\Sigma_{k}\in\mathscr{I}^{M}(\Sigma)$, we inductively use Lemma 4.10 to
find a component $M_{1}$ of $M\setminus\tilde{\Sigma}_{k}$ whose boundary
filling is homeomorphic to $\hat{M}_{0}$. Since $\tilde{\Sigma}_{k}$ is
isotopic to $S_{k}$ for $k$ large enough, there is a diffeomorphism $\phi$ of
$M$ with $\phi(\tilde{\Sigma}_{k})=S_{k}$. Therefore, there is a component
$M^{\prime}_{1}=\phi(M_{1})$ of $M\setminus S_{k}$ whose boundary filling is
homeomorphic to $\hat{M}_{0}$.

Since each $\Sigma^{(i)}$ is $\mathbb{S}^{2}$ or 1-sided $\mathbb{R}P^{2}$,
then $\text{Int}~{}B(\Sigma^{(i)},{m_{i}}/{k})$ is homeomorphic to
$\mathbb{S}^{2}\times(0,1)$ or $\mathbb{R}P^{3}$ with a puncture for $k$ large
enough. In addition, each component of
$\text{Int}~{}B(\Sigma^{(i)},{m_{i}}/{k})\setminus\Sigma^{(i)}$ is a cylinder.

We have that
$M^{\prime}_{1}\cap\amalg_{i}B(\Sigma^{(i)},{m_{i}}/{k})=\emptyset$. (If not,
$M^{\prime}_{1}\subset\text{Int}~{}B(\Sigma^{(i)},{m_{i}}/{k})$ for some $i$.
Then, from the last paragraph, the boundary filling of $M^{\prime}_{1}$ is
homeomorphic to $\mathbb{S}^{3}$ or $\mathbb{R}P^{3}$, a contradiction.)

Choose $M^{\prime}$ as the component of
$M\setminus\amalg_{i=1}^{R}\Sigma^{(i)}$ containing $M^{\prime}_{1}$. The set
$\text{Int}~{}(M^{\prime}\setminus M^{\prime}_{1})$ is a subset of
$\amalg_{i}~{}\text{Int}~{}(B(\Sigma^{(i)},m_{i}/k)\setminus\Sigma^{(i)})$
(each component is a cylinder). We have that $M^{\prime}$ is homeomorphic to
$M^{\prime}_{1}$ (or $M_{1}$). That is to say, the boundary filling of
$M^{\prime}$ is homeomorphic to $\hat{M}_{0}$. ∎

######  Remark 4.12.

(1) We have that $\pi_{1}({M}^{\prime})$ is isomorphic to $\pi_{1}(M_{0})$.
Since the fundamental group remains unchanged under the $\gamma$-reduction,
then the isomorphism comes from an isotopy of diffeomorphism of $M$ .

(2) We have that $\pi_{1}(M^{\prime})$ is non-trivial. The reason is as
follows:

Since $\Sigma$ is not homotopically trivial in $M$, then $M_{0}$ is not
$3$-ball. Say, $\hat{M}_{0}$ is not simply-connected. From (1), we have that
$\pi_{1}(M^{\prime})\cong\pi_{1}(\hat{M}_{0})\neq\\{1\\}$.

(3) Let $S\subset M^{\prime}$ be an embedded $2$-sphere. Then there is a
unique component of $M^{\prime}\setminus S$ which is homeomorphic to a
punctured $3$-sphere.

The sphere $S$ can be considered as a 2-sphere in the irreducible $3$-manifold
$\hat{M}^{\prime}$, where $\hat{M}^{\prime}$ is the boundary filling of
$M^{\prime}$. It bounds a $3$-ball $B\subset\hat{M}^{\prime}$. Thus, $B\cap
M^{\prime}$ is a punctured $3$-sphere. It is a component of
$M^{\prime}\setminus S$.

It remains to show the uniqueness. Let $M_{1}$ and $M_{2}$ be two components
of $M^{\prime}\setminus S$. If they are two punctured $3$-spheres, then
$\pi_{1}(M_{i})=\\{1\\}$ for $i=1,2$. Van-Kampen’s theorem shows that
$\pi_{1}(M^{\prime})\cong\pi_{1}(M_{1})\ast\pi_{1}(M_{2})$ is trivial, a
contradiction with the fact that $\pi_{1}(M^{\prime})\neq\\{1\\}$.

######  Corollary 4.13.

Let $M^{\prime}$ and $M_{0}$ be assumed as in Lemma 4.11. Then any closed
curve $\gamma\subset M^{\prime}\setminus M_{0}$ is contractible in
$M^{\prime}$.

###### Proof.

Suppose that a closed curve $\gamma\subset M^{\prime}\setminus M_{0}$ is non-
contractible in $M^{\prime}$. Since $\pi_{1}(M^{\prime})$ is isomorphic to
$\pi_{1}(M_{0})$ and the isomorphism comes from the isotopy (see Remark 4.12),
then there is a circle $\gamma^{\prime}\subset M_{0}$ and an annulus $A\subset
M$ so that

  * •

$\gamma^{\prime}$ is not contractible in $M_{0}$;

  * •

$\partial A=\gamma\amalg\gamma^{\prime}$.

We may assume that $\Sigma:=\partial M_{0}$ intersects $A$ transversally and
$A\cap\Sigma$ is a disjoint union of circles $\\{c_{l}\\}_{l}$. The component
$A_{0}$ of $A\cap M_{0}$ containing $\gamma^{\prime}$ is a disc with finitely
punctures with $\partial A_{0}\subset\gamma^{\prime}\amalg_{l}c_{l}$. Since
$\Sigma$ is a $2$-sphere, then each $c_{l}\subset\Sigma$ is contractible in
$\overline{M}_{0}$. So $\gamma^{\prime}$ is contractible in
$\overline{M}_{0}$. However, it is non-contractible in $M_{0}$, a
contradiction. ∎

##  5\. Isotopy classes and Mixed boundary

In this section, we introduce the mixed boundary condition and show the
existence of $3$-submanifold with mixed boundary in a complete $3$-manifold
with scalar curvature $\kappa\geq 1$ (see Theorem 5.3).

######  Definition 5.1.

An orientable $3$-manifold $(X,g)$ has _mixed boundary_ if it satisfies that

  * •

the boundary $\partial X$ consists of $2$-spheres, $\\{S_{j}\\}_{j\in
J^{\prime}\amalg J^{\prime\prime}}$;

  * •

$S_{j}$ is minimal for $g$, where $j\in J^{\prime}$;

  * •

there are disjoint punctured $3$-spheres $\\{\Omega_{l}\\}_{l}$ with
$\amalg_{j\in J^{\prime\prime}}S_{j}\subset\amalg_{l}\partial\Omega_{l}$;

  * •

for each $l$, $\Omega_{l}\cap\amalg_{j\in J^{\prime\prime}}S_{j}$ is non-
empty;

  * •

the set $\hat{S}_{l}:=\partial\Omega_{l}\cap\text{Int}~{}~{}X$ is a $2$-sphere
and $d(\hat{S}_{l},\partial\Omega_{l}\cap\amalg_{j\in
J^{\prime\prime}}S_{j})>4\pi$.

For example, if the boundary of $3$-manifold is a union of $2$-spheres and
minimal , it has mixed boundary.

######  Remark 5.2.

If $(X,g)$ has mixed boundary, then the cover space $(\tilde{X},\tilde{g})$
also has mixed boundary.

######  Theorem 5.3.

Let $(M,g)$ be a complete (non-compact) orientable $3$-manifold with scalar
curvature $\kappa\geq 1$ and $M_{0}\subset M$ a compact subset satisfying that

  * •

the boundary $\Sigma:=\partial M_{0}$ is a $2$-sphere;

  * •

the boundary filling $\hat{M}_{0}$ of $M_{0}$ is irreducible;

  * •

$\hat{M}_{0}$ is neither $\mathbb{S}^{3}$ nor $\mathbb{R}P^{3}$;

Then there is an open $3$-manifold $M^{\prime}\subset M$ so that

  * •

$M^{\prime}$ is homeomorphic to $\hat{M}_{0}$ with finitely many punctures;

  * •

the closure of $(M^{\prime},g|_{M^{\prime}})$ is a compact $3$-manifold with
mixed boundary.

