# Stable free boundary minimal hypersurfaces in a wedge domain

Zetian Yan  Department of Mathematics
UC Santa Barbara
Santa Barbara
CA 93106
USA<EMAIL_ADDRESS>
###### Abstract.

We prove that a stable $C^{1,1}$-to-edge properly embedded free boundary
minimal hypersurface $\Sigma^{3}$ of a $4$-dimensional wedge domain
$\Omega^{4}_{\theta}$ with angle $\theta\in(0,\pi]$ is flat.

###### Key words and phrases:

Bernstein problem; Free boundary minimal surfaces; $\mu$-bubbles

###### 2020 Mathematics Subject Classification:

Primary 53A10; Secondary 49Q05, 49Q10, 58E12.

##  1\. Introduction

Classical Bernstein problem. For an entire minimal graph, i.e., the graph
$\Gamma(u)$ of a function $u:\mathbb{R}^{n}\to\mathbb{R}$ satisfying the
minimal hypersurface equation, does it have to be necessarily a hyperplane in
$\mathbb{R}^{n+1}$?

In 1914, Sergei Bernstein solved this problem in the case $n=2$. In 1962,
Fleming [Fle62] provided an alternative proof. In other low dimensional cases,
the classical Bernstein problem was solved by De Giorgi [DG65](n=3), Almgren
[Alm66](n=4) and Simons [Sim68] ($5\leqslant n\leqslant 7$), respectively.
Note that a minimal graph $\Gamma(u)$ in $\mathbb{R}^{n+1}$ satisfy the volume
growth condition: for the ball $B_{r}(0)$ in $\mathbb{R}^{n+1}$, we have

(1.1)
$\mathrm{Vol}(B_{r}(0)\cap\Gamma(u))\leqslant\frac{\mathrm{Vol}(\mathbb{S}^{n})}{2}r^{n},$

where $\mathbb{S}^{n}$ is the unit sphere in $\mathbb{R}^{n+1}$. Moreover,
two-dimensional minimal graphs satisfy the area-minimizing property
automatically. Combining these with the log-cutoff trick yields the flatness;
see [CM11, Chapter 1] for more details.

A natural generalization of the classical Bernstein problem is the following:

Stable Bernstein problem. Is every complete orientable immersed stable minimal
hypersurface a hyperplane?

When $n=2$, the stable Bernstein problem was confirmed by do Carmo and Peng
[dCP79], Fischer-Colbrie and Schoen [FCS80] and Pogorelov [Pog81],
respectively. As for $n\leqslant 5$, it is also true with some additional
assumptions; see [SSY75, dCP82, CSZ97, Che01, NS07] and references therein. In
particular, under the assumption on the volume growth of geodesic balls,
Schoen, Simon and Yau [SSY75] proved the flatness by estimating the $L^{p}$
bounds on the second fundamental form.

In high dimensional cases, both classical and stable Bernstein problems are
not true. For $n\geqslant 8$, Bombieri, De Giorgi and Giusti [BDGG69] showed
that there are minimal entire graphs that are not hyperplanes. Moreover, in
$\mathbb{R}^{8}$, there are non-flat area-minimizing complete orientable
hypersurfaces constructed in [HS85].

The stable Bernstein problem in the remaining cases without additional
hypothesis were still open. In 2021, Chodosh and Li [CL21] gave the positive
answer in $\mathbb{R}^{4}$ by estimating the quantity

$F(t)=\int_{\Sigma_{t}}|\nabla u|^{2}$

and relating it to

$A(t)=\int_{\Sigma_{t}}|A_{M}|^{2},$

where $u$ is the minimal positive Green’s function, $\Sigma_{t}$ is the
$t$-level set of $u$ and $A_{M}$ is the second fundamental form of $M$. Later,
they gave an alternative proof [CL23] by using the $\mu$-bubble to control
volume growth and combining it with the argument in [SSY75]. Recently,
Chodosh, Li, Minter and Stryker [CLMS24] generalized the $\mu$-bubble argument
and solved the stable Bernstein problem in $\mathbb{R}^{5}$. In
$\mathbb{R}^{4}$, we also note that Catino, Mastrolia, Roncoroni [CMR24]
provided a different proof by the conformal transformation.

It is interesting to consider Bernstein-type problems in less regular domains.
In this paper, we consider the free boundary stable Bernstein problem in a
wedge domain $\Omega^{m}_{\theta}$ with angle $\theta\in(0,\pi]$, where the
wedge domain $\Omega^{m}_{\theta}$ is in the form of

$\Omega^{m}=\Omega^{2}_{\theta}\times\mathbb{R}^{m-2}=\mathrm{Clos}\left(\left\\{x\in\mathbb{R}^{m}:x_{1}>0,x_{2}\in(\tan(-\frac{\theta}{2})x_{1},\tan(\frac{\theta}{2})x_{1})\right\\}\right).$

In the 1990s, Hildebrandt and Sauvigny [HS97a, HS97b, HS99a, HS99b] were among
the first to investigate free boundary minimal surfaces in a wedge domain
$\Omega^{2}_{\theta}$. From their work, we note that the wedge angle $\theta$
affects the behavior of minimal surfaces crucially. In particular, when
$\theta=\pi$, by the reflection principle [GLZ20], the free boundary stable
Bernstein problem in $\mathbb{R}^{n+1}_{+}$ is equivalent to that in
$\mathbb{R}^{n+1}$. While, for $\theta\in(0,\pi)$, the global reflection
principle is no longer available, and the regularity of free boundary minimal
hypersurfaces in $\Omega^{m}_{\theta}$ is subtle.