We first construct the required manifold $M^{\prime}$ in Lemma 5.4 and then
show the manifold $(M^{\prime},g|_{M^{\prime}})$ has mixed boundary in Lemma
5.6.

###  5.1. Find the required manifold $M^{\prime}$

In the following, let us consider an orientable $3$-manifold $(M,g)$ and a
subset $K\subset M$. We will use the notions:

  * •

$B^{M}(K,r_{1}):=\\{x~{}|~{}d(x,K)\leq r_{1}\\}$;

  * •

$A^{M}(K,r_{1},r_{2}):=\overline{B^{M}(K,r_{2})\setminus B^{M}(K,r_{1})}$,
where $r_{2}>r_{1}$.

######  Lemma 5.4.

Let $(M,g)$, $\Sigma$ and $M_{0}$ be assumed as in Theorem 5.3. Then there is
an open $3$-manifold $M^{\prime}\subset M$ satisfying that

  * •

$M^{\prime}$ is homeomorphic to $\hat{M}_{0}$ with finitely many punctures;

  * •

the boundary of $(M^{\prime},g|_{M^{\prime}})$ consists of some $2$-spheres
$\\{S_{j}\\}_{j\in J}$.

  * •

Each $S_{j}$ holds one of the following:

    1. (1)

$S_{j}$ is stable minimal for $g$ and $\text{Diam}(S_{j})\leq 2\pi/\sqrt{3}$;

    2. (2)

$d(S_{j},M_{0}\cap M^{\prime})\geq 16\pi$,

where $d$ is the distance function induced from the closure of
$({M}^{\prime},g|_{{M}^{\prime}})$.

######  Remark 5.5.

Let $J^{\prime}:=\\{j\in J~{}|~{}S_{j}$ is stable minimal for $g\\}$ and
$J^{\prime\prime}=J\setminus J^{\prime}$. Proposition 4.1 shows that
$\text{Diam}(S_{j})\leq 2\pi/\sqrt{3}$ for $j\in J^{\prime}$.

In addition, we have that $d(S_{j},M_{0}\cap M^{\prime})\geq 16\pi$ for $j\in
J^{\prime\prime}$.

###### Proof.

Consider the compact set $B^{M}(M_{0},18\pi+\epsilon)$, where $\epsilon>0$. We
may assume that its boundary consists of smooth surfaces. From Section 4.2,
there is a metric $g^{\prime}$ on $B^{M}(M_{0},18\pi+\epsilon)$ so that

  * •

$g^{\prime}$ is equal to $g$ on $B^{M}(M_{0},18\pi)$;

  * •

the mean curvature of $\partial B^{M}(M_{0},18\pi+\epsilon)$ is positive for
$g^{\prime}$.

Thus, $(B^{M}(M_{0},18\pi+\epsilon),g^{\prime})$ has mean convex boundary.
Remark that the scalar curvature of $g^{\prime}$ may be negative on
$A^{M}(M_{0},18\pi,18\pi+\epsilon)$.

We have that $\Sigma$ is not homotopically trivial in $M$. If not, we use
Proposition 2.1 to find a $3$-ball $B\subset M$ with boundary $\Sigma$. Since
$M_{0}$ is not a $3$-ball, then we have that $B\cap M_{0}=\Sigma$. It gives a
closed manifold $M_{0}\cup_{\Sigma}B\subset M$, a contradiction with the fact
that $M$ is connected and non-compact.

We consider an area-minimizing sequence $\\{\Sigma_{k}\\}$ of the isotopy
class $\mathscr{I}^{B^{M}(M_{0},18\pi+\epsilon)}(\Sigma)$ in the manifold
$(B^{M}(M_{0},18\pi+\epsilon),g^{\prime})$. From Theorem 4.5, after passing to
a subsequence, there are positive integers, $R$, $n_{1}$,$\cdots$, $n_{R}$ and
pairwise disjoint stable minimal surfaces $\Sigma^{(1)}$, $\cdots$,
$\Sigma^{(R)}$ such that

$\Sigma_{k}\rightarrow
n_{1}\Sigma^{(1)}+n_{2}\Sigma^{(2)}+\cdots+n_{R}\Sigma^{(R)}$

in the sense of varifolds.

Since $\hat{M}_{0}$ is irreducible and $\hat{M}_{0}$ is not $\mathbb{R}P^{3}$,
then we use Corollary 4.11 to find an open $3$-manifold $M^{\prime}\subset M$
satisfying that it is homeomorphic to $\hat{M}_{0}$ with finitely many
punctures.

Remark that $M^{\prime}$ is one of components of
$B^{M}(M_{0},18\pi+\epsilon)\setminus\amalg_{i=1}^{R}\Sigma^{(i)}$. Its
boundary consists of some $2$-spheres $\\{S_{j}\\}_{j\in J}$ coming from
$\\{\Sigma^{(i)}\\}_{i=1}^{R}$.

It remains to show the property of the boundary $\\{S_{j}\\}_{j\in J}$ of
$(M^{\prime},g|_{M^{\prime}})$. We begin with the properties of
$\\{\Sigma^{(i)}\\}$ and then use the properties to get the information on
$\\{S_{j}\\}_{j\in J}$.

We have that each $\Sigma^{(i)}$ satisfies one of the following:

  * •

$\Sigma^{(i)}$ is stable minimal for $g$ and $\text{Diam}(\Sigma^{(i)})\leq
2\pi/\sqrt{3}$;

  * •

$d_{0}(\Sigma^{(i)},M_{0})\geq 16\pi$.

where $d_{0}$ is the distance function on $(M,g)$.

The reason is as follows: If $\Sigma^{(i)}$ is stable minimal for $g$, we use
Proposition 4.1 and the scalar curvature $\kappa$ of $g$ is not less than $1$
to have that $\text{Diam}(\Sigma^{(i)})\leq 2\pi/\sqrt{3}$.

If not, then $\Sigma^{(i)}$ intersects $A^{M}(M_{0},18\pi,18\pi+\epsilon)$.
Each component $\Sigma^{(i)}_{s}$ of $\Sigma^{(i)}\cap B^{M}(M_{0},18\pi)$ is
stable minimal for the original metric $g$ and
$\partial\Sigma^{(i)}_{s}\subset\partial B^{M}(M_{0},18\pi)$. Proposition 4.1
shows that for $x\in\Sigma^{(i)}_{s}$, $d(x,\partial\Sigma^{(i)}_{s})\leq
2\pi/\sqrt{3}$. We conclude that $d(\Sigma^{(i)},M_{0})\geq 16\pi$.

Since the boundary $\\{S_{j}\\}_{j}$ of $(M^{\prime},g_{M^{\prime}})$ comes
from $\\{\Sigma^{(i)}\\}$, then each $S_{j}$ satisfies one of the following:

  * •

$S_{j}$ is stable minimal for $g$ and $\text{Diam}(S_{j})\leq 2\pi/\sqrt{3}$;

  * •

$d(S_{j},M_{0}\cap M^{\prime})\geq 16\pi$.

The manifold $M^{\prime}$ is the required candidate in our assertion. ∎

###  5.2. Mixed boundary and punctured $3$-spheres

In the following, we show that $(M^{\prime},g|_{M^{\prime}})$ constructed in
Lemma 5.4 has mixed boundary.

######  Lemma 5.6.

Let $(M,g)$ , $\Sigma$, $M_{0}$ be assumed as in Theorem 5.3 and $M^{\prime}$
constructed as in Lemma 5.4. Then $(M^{\prime},g|_{M^{\prime}})$ has mixed
boundary.

###### Proof.

As in the proof of Lemma 5.4, there is an open $3$-manifold $M^{\prime}\subset
M$ satisfying that

  * •

$M^{\prime}$ is homeomorphic to $\hat{M}_{0}$ with finitely many punctures;

  * •

the boundary of $(M^{\prime},g|_{M^{\prime}})$ consists of some $2$-spheres
$\\{S_{j}\\}_{j\in J}$.

  * •

$J^{\prime}:=\\{~{}j\in J~{}|~{}S_{j}$ is stable minimal for $g\\}$ and
$J^{\prime\prime}=:J\setminus J^{\prime}$

  * •

For $j\in J^{\prime}$, $\text{Diam}(S_{j})\leq 2\pi/\sqrt{3}$;

  * •

For $j\in J^{\prime\prime}$, $d(S_{j},M_{0}\cap M^{\prime})\geq 16\pi$;

where $d$ is the distance function induced from the closure of
$(M^{\prime},g|_{M^{\prime}})$.

In the following, we first find a collection of $2$-spheres
$\\{\hat{S}_{l}\\}$ satisfying that no component of
$M^{\prime}\setminus\amalg_{l}\hat{S}_{l}$ meets both $M_{0}\cap M^{\prime}$
and $\amalg_{j\in J^{\prime\prime}}S_{j}$. Then we use these $2$-spheres to
find punctured $3$-spheres and verify the mixed boundary condition.

Step 1: Find the required $2$-spheres $\\{\hat{S}_{l}\\}\subset M^{\prime}$.

Choose a region $Y\subset M$ satisfying that

  * •

$A^{M^{\prime}}(M_{0}\cap M^{\prime},4\pi,8\pi)\subset Y\subset
A^{M^{\prime}}(M_{0}\cap M^{\prime},2\pi,10\pi)$;

  * •

$\partial Y=\partial_{1}Y\amalg\partial_{2}Y\amalg\\{S_{j}\\}_{j\in J_{1}}$,
where $J_{1}\subset J^{\prime}$

  * •

$d(\partial_{1}Y_{1},\partial_{2}Y)\geq 4\pi$.

Let me explain the construction of $Y$. We begin with the set
$Y_{0}=A^{M^{\prime}}(M^{\prime}\cap M_{0},4\pi,8\pi)$. If
$(\partial_{1}Y_{0}\amalg\partial_{2}Y_{0})\cap\\{S_{j}\\}_{j\in
J}=\emptyset$, we choose $Y=Y_{0}$, where $\partial_{1}Y_{0}$ is the closure
of $\partial B^{M^{\prime}}(M_{0}\cap
M^{\prime},8\pi)\cap\text{Int}~{}M^{\prime}$ and $\partial_{2}Y_{0}$ is the
closure of $\partial B^{M^{\prime}}(M_{0}\cap
M^{\prime},4\pi)\cap\text{Int}~{}M^{\prime}$.

If not, the set $J_{0}=:\\{j~{}|~{}S_{j}$ meets
$(\partial_{1}Y_{0}\amalg\partial_{2}Y_{0})~{}j\in J\\}$ is non-empty. Since
$d(S_{j},M_{0}\cap M^{\prime})\leq 8\pi$ for $j\in J_{0}$, then $S_{j}$ is
stable minimal for $g$ and $\text{Diam}(S_{j})\leq 2\pi/\sqrt{3}$, where $j\in
J_{0}$. Namely, $J_{0}\subset J^{\prime}$.

Consider the tubular neighborhood $B^{M^{\prime}}(S_{j},\epsilon_{j})$ of
$S_{j}$ in $M^{\prime}$ satisfying that
$\\{B^{M^{\prime}}(S_{j},\epsilon_{j})\\}_{j\in J_{0}}$ are disjoint, where
$\epsilon_{j}<1/100$. Consider the set

$Y=Y_{0}\cup_{j\in J_{0}}B^{M^{\prime}}(S_{j},\epsilon_{j})\subset
M^{\prime}.$

We have that $B^{M^{\prime}}(S_{j},\epsilon_{j})\subset
A^{M^{\prime}}(M^{\prime}\cap M_{0},2\pi,10\pi)$ for $j\in J_{0}$. Namely,
$Y\subset A^{M^{\prime}}(M^{\prime}\cap M_{0},2\pi,10\pi)$.

$\partial_{2}Y$$\partial_{1}Y$$S_{1}$$S_{2}$$M^{\prime}\cap M_{0}$ Figure 6.

We apply Lemma 3.10 to the manifold $(Y,g|_{Y})$ and find a finite collection
$\\{\hat{S}_{l}\\}$ of disjoint $2$-spheres in $\text{Int}~{}Y$ so that

  * •

no component of $Y\setminus\amalg\hat{S}_{l}$ intersects both $\partial_{1}Y$
and $\partial_{2}Y$;

  * •

$\hat{S}_{l}\subset Y\subset\text{Int}~{}~{}B^{M^{\prime}}(M_{0}\cap
M^{\prime},10\pi)$ for each $l$.

Further, we could conclude that no component of
$M^{\prime}\setminus\amalg_{l}\hat{S}_{l}$ meets both $M_{0}\cap M^{\prime}$
and $\amalg_{j\in J^{\prime\prime}}S_{j}$. If not, there is a component of
$Y\setminus\amalg_{l}\hat{S}_{l}$ meeting both $\partial_{1}Y$ and
$\partial_{2}Y$, a contradiction with the last paragraph.

Step 3: Find disjoint punctured $3$-spheres, $\\{\Omega_{l}\\}_{l}$, in
Definition 5.1.

For each $l$, there is a unique component $\Omega_{l}$ of
$M^{\prime}\setminus\hat{S}_{l}$ which is a punctured $3$-sphere (see Remark
4.12). Since $\hat{S}_{l}\subset\text{Int}~{}~{}B^{M^{\prime}}(M_{0}\cap
M^{\prime},10\pi)$ and $d(S_{j},M_{0}\cap M^{\prime})\geq 16\pi$ for $j\in
J^{\prime\prime}$, then $d(\hat{S}_{l},\partial\Omega_{l}\cap\amalg_{j\in
J^{\prime\prime}}S_{j})>4\pi$.

Claim: For each $l$ and $l^{\prime}$, one of the following holds: (1)
$\Omega_{l}\cap\Omega_{l^{\prime}}=\emptyset$; (2)
$\Omega_{l}\subset\Omega_{l^{\prime}}$; (3)
$\Omega_{l^{\prime}}\subset\Omega_{l}$.

Suppose that $\Omega_{l}\cap\Omega_{l^{\prime}}$ and
$\Omega_{l}\setminus\Omega_{l^{\prime}}$ are both nonempty. Say, there are two
points, $p_{1}\in\Omega_{l}\setminus\Omega_{l^{\prime}}$ and
$p_{2}\in\Omega_{l^{\prime}}\cap\Omega_{l}$. It is sufficient to show that
$\Omega_{l^{\prime}}\subset\Omega_{l}$.

First, $\hat{S}_{l^{\prime}}$ is a subset of $\Omega_{l}$. The reason is as
follows: there is a curve $\gamma\subset\Omega_{l}$ connecting $p_{1}$ and
$p_{2}$. The curve $\gamma$ intersects $\hat{S}_{l^{\prime}}$ at some point.
Namely, $\hat{S}_{l^{\prime}}\cap\Omega_{l}\neq\emptyset$. Since
$\hat{S}_{l}\cap\hat{S}_{l^{\prime}}=\emptyset$, then $\hat{S}_{l^{\prime}}$
is contained in one of components of $M^{\prime}\setminus\hat{S}_{l}$. Thus ,
we have that $\hat{S}_{l^{\prime}}$ is a subset of $\Omega_{l}$.

Remark that $\hat{S}_{l^{\prime}}$ cuts $\Omega_{l}$ into two punctured
$3$-spheres, since $\Omega_{l}$ is a punctured $3$-sphere.

Then, take the component $B$ of $\Omega_{l}\setminus\hat{S}_{l^{\prime}}$ with
$\partial B\cap\hat{S}_{l}=\emptyset$. It is a component of
$M^{\prime}\setminus\hat{S}_{l^{\prime}}$ and a punctured $3$-sphere . From
the uniqueness of $\Omega_{l^{\prime}}$ (see Remark 4.12), we can conclude
that $B$ is equal to $\Omega_{l^{\prime}}$. Namely,
$\Omega_{l^{\prime}}\subset\Omega_{l}$.

It finishes the proof of the claim.

Consider the partially ordered set $(\\{\Omega_{l}\\}_{l},\subset)$. It is a
finite set. In the following, we abuse the notions and write
$\\{\Omega_{l}\\}_{l}$ for the maximal elements in
$(\\{\Omega_{l}\\}_{l},\subset)$. The sets $\\{\Omega_{l}\\}$ are disjoint.

Remark that $M^{\prime\prime}:=M^{\prime}\setminus\amalg_{l}\Omega_{l}$ is a
component of $M^{\prime}\setminus\amalg_{l}\hat{S}_{l}$. Since each
$\Omega_{l}$ is simply-connected, then we use Van-Kampen’s theorem to have
that $\pi_{1}(M^{\prime\prime})\cong\pi_{1}(M^{\prime})\neq\\{1\\}$. The
isomorphism comes from the inclusion.