For $3\leqslant n+1\leqslant 6$, Mazurowski and Wang [MW23] proved a
Bernstein-type theorem under the assumption on the volume growth (1.1). We use
the $\mu$-bubble technique to obtain the volume estimate and confirm this
problem without additional assumptions in dimension $4$.

######  Theorem 1.1.

Suppose that
$\left(\Sigma^{3},\left\\{\partial_{3}\Sigma\right\\}\right)\subset\left(\Omega^{4},\left\\{\partial_{4}\Omega\right\\}\right)$
is a stable $C^{1,1}$-to-edge properly embedded free boundary minimal
hypersurface. There exists an explicit constant $C$ such that

(1.2) $|\Sigma\cap B_{\mathbb{R}^{4}}(0,\rho)|_{g}\leqslant C\rho^{3}.$

In particular, $\Sigma=\Omega\cap P$ where $P\in\mathbb{R}^{4}$ is a
hyperplane.

We adapt the construction of weighted free boundary $\mu$-bubbles in [CL23].
First of all, in the same spirit in [CL23], we carry out the conformal
deformation technique used by Gulliver–Lawson [GL86]. By the local reflection
argument, we find that free boundaries are totally geodesic with respect to
the blow-up metric. Besides, in our setting, the weighted function $u$ should
satisfy the Neumann condition on the boundary and have $C^{1,\alpha}$
regularity near the edge.

Note that in [CL23, Lemma 21], the properly embeddedness of free boundary
$\mu$-bubbles comes from the maximum principle because both constrained
boundaries and free boundary $\mu$-bubbles are minimizers of the same
functional. However, due to the existence of the edge, the properly
embeddedness here is a delicate issue. Suppose that $\Xi$ is a stable free
boundary $\mu$-bubble in $\Sigma$. Touching phenomenon consists of two cases
[MW23]:

  * •

the interior $\mathring{\Xi}$ touches the face $\partial^{F}\Sigma$;

  * •

the face $\partial^{F}\Xi$ touches the edge $\partial^{E}\Sigma$.

By the Neumann condition on $u$ and the totally geodesic property of free
boundaries, we find that the first case cannot happen. As for the second case,
if happened, we can construct a vector field along which the first variation
of $\Xi$ is negative, which violates the stability of $\Xi$.

Finally, combining the log-cutoff trick and the fact that free boundaries are
totally geodesic, boundary integrals in the second variation of a $\mu$-bubble
vanish and we obtain the volume estimate following [CL23].

The remainder of the paper is organized as follows. In Section 2, we review
several basic concepts which is borrowed from [MW23]. In Section 3, by the
local reflection argument, we prove the one-end theorem following [CSZ97]. In
Section 4 and 5, we construct a free boundary $\mu$-bubble and obtain the
volume control.

Acknowledgements. I would like to thank Dr. Tongrui Wang, Dr. Liangjun Weng,
Nan Wu and Dr. Xingyu Zhu for their encouragement and useful discussion. Thank
Dr. Yaoting Gui for reviewing the draft.

##  2\. Preliminary

###  2.1. Locally wedge-shaped hypersurfaces.

To begin with, let us fix some notations in the Euclidean space. Let
$H^{m}_{+}$ and $H^{m}_{-}$ be two closed half spaces in $\mathbb{R}^{m}$,
where $m\in\\{2,3,\cdots\\}$. $\Omega^{m}:=H^{m}_{+}\cap H^{m}_{-}$ is called
an $m$-dimensional wedge domain if it has non-empty interior. By rotating, we
can always write $\Omega^{m}$ in the form of

$\Omega^{m}=\Omega^{2}_{\theta}\times\mathbb{R}^{m-2}=\mathrm{Clos}\left(\left\\{x\in\mathbb{R}^{m}:x_{1}>0,x_{2}\in(\tan(-\frac{\theta}{2})x_{1},\tan(\frac{\theta}{2})x_{1})\right\\}\right),$

where $\theta\in(0,\pi]$ is called the wedge angle of $\Omega^{m}$. We usually
use the notation $\Omega^{m}_{\theta}$ to emphasize its angle.