Step 3: Show that $\amalg_{j\in
J^{\prime\prime}}S_{j}\subset\amalg_{l}\partial\Omega_{l}$.

We have that $M_{0}\cap M^{\prime}\subset\amalg\Omega_{l}$ or $\amalg_{j\in
J^{\prime\prime}}S_{j}\subset\amalg\Omega_{l}$. If not,
$M^{\prime\prime}\subset M\setminus\amalg_{l}\hat{S}_{l}$ meets $\amalg_{j\in
J^{\prime\prime}}S_{j}$ and $M_{0}\cap M$, which is in contradiction with the
fact that no component of $M^{\prime}\setminus\amalg_{l}\hat{S}_{l}$ meeting
both $M_{0}\cap M^{\prime}$ and $\amalg_{j\in J^{\prime\prime}}S_{j}$.

If $M_{0}\cap M^{\prime}\subset\amalg_{l}\Omega_{l}$, we have that
$M^{\prime\prime}$ is a subset of $M^{\prime}\setminus M_{0}$. There is a non-
contractible circle $c\subset M^{\prime\prime}$ in $M^{\prime}$ (see the above
remark). However, Corollary 4.13 shows that $c\subset M^{\prime\prime}\subset
M^{\prime}\setminus M_{0}$ is contractible in $M^{\prime}$, which leads to a
contradiction.

We can conclude that $\amalg_{j\in
J^{\prime\prime}}S_{j}\subset\amalg_{l}\partial\Omega_{l}$. Namely,
$(M^{\prime},g|_{M^{\prime}})$ has mixed boundary. ∎

######  Remark 5.7.

Let $(M,g)$ be an orientable (non-compact) $3$-manifold with scalar curvature
$\kappa\geq 1$ whose boundary is a union of some $2$-spheres and minimal for
$g$. Theorem 5.3 is also true for such $3$-manifolds.

##  6\. Classification of $3$-manifold with mixed boundary

In this section, we classify $3$-manifold with mixed boundary and with
uniformly positive scalar curvature. In the following, we use the notions:

  * •

Assume that $(M,g)$ is a $3$-manifold with mixed boundary;

  * •

The boundary $\partial M$ consists of $2$-spheres, $\\{S_{j}\\}_{j\in J}$;

  * •

$J^{\prime}:=\\{j|$ $S_{j}$ is minimal for $g\\}$ and
$J^{\prime\prime}:=J\setminus J^{\prime}$;

  * •

$\\{\Omega_{l}\\}_{l}$ are disjoint punctured $3$-spheres in $M$ with
$\amalg_{j\in J^{\prime\prime}}S_{j}\subset\amalg_{l}~{}\partial\Omega_{l}$;

  * •

$J_{l}:=\\{~{}j\in J~{}|~{}S_{j}\subset\partial\Omega_{l}\\}$,
$J^{\prime\prime}_{l}:=J_{l}\cap J^{\prime\prime}$ and
$J^{\prime\prime}=\amalg_{l}J^{\prime\prime}_{l}$

  * •

$\hat{S}_{l}:=\partial\Omega_{l}\cap\text{Int}~{}M$ is a $2$-sphere with
$d(\hat{S}_{l},\amalg_{j\in J_{l}^{\prime\prime}}S_{j})>4\pi$,

  * •

$\pi_{1}(\hat{M})\cong\pi_{1}(M)$, where $\hat{M}:=M\cup_{j\in J}B_{j}$ and
$B_{j}$ is a $3$-ball.

######  Remark 6.1.

(1) If $\pi_{2}(M)$ is generated by the spheres in $\partial M$, then
$\pi_{2}(\hat{M})$ is trivial.

(2) Each $\hat{S}_{l}$ can be considered as a subset in $\hat{M}$. Proposition
2.1 shows that it bounds a $3$-ball $\hat{\Omega}_{l}\subset\hat{M}$, where
$\hat{\Omega}_{l}:=\Omega_{l}\bigcup_{j\in J_{l}}(\cup_{S_{j}}B_{j})$.

(3) The $3$-balls, $\\{\hat{\Omega}_{l}\\}_{l}$ and $\\{B_{j}\\}_{j\in
J^{\prime}\setminus\amalg_{l}J_{l}}$ are disjoint in $\hat{M}$.

We now find a new metric $g^{\prime}$ on $M$ so that $(M,g^{\prime})$ has mean
convex boundary.

$S_{j_{1}}$$S_{j_{2}}$$S_{j_{3}}$$S^{\prime}_{j_{1}}$$S^{\prime}_{j_{3}}$$\hat{S}_{l}$$\Omega_{l}$$B(S_{j_{1}},\epsilon_{j_{1}})$$B(S_{j_{3}},\epsilon_{j_{3}})$

Figure 7.

######  Lemma 6.2.

Let $(M^{3},g)$ be a manifold with mixed boundary and $\\{S_{j}\\}_{j\in
J^{\prime}\amalg J^{\prime\prime}}$, $\\{\Omega_{l}\\}_{l}$ and $\hat{S}_{l}$
be assumed as above. Then, there is a metric $g^{\prime}$ on $M$ so that

  * •

$(M,g^{\prime})$ has mean convex boundary;

  * •

For each $j\in J^{\prime\prime}$, there is a neighborhood
$B(S_{j},\epsilon_{j})$ of $S_{j}$ in $M$ so that
$\\{B(S_{j},\epsilon_{j})\\}_{j\in J^{\prime\prime}}$ and $\\{S_{j}\\}_{j\in
J^{\prime}}$ are disjoint;

  * •

$d(S^{\prime}_{j},\hat{S}_{l})>4\pi$ for $j\in J^{\prime\prime}_{l}$;

  * •

$g^{\prime}$ is equal to $g$ on $M\setminus\amalg_{j\in
J^{\prime\prime}}B(S_{j},\epsilon)$,

where $S^{\prime}_{j}:=\partial B(S_{j},\epsilon_{j})\cap\text{Int}~{}M$ (see
Figure 7).

###### Proof.

For each $j\in J^{\prime\prime}$, choose $\epsilon_{j}>0$ small enough so that

  * •

$\\{B(S_{j},\epsilon_{j})\\}_{j\in J^{\prime\prime}}$ and $\\{S_{j}\\}_{j\in
J^{\prime}}$ are disjoint;

  * •

$d(\hat{S}_{l},S^{\prime}_{j})>4\pi$ and
$B(S_{j},\epsilon_{j})\subset\Omega_{l}$ for $j\in J^{\prime\prime}_{l}$,

where $S^{\prime}_{j}=\text{Int}~{}M\cap\partial B(S_{j},\epsilon_{j})$ (see
Figure 7).

We apply the metric deformation in Section 4.2 to each $B(S_{j},\epsilon)$ and
obtain a metric $g^{\prime}$ on $M$ so that

  * •

$S_{j}$ is mean convex for $g^{\prime}$ and $j\in J^{\prime\prime}$;

  * •

$g^{\prime}$ is equal to $g$ on $M^{\prime}\setminus\amalg_{j\in
J^{\prime\prime}}B(S_{j},\epsilon_{j})$.

Namely, $S_{j}$ is also minimal for $g^{\prime}$ and $j\in J^{\prime}$. Thus,
$(M,g^{\prime})$ has mean convex boundary. ∎

Remark that if the scalar curvature of $(M,g)$ is uniformly positive, the
scalar curvature of $g^{\prime}$ may be negative on $B(S_{j},\epsilon_{j})$
for some $j$.

In the following, we study the fundamental group of a $3$-manifold with mixed
boundary and with uniformly positive scalar curvature

######  Lemma 6.3.

Let $(M,g)$ be a $3$-manifold with mixed boundary satisfying that $\pi_{2}(M)$
is generated by the spheres in $\partial M$. If $\pi_{1}(M)$ has a subgroup
which is isomorphic to $\mathbb{Z}$, then the scalar curvature $\kappa$ of
$(M,g)$ can not be bounded below by one.

###### Proof.

We may assume that $\pi_{1}(M)$ is isomorphic to $\mathbb{Z}$. If not, we
choose a cover of $M$ whose fundamental group is isomorphic to $\mathbb{Z}$
and replace $M$ by it. (By Remark 5.2, the cover is also a manifold with mixed
boundary.)

In the following, we assume that $B_{j}$, $\hat{M}$, $\hat{\Omega}_{l}$,
$J_{l}$, $S^{\prime}_{j}$, $M_{1}$ and $g^{\prime}$ are defined as above.
Suppose that the scalar curvature $\kappa$ of $g$ is bounded below by one
(i.e. $\kappa\geq 1$).

We have $\pi_{2}(\hat{M})=\\{1\\}$ (see Remark 6.1). Namely, $\hat{M}$ is
irreducible. Since $\pi_{1}(\hat{M})$ is infinite, the universal cover of
${\hat{M}}$ is contractible (see Corollary 3.9 of [Hat00] on Page 51). Thus,
$\hat{M}$ is a $K(\mathbb{Z},1)$-space. Namely, $\hat{M}$ is homotopically
equivalent to $\mathbb{S}^{1}$.

Let $\gamma\subset M$ be a circle generating $\pi_{1}(\hat{M})$. We may assume
that $\gamma$ is contained in $M\setminus\amalg_{l}\Omega_{l}$ (since each
$\Omega_{l}$ is a punctured $3$-sphere). By Corollary 2.8, there is a closed
curve $\sigma\subset{M}$ so that

  * •

it is contractible in $\hat{M}$ but non-contractible in
$\hat{M}\setminus\gamma$;

  * •

$d(\sigma,\gamma)>4\pi$.

Since there are disjoint $3$-balls, $\\{\hat{\Omega}_{l}\\}$ and
$\\{B_{j}\\}_{j\in J^{\prime}\setminus\amalg_{l}J_{l}}$, we may assume that
$\sigma$ is a subset of
$M\setminus\amalg_{l}\Omega_{l}=\hat{M}\setminus(\amalg_{l}\hat{\Omega}_{l}\cup\amalg_{j\in
J^{\prime}\setminus\amalg_{l}J_{l}}B_{j})$. In addition, we have that

  * •

$\sigma$ is contractible in $M$;

  * •

it is non-contractible in $M\setminus\gamma$.

We choose the metric $g^{\prime}$ constructed from Lemma 6.2. The manifold
$(M,g^{\prime})$ has mean convex boundary. We use the solution of Plateau
problem to find an area-minimizing disc $D\subset M$ with boundary $\sigma$.
It is stable minimal for $g^{\prime}$.

Remark that $g^{\prime}=g$ on $M\setminus\amalg_{j\in
J^{\prime\prime}}B(S_{j},\epsilon_{j})$.

Claim: $D$ is also stable minimal for $g$.

It is sufficient to show that $D$ is a subset of $M\setminus\amalg_{j\in
J^{\prime\prime}}B(S_{j},\epsilon)$. We argue by contradiction. Suppose that
$D$ intersects some $B(S_{j_{0}},\epsilon_{j_{0}})$ for some $j_{0}\in
J^{\prime\prime}_{l}\subset J^{\prime\prime}$. Say,
$D\cap{S}^{\prime}_{j_{0}}\neq\emptyset$.

Since $\sigma=\partial D$ is away from $\Omega_{l}$, then there is a component
$D_{s}$ of $D\cap(\Omega_{l}\setminus\amalg_{j\in
J^{\prime\prime}_{l}}B(S_{j},\epsilon_{j}))$ connecting $\hat{S}_{l}$ and
$\amalg_{j\in J^{\prime\prime}_{l}}S^{\prime}_{j}$.

Since the scalar curvature $\kappa$ of $(M,g)$ $\geq 1$, Corollary 4.2 implies
that $d(\amalg_{j\in J^{\prime\prime}_{l}}S^{\prime}_{j},\hat{S}_{l})\leq
4\pi$. However, $d(\amalg_{j\in
J^{\prime\prime}_{l}}S^{\prime}_{j},\hat{S}_{l})>4\pi$ (see Lemma 6.2), which
leads to a contradiction. It completes the proof of the claim.

We now use the claim to finish the proof. Since $\sigma$ is non-contractible
in $M\setminus\gamma$, then $D\cap\gamma$ is non-empty. Consider the set
$B(\gamma,4\pi):=\\{x\in M~{}|~{}d(\gamma,x)\leq 4\pi\\}$ and a component
$D^{\prime}$ of $D\cap B(\gamma,4\pi)$ intersecting $\gamma$. The boundary
$\partial D^{\prime}$ is a subset $\partial B(\gamma,4\pi)$.

Because $\kappa\geq 1$, Theorem 4.1 gives that

$D^{\prime}~{}\text{is contained in
the}~{}\frac{2\pi}{\sqrt{3}}\text{-neighborhood of }\partial
D^{\prime}~{}\text{in}~{}D^{\prime}.$

Namely, $d(\partial D^{\prime},\gamma)<2\pi$. However, $d(\gamma,\partial
D^{\prime})\geq 4\pi$ (since $\partial D^{\prime}\subset\partial
B(\gamma,4\pi)$), a contradiction. We complete the proof. ∎

In the following, we use Lemma 6.3 to finish the proof for Theorem 1.4

Theorem 1.4 _Let $(M,g)$ be a compact $3$-manifold with scalar curvature
$\kappa\geq 1$ and with mixed boundary. If $\pi_{2}(M)$ is generated by the
spheres in $\partial M$, then $\text{Int}~{}M$ is homeomorphic to a spherical
$3$-manifold with finitely many punctures. _

######  Remark.

If $M$ is non-compact, it is homeomorphic to $\mathbb{R}^{3}$ with (at most)
countably many disjoint open $3$-balls removed.

###### Proof.

Let $\hat{M}$ be obtained from $M$ by filling the boundary with some
$3$-balls. Then it is a closed manifold and $\pi_{2}(\hat{M})=\\{1\\}$. We
have that $\pi_{1}(\hat{M})$ is finite. The reason is as follows:

If not, the universal cover of $\hat{M}$ is contractible (see Corollary 3.9 of
[Hat00] on Page 51). Namely, the compact manifold $\hat{M}$ is a
$K(\pi,1)$-space and $\pi_{1}(\hat{M})$ is non-trivial. From Lemma 4.1 of
[Lüc12] on Page 4, $\pi_{1}(\hat{M})$ is torsion-free.

Therefore, $\pi_{1}(M)$ has a subgroup which is isomorphic to $\mathbb{Z}$.
However, the scalar curvature $\kappa\geq 1$, which contradicts with Lemma
6.3.

Since $\pi_{1}(\hat{M})$ is finite, the universal cover of $\hat{M}$ is
compact. The Poincaré conjecture (see [BBB+10] ,[MT07] and [CZ06]) shows that
the universal cover is homeomorphic to $\mathbb{S}^{3}$. Namely, $\hat{M}$ is
a spherical $3$-manifold. Thus, $\text{Int}~{}M$ is a spherical $3$-manifold
with finitely many punctures. ∎

Theorem 1.3 shows that a complete orientable $3$-manifold with uniformly
positive scalar curvature has a prime decomposition. We now use Theorem 1.4 to
complete the proof of Theorem 1.1. It is sufficient to show that each prime
factor in the decomposition is a spherical $3$-manifold or
$\mathbb{S}^{1}\times\mathbb{S}^{1}$.

###### Proof.

Let $(M,g)$ be a complete orientable $3$-manifold with scalar curvature
$\kappa\geq 1$. From Theorem 1.3, $M$ is homeomorphic to a (possibly) infinite
connected sum of closed $3$-manifolds, $\\{\hat{M}_{l}\\}_{l}$, where
$\hat{M}_{l}$ is a closed and prime $3$-manifold.

Remark that for each $l$, $M\cong\hat{M}_{l}\\#M^{\prime}_{l}$, where
$M^{\prime}_{l}$ is a non-compact $3$-manifold.

It is sufficient to show that each $\hat{M}_{l}$ is homeomorphic to
$\mathbb{S}^{1}\times\mathbb{S}^{2}$ or a spherical 3-manifold.

For each $\hat{M}_{l}$, there is an embedded $2$-sphere $\Sigma$ satisfying
that

  * •

it bounds a compact $3$-manifold $M_{l}\subset M$;

  * •

$\text{Int}~{}M_{l}$ is homeomorphic to $\hat{M}_{l}$ with a puncture.

We assume that $\hat{M}_{l}$ is not homeomorphic to one of $\mathbb{S}^{3}$,
$\mathbb{S}^{1}\times\mathbb{S}^{2}$ and $\mathbb{R}P^{3}$. Then,
$\hat{M}_{l}$ is irreducible (see Proposition 1.4 of [Hat00] on Page 4). It
remains to prove that $\hat{M}_{l}$ is a spherical $3$-manifold.

By Theorem 5.3, there is an open $3$-manifold $M^{\prime}_{l}\subset M$ so
that

(1) $M^{\prime}_{l}$ is homeomorphic to $\hat{M}_{l}$ with finitely many
punctures;

(2) the closure of $(M^{\prime}_{l},g|_{M^{\prime}_{l}})$ is a compact
$3$-manifold with mixed boundary.

Since $\hat{M}_{l}$ is irreducible, $\pi_{2}(M^{\prime}_{l})$ is generated by
the $2$-spheres in $\partial M^{\prime}_{l}$. Because the scalar curvature
$\kappa\geq 1$, we use Theorem 1.4 to show that $M^{\prime}_{l}$ is a
spherical $3$-manifold with finitely many punctures. Namely, $\hat{M}_{l}$ is
homeomorphic to a spherical $3$-manifold. ∎

######  Corollary 6.4.

Let $(M,g)$ be an orientable $3$-manifold with scalar curvature $\kappa\geq
1$. If $\partial M$ is a union of $2$-spheres and minimal for $g$, then $M$ is
homeomorphic to an infinite connected sum of spherical $3$-manifolds,
$3$-balls and some copies of $\mathbb{S}^{2}\times\mathbb{S}^{1}$.

The proof is the same as the proof of Theorem 1.1 but use Corollary 3.11 and
Remark 5.7 instead of Theorem 1.3 and Theorem 5.3.

##  7\. Proof of Theorem 1.2

###  7.1. Geometric version of Loop Lemma

######  Proposition 7.1.

Let $(M,g)$ be a $3$-manifold with scalar curvature $\kappa\geq 1$ and with
mean convex boundary. Assume that the boundary of $M$ is a union of closed
surfaces, $\\{S_{j}\\}_{j}$. If $\pi_{1}(S_{j})\rightarrow\pi_{1}(M)$ is not
injective for some $j$, there is an embedded stable minimal disc $(D,\partial
D)\subset(M,S_{j})$ whose boundary is not null-homotopic in $S_{j}$.

###### Proof.

Let $\sigma\subset S_{j}$ be a simple circle, which is in the kernel of
$\pi_{1}(S_{j})\rightarrow\pi_{1}(M)$. Choose a connected compact set
$K\subset M$ containing $S_{j}$ so that

  * •

$\sigma$ is homotopically trivial in $K$;

  * •

$\partial^{*}K:=\partial K\cap\text{Int}~{}M$ is a disjoint union of some
closed surfaces,

  * •

$d(S_{j},\partial^{*}K)>4\pi$.

The sets $\partial^{*}K$ and $\partial M$ are disjoint.

Step1: Find a metric $g^{\prime}$ on $K$ so that $(K,g^{\prime})$ has mean
convex boundary.

Choose $\epsilon>0$ sufficiently small satisfying that

  * •

$B(\partial^{*}K,\epsilon)$ and $\partial M$ are disjoint;

  * •

$d(\partial^{*}K^{\prime},S_{j})>4\pi$,

where $K^{\prime}:=K\setminus B(\partial^{*}K,\epsilon)$ and
$\partial^{*}K^{\prime}=\partial K^{\prime}\cap\text{Int}~{}M$.

The metric deformation in Section 4 gives a metric $g^{\prime}$ on $K$ so that
(1) $g^{\prime}=g$ on $K^{\prime}$; (2) the mean curvature of $\partial^{*}K$
is positive for $g^{\prime}$.

Since $(M,g)$ has mean convex boundary, the mean curvature of $\partial M\cap
K$ is still non-negative for $g^{\prime}$. Therefore, $(K,g^{\prime})$ has
mean convex boundary.

We use Theorem 4.3 to find an embedded disc $D\subset K$ with boundary
$\sigma$. It is stable minimal for $g^{\prime}$.

Step 2: Show that $D$ is stable minimal for $g$.

It is sufficient to show that $D$ is a subset of $K^{\prime}$.

Suppose that $D$ is not contained in $K^{\prime}$. Namely,
$D\cap\partial^{*}K^{\prime}\neq\emptyset$. Consider the component
$D^{\prime}$ of $D\cap K^{\prime}$ connecting $S_{j}$ and
$\partial^{*}K^{\prime}$. Since $g^{\prime}=g$ on $K^{\prime}$, $D^{\prime}$
is stable minimal for $g$. Corollary 4.2 gives that
$d(S_{j},\partial^{*}K^{\prime})\leq 4\pi$. However,
$d(\partial^{*}K^{\prime},S_{j})>4\pi$, a contradiction.

We can conclude that $D$ is in $K^{\prime}$. It is stable minimal for $g$. ∎

######  Remark 7.2.

Let $M$ and $D$ be as in Proposition 7.1. By Definition 2.3, $M$ is
homeomorphic to a boundary connected sum of the components of $M\setminus D$
(with itself).

We inductively use Proposition 7.1 to obtain the following.

######  Corollary 7.3.

Let $(M,g)$ and $\\{S_{j}\\}_{j\in J}$ be as in Proposition 7.1. For each $j$,
there are finitely many embedded disks $\\{D^{l}_{j}\\}^{L_{j}}_{l=1}$ so that

  * •

each $(D^{l}_{j},\partial D^{l}_{j})\subset(M,S_{j})$ is stable minimal;

  * •

for each component $M_{s}$ of $M\setminus\amalg^{L_{j}}_{l=1}D^{l}_{j}$, each
closed surface in $\partial M_{s}$ coming from $S_{j}$ is incompressible in
$M_{s}$.

######  Corollary 7.4.

Let $(M,g)$ and $\\{S_{j}\\}_{j\in J}$ be as in Proposition 7.1. Then there
are a family of stable minimal discs $\\{D^{l}_{j}\\}_{j\in J,1\leq l\leq
L_{j}}$ so that

  * (a)

each component of $M\setminus\amalg_{j\in J}(\amalg_{l}D^{l}_{j})$ is a
$3$-manifold with mean convex boundary;

  * (b)

the boundary of a component $M_{s}$ of $M\setminus\amalg_{j\in
J}(\amalg_{l}D^{l}_{j})$ is a union of some closed surfaces;

  * (c)

Each component of the boundary of $M_{s}$ is incompressible in $M_{s}$.

###### Proof.

We inductively apply Corollary 7.3 to each $S_{j}$ and get finitely many
disjoint stable minimal discs $\\{D_{j}^{l}\\}^{L_{j}}_{l=1}$. The discs,
$\\{D_{j}^{l}\\}_{j\in J,1\leq l\leq L_{j}}$ are also disjoint.

It remains to show that the discs, $\\{D^{l}_{j}\\}_{j\in J,1\leq l\leq
L_{j}}$ are locally finite.

For any compact set $K\subset M$, consider a compact set
$B(K,4\pi):=\\{x~{}|~{}d(x,K)\leq 4\pi\\}\subset M$. Since the family
$\\{S_{j}\\}_{j\in J}$ is locally finite, there are (at most) finitely many
non-empty sets in $\\{B(K,4\pi)\cap S_{j}\\}_{j\in J}$.

Each $D^{l}_{j}$ is stable minimal for $g$ and the scalar curvature
$\kappa\geq 1$. Theorem 4.1 shows that if $D^{l}_{j}\cap K\neq\emptyset$, we
have that

  * •

$d(K,\partial D^{l}_{j})\leq\frac{2\pi}{\sqrt{3}}$.

  * •

$d(K,S_{j})\leq 2\pi$. Namely, $B(K,4\pi)\cap S_{j}\neq\emptyset$.

Therefore, there are finitely many non-empty sets in
$\\{K\cap(\amalg^{L_{j}}_{l=1}D^{l}_{j})\\}_{j}$ (i.e. $\\{D^{l}_{j}\\}_{j\in
J,1\leq l\leq L_{j}}$ are locally finite). If not, there are infinitely many
non-empty sets in $\\{B(K,4\pi)\cap S_{j}\\}_{j\in J}$, a contradiction.

The local finiteness of $\\{D^{l}_{j}\\}_{j\in J,1\leq l\leq L_{j}}$ shows
that each component of $M\setminus\amalg_{j}(\amalg_{l}D^{l}_{j})$ is a
$3$-manifold. Its boundary comes from minimal discs and $\\{S_{j}\\}_{j\in
J}$. Thus, each component is a $3$-manifold with mean convex boundary.

We could use Corollary 7.3 to get that each component also satisfies (b) and
(c). ∎

###  7.2. Incompressible surfaces and Positive scalar curvature

In this part, we talk about incompressible surfaces in a $3$-manifold with
positive scalar curvature.

######  Lemma 7.5.

Let $(X,g)$ be a (non-compact) $3$-manifold with mean convex boundary and
scalar curvature $\kappa\geq 1$. If the boundary $\partial X$ is a disjoint
union of closed surfaces, $\\{S_{j}\\}_{j}$, then the doubling of $X$ carries
a complete metric of positive scalar curvature.

The proof for the compact case is the same as [Theorem 1.1, Page 75] in
[Alm85]. Since the proof of [Theorem 1.1, Page 75] in [Alm85] only depends on
the local computations, it remains valid for the non-compact case.

We use Lemma 7.5 to get the following result.

######  Theorem 7.6.

Let $(X,g)$ be a $3$-manifold with scalar curvature $\kappa\geq 1$ and with
mean convex boundary. If each component of $\partial X$ is a closed
incompressible surface in $X$, then each component of $\partial X$ is
homeomorphic to $\mathbb{S}^{2}$.

###### Proof.

Suppose that there is an incompressible surface $S_{j}\subset\partial X$ with
the genus $g(S_{j})\geq 1$. Since the map $\pi_{1}(\partial
X)\rightarrow\pi_{1}(X)$ is injective, then $S_{j}$ is also an incompressible
surface in the doubling of $X$ (see Theorem 11.60 of [Rot12] on Page 386).

By Lemma 7.5, the doubling of $X$ has a complete metric with positive scalar
curvature. However, it is well-known from Theorem 4 of [SY82] on Page 225 and
Theorem 8.4 of [GL83] on Page 164 that if a $3$-manifold has an incompressible
closed surface whose genus is greater than zero, it has no complete metric
with positive scalar curvature. It leads to a contradiction.

We can conclude that each component of $\partial X$ is a $2$-sphere. ∎

###  7.3. Complete the proof of Theorem 1.2

We begin with a metric deformation and then use it to complete the proof of
Theorem 1.2.

######  Theorem 7.7.

Let $(X,g)$ be a $3$-manifold with scalar curvature $\kappa\geq 1$ and with
mean convex boundary. If $\partial X$ has a closed surface $S$, then for any
$\epsilon>0$, there is a metric $g^{\prime}$ on $X$so that

  * •

the scalar curvature of $g^{\prime}$ is not less than $\frac{1}{2}$;

  * •

the boundary $\partial X$ is minimal for $g^{\prime}$;

  * •

$g^{\prime}$ is equal to $g$ on $X\setminus B(S,\epsilon)$.

where $B(S,\epsilon)$ is the tubular neighborhood of $S$ with radius
$\epsilon$.

The metric $g^{\prime}$ comes from a local metric deformation on
$B(S,\epsilon)$. It was proved by Theorem 27 and Remark 31 in [BH20] and also
by [CL19].

######  Corollary 7.8.

Let $(X,g)$ be a $3$-manifold with scalar curvature $\kappa\geq 1$ and with
mean convex boundary. If $\partial X$ is a union of closed surfaces
$\\{S_{j}\\}_{j}$, there is a metric $g^{\prime}$ on $X$ satisfying that

  * •

the scalar curvature of $g^{\prime}$ is not less than $\frac{1}{2}$;

  * •

the boundary of $\partial X$ is minimal for $g^{\prime}$;

######  Remark 7.9.

After scaling the metric $g^{\prime}$, we have a metric $g^{\prime\prime}$ so
that (1) the scalar curvature of $g^{\prime\prime}$ is bounded below by one
(2) $\partial X$ is minimal for $g^{\prime\prime}$.

###### Proof.

For each $j$, there is a positive constant $\epsilon_{j}$ so that
$\\{B(S_{j},\epsilon_{j})\\}_{j}$ are disjoint. We apply Theorem 7.7 to each
$S_{j}$ and deform the metrics on each $B(S_{j},\epsilon_{j})$. Thus, we could
obtain a metric $g^{\prime}$ such that (1) the scalar curvature of
$g^{\prime}$ is not less than $\frac{1}{2}$ (2) $\partial X$ is minimal for
$g^{\prime}$. ∎

We now complete the proof of Theorem 1.2.

###### Proof.

Let $(M,g)$ be an orientable $3$-manifold with mean convex boundary and with
scalar curvature $\kappa\geq 1$. The boundary $\partial M$ consists of some
closed surfaces, $\\{S_{j}\\}_{j\in J}$.

We use Corollary 7.4 to get a family of stable minimal discs
$\\{D^{l}_{j}\\}_{j\in J,1\leq l\leq L_{j}}$ so that

  * •

they cut $M$ into some connected $3$-manifolds, $\\{M_{s}\\}$.

  * •

Each component of $\partial M_{s}$ is incompressible in $M_{s}$ and a closed
surface.

  * •

$(M_{s},g|_{M_{s}})$ has mean convex boundary and the scalar curvature
$\kappa\geq 1$.

From Theorem 7.6, each component of $\partial M_{s}$ is a $2$-sphere.

We use Remark 7.9 to find a metric on $M_{s}$ so that (1) the scalar curvature
is bounded below by one; (2) $\partial X_{s}$ is minimal for the new metric.
By Corollary 6.4, $M_{s}$ is an infinite connected sum of spherical
$3$-manifolds, $3$-balls and some copies of
$\mathbb{S}^{1}\times\mathbb{S}^{2}$.

By Definition 2.3, $M$ is homeomorphic to a boundary connected sum of
$\\{M_{s}\\}_{s}$ (with themselves) (see Remark 7.2). We could do the
operation of the boundary connected sum on each $3$-ball (at most) finitely
many times and obtain some handlebodies (see Remark 2.4).

Therefore, $M$ is an infinite connected sum of spherical $3$-manifolds,
handlebodies and some copies of $\mathbb{S}^{1}\times\mathbb{S}^{2}$. ∎

## Appendix A: Proofs of Lemma 2.5 and Lemma 2.7

We will complete the proofs of Lemma 2.5 and Lemma 2.7

Lemma 2.5 _Let $M$ be an open $3$-manifold which is homotopically equivalent
to $\mathbb{S}^{1}$. Then a closed surface $\Sigma\subset M$ cuts $M$ into a
non-compact part and a compact part._

###### Proof.

Suppose that $M\setminus\Sigma$ is connected. Namely, there is a closed curve
$\sigma$ which intersects $\Sigma$ transversally at one point. The
intersection number of $\Sigma$ and $\sigma$ is $\pm 1$.

However, since $H_{2}(M)=\\{1\\}$, then there is a compact connected
$3$-manifold $M_{0}\subset M$ containing $\Sigma$ and $\sigma$ satisfying that
$[\Sigma]=0$ in $H_{2}(M_{0})$. The Poincaré duality shows that

$\begin{split}H_{1}(M_{0},\partial M_{0})\times
H_{2}(M_{0})&\longrightarrow\mathbb{Z}\\\
\sigma\quad\quad\times\quad\Sigma\quad&\mapsto(\sigma,\Sigma).\\\ \end{split}$

Therefore, the intersection number $(\sigma,\Sigma)$ is equal to zero, which
contradicts with the last paragraph. We can conclude that $\Sigma$ cuts $M$
into two components, $M_{1}$ and $M_{2}$.

Remark that $\Sigma$ is the boundary of $\overline{M}_{i}$ for $i=1,2$. If
$M_{i}$ is non-compact, $[\Sigma]\neq 0$ in $H_{2}(\overline{M}_{i})$.

It remains to show that one of components of $M\setminus\Sigma$ is compact.

The Mayer-Vietoris sequence shows

$H_{3}(M)\rightarrow H_{2}(\Sigma)\rightarrow H_{2}(\overline{M_{1}})\oplus
H_{2}(\overline{M_{2}})\rightarrow H_{2}(M).$

Since $H_{3}(M)\cong H_{2}(M)\cong\\{1\\}$, Hence,
$H_{2}(\overline{M}_{1})\oplus H_{2}(\overline{M}_{2})\cong\mathbb{Z}$. One of
$H_{2}(\overline{M}_{1})$ and $H_{2}(\overline{M}_{2})$ is isomorphic to
$\mathbb{Z}$ and generated by $[\Sigma]$.

We may assume that $H_{2}(\overline{M}_{1})$ is isomorphic to $\mathbb{Z}$ and
generated by $[\Sigma]$. Thus, $H_{2}(\overline{M}_{2})$ is trivial. From the
above remark, $\overline{M}_{2}$ is compact. Since $M$ is non-compact,
$\overline{M}_{1}$ is non-compact. We finish the proof of the lemma. ∎

Before proving Lemma 2.7, we introduce two types of surgeries on a compact
set. Let $M$ and $\gamma$ be assumed as in Lemma 2.7. Assume that an embedded
surface $\Sigma\subset M$ bounds a compact set $M_{\Sigma}\subset M$ and
$\gamma\subset M_{\Sigma}$.

Consider an embedded disc $(D,\partial D)\subset(M\setminus\gamma,\Sigma)$
with $D\cap\Sigma=\partial D$, where $\partial D$ is non-contractible in
$\Sigma$.

Type I: If $D$ is a subset of $M_{\Sigma}$, then we consider an open tubular
neighborhood $N(D,\epsilon)\subset M_{\Sigma}\setminus\gamma$ of $D$. We then
has two cases:

Case (1): If $N(D,\epsilon)$ cuts $M_{\Sigma}$ into two components, then we
choose the compact set $M_{\Sigma^{\prime}}$ as the component of
$M_{\Sigma}\setminus N(\Sigma,\epsilon)$ containing $\gamma$, where
$\Sigma^{\prime}=\partial M_{\Sigma^{\prime}}$ is a closed surface.

Case (2): If $M_{\Sigma}\setminus N(D,\epsilon)$ is connected, then we choose
the compact set $M_{\Sigma^{\prime}}$ as $M_{\Sigma}\setminus N(D,\epsilon)$,
where $\Sigma^{\prime}=\partial M_{\Sigma^{\prime}}$ is a closed surface.

Type II: If $D$ is a subset of $\overline{M\setminus M_{\Sigma}}$, then we
consider an open tubular neighborhood $N(D,\epsilon)$ of $D$ in
$\overline{M\setminus M_{\Sigma}}$. We also have two cases.

Case (1): If $\Sigma\setminus\partial D$ is connected, we choose
$M_{\Sigma^{\prime}}$ as $M_{\Sigma}\cup N(D,\epsilon)$, where
$\Sigma^{\prime}=\partial M_{\Sigma^{\prime}}$ is a closed surface.

Case (2): If $\Sigma\setminus\partial D$ is disconnected, then the boundary of
$M_{\Sigma}\cup N(D,\epsilon)$ has two components. Choose the component
$\Sigma^{\prime}$ satisfying that

  * •

it bounds a compact set $M_{\Sigma^{\prime}}\subset M$;

  * •

$M_{\Sigma}$ is a subset of $M_{\Sigma^{\prime}}$.

Its existence is ensured by the proof of Corollary 2.6.

Remark: Let $\Sigma^{\prime}$ be a closed surface constructed from Type I or
Type II surgery. We have that

(1) $\gamma$ is a subset of $M_{\Sigma^{\prime}}$;

(2) the genus of $\Sigma^{\prime}$ is less than the genus of $\Sigma$;

(3) $\Sigma^{\prime}$ is a union of $\Sigma^{\prime}\cap\Sigma$ and some
disjoint discs. .

Lemma 2.7 _Let $M$ be an open $3$-manifold which is honmotopically equivalent
to $\mathbb{S}^{1}$ and $\gamma$ a circle generating $\pi_{1}(M)$. Assume that
an embedded closed surface $\Sigma$ bounds a compact set $M_{\Sigma}\subset
M$. If $\gamma$ is a subset of $M_{\Sigma}$, then there is a circle
$\sigma\subset\Sigma$ so that it is contractible in $M$ but non-contractible
in $M\setminus\gamma$. _

Remark that $g(\Sigma)>0$. If not, $\Sigma$ is a $2$-sphere. Since $M$ is
irreducible, then it bounds a $3$-ball $M_{\Sigma}\subset M$. Namely, $\gamma$
is contractible in $M_{\Sigma}\subset M$, a contradiction.

###### Proof.

We argue by induction on the genus $g(\Sigma)$.

When $g(\Sigma)=1$, then $\pi_{1}(\Sigma)\rightarrow\pi_{1}(M\setminus\gamma)$
is injective. If not, there is an embedded disc $D\subset M\setminus\gamma$
with $D\cap\Sigma=\partial D$. The boundary $\partial D\subset\Sigma$ is non-
contractible in $\Sigma$. We use Type I or Type II surgery to get a closed
surface $\Sigma^{\prime}$ so satisfying that

  * •

it bounds a compact set $M_{\Sigma^{\prime}}\subset M$ and $\gamma\subset
M_{\Sigma^{\prime}}$;

  * •

$g(\Sigma^{\prime})<g(\Sigma)$.

Thus, $g(\Sigma^{\prime})$ is equal to zero. Say, $\Sigma^{\prime}$ is a
$2$-sphere and $M_{\Sigma^{\prime}}$ is a $3$-ball. We get that $\gamma$ is
contractible in $M_{\Sigma^{\prime}}\subset M$, which leads to a
contradiction.

Since $\pi_{1}(M)\cong\mathbb{Z}$, then the map
$\pi_{1}(\Sigma)\rightarrow\pi_{1}(M)$ is not injective. We choose a circle
$\sigma\subset\Sigma$ so that $[\sigma]$ is a non-trivial element in
$\text{Ker}(\pi_{1}(\Sigma)\rightarrow\pi_{1}(M))$. Namely, $\sigma$ is
contractible in $M$.

Because $\pi_{1}(\Sigma)\rightarrow\pi_{1}(M\setminus\gamma)$ is injective,
$\sigma$ is non-contractible in $M\setminus\gamma$. It is the required
candidate in our assertion.

Suppose that it is true when $g(\Sigma)\leq k$.

When $g(\Sigma)=k+1$, we can conclude that the map
$\pi_{1}(\Sigma)\rightarrow\pi_{1}(M\setminus\gamma)$ is injective. The reason
is as follows:

If not, there is an embedded disc $D\subset M\setminus\gamma$ with
$D\cap\Sigma=\partial D$. The boundary $\partial D\subset\Sigma$ is non-
contractible in $\Sigma$. We use Type I or Type II surgery to get a closed
surface $\Sigma^{\prime}$ satisfying that

  * (a)

it bounds a compact set $M_{\Sigma^{\prime}}\subset M$ and $\gamma\subset
M_{\Sigma^{\prime}}$;

  * (b)

$\Sigma^{\prime}$ is a union of $\Sigma^{\prime}\cap\Sigma$ and some disjoint
discs;

  * (c)

$g(\Sigma^{\prime})<g(\Sigma)$.

From (c), we apply the inductive hypothesis to $\Sigma^{\prime}$ and get a
circle $\sigma_{1}\subset\Sigma^{\prime}$. From (b), we may assume that
$\sigma_{1}$ is a subset of $\Sigma\cap\Sigma^{\prime}\subset\Sigma$. It is
the required circle in our assertion.

Since $\pi_{1}(M)\cong\mathbb{Z}$, then the map
$\pi_{1}(\Sigma)\rightarrow\pi_{1}(M)$ is not injective. We choose a circle
$\sigma\subset\Sigma$ so that $[\sigma]$ is a non-trivial element in
$\text{Ker}(\pi_{1}(\Sigma)\rightarrow\pi_{1}(M))$. Namely, $\sigma$ is
contractible in $M$.

Because $\pi_{1}(\Sigma)\rightarrow\pi_{1}(M\setminus\gamma)$ is injective,
$\sigma$ is non-contractible in $M\setminus\gamma$. It is the required
candidate in our assertion. It finishes the proof of Lemma 2.7 ∎

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