######  Definition 2.1 (Stratification).

For a non-trivial $m$-dimensional wedge domain $\Omega^{m}_{\theta}$; i.e.
$\theta\in(0,\pi)$, define the stratification
$\partial_{m-2}\Omega\subset\partial_{m-1}\Omega\subset\partial_{m}\Omega$ of
$\Omega^{m}_{\theta}$ by

$\partial_{m}\Omega:=\Omega,\quad\partial_{m-1}\Omega:=\partial\Omega,\quad\partial_{m-2}\Omega:=\\{0\\}\times\mathbb{R}^{m-2}.$

Then we define

  * •

$\mathring{\Omega}:=\partial_{m}\Omega\backslash\partial_{m-1}\Omega$ to be
the interior of $\Omega$;

  * •

$\partial^{F}\Omega:=\partial_{m-1}\Omega\backslash\partial_{m-2}\Omega$ to be
the face of $\Omega$;

  * •

$\partial^{E}\Omega:=\partial_{m-2}\Omega$ to be the edge of $\Omega$.

Next, we introduce the notations for hypersurfaces that are locally modeled by
wedge domains.

######  Definition 2.2 (Locally wedge-shaped hypersurfaces).

We say that $M^{n}\subset\Omega^{n+1}_{\theta}$ is an embedded locally wedge-
shaped hypersurface of $\Omega^{n+1}_{\theta}$, if for any $p\in M^{n}$, there
exists $R=R(p)>0$ and a diffeomorphism
$\psi=\psi_{p}:\mathbb{B}^{n+1}_{R}(0)\to\mathbb{B}^{n+1}_{R}(p)$ so that

  * •

$\psi(0)=p$ and the tangent map $(D\psi)_{0}\in O(n+1)$ is an orthogonal
transformation;

  * •

$\psi\left(\left(\Omega(p)\times\\{0\\}\right)\cap\mathbb{B}^{n+1}_{R}(0)\right)=M\cap\mathbb{B}^{n+1}_{R}(p)$,
where

$\Omega(p)=\left\\{\begin{array}[]{cc}\mathbb{R}^{n},&p\in\mathring{M}^{n},\\\
\Omega^{n}_{\theta}(p),&p\in\partial M^{n},\end{array}\right.$

for some $n$-dimensional wedge domain $\Omega^{n}_{\theta}(p)$ with
$\theta(p)\in(0,\pi]$.

We call $(\psi_{p},\mathbb{B}^{n+1}_{R}(p),\Omega(p))$ a local model of
$M^{n}$ around $p$, and call $\theta$ the wedge angle of $M^{n}$ at
$p\in\partial M$. Additionally, given $l\in\mathbb{Z}_{+}$ and
$\alpha\in(0,1]$, if for any $p\in\partial M$ with $\theta(p)\in(0,\pi)$,
$\psi_{p}$ is globally a $C^{l,\alpha}$-diffeomorphism, and is a
$C^{\infty}$-diffeomorphism in
$(\mathbb{R}^{n}\times\\{0\\})\backslash\partial^{E}\Omega(p)$, then we say
$M^{n}$ is a $C^{l,\alpha}$-to-edge locally wedge-shaped $n$-hypersurface; see
[MW23, Section 2] for more details.

Similarly, an embedded locally wedge-shaped hypersurface can be stratified
using its local models.

######  Definition 2.3.

Let $M^{n}\subset\Omega^{n+1}$ is an embedded locally wedge-shaped
hypersurface of $\Omega^{n+1}$. We define the stratification
$\partial_{m-2}M\subset\partial_{m-1}M\subset\partial_{m}M$ of $M$ by

$\partial_{n}M:=M,\quad\partial_{n-1}M:=\partial
M,\quad\partial_{n-2}M:=\bigcup_{p\in\partial
M}\psi_{p}\left(\left(\partial_{n-2}\Omega^{n}_{\theta}(p)\times\\{0\\}\right)\cap\mathbb{B}_{R}^{n+1}(0)\right),$

where $(\psi_{p},\mathbb{B}^{n+1}_{R}(p),\Omega(p))$ is a local model of
$M^{n}$ around $p$. Moreover, we define

  * •

$\mathring{M}:=\partial_{n}M\backslash\partial_{n-1}M$ to be the interior of
$M$;

  * •

$\partial^{F}M:=\partial_{n-1}M\backslash\partial_{n-2}M$ to be the face of
$M$;

  * •

$\partial^{E}M:=\partial_{n-2}M$ to be the edge of $M$.

Using the stratification of locally wedge-shaped hypersurfaces, we now
introduce the definition of almost properly embedding.

######  Definition 2.4 (Almost properly embedding).

Let $M^{n}\subset\Omega^{n+1}$ is an embedded locally wedge-shaped
hypersurface of $\Omega^{n+1}$. We say that $M^{n}$ is almost properly
embedded in $\Omega^{n+1}$, denoted by
$(M^{n},\\{\partial_{n}M^{n}\\})\subset(\Omega^{n+1},\\{\partial_{n+1}\Omega^{n+1}\\})$,
if

$\partial_{i}M^{n}\subset\partial_{i+1}\Omega^{n+1},\quad
i\in\\{n-2,n-1,n\\}.$

In particular, if $\partial^{F}M^{n}=\partial^{F}\Omega\cap M$ and
$\partial^{E}M^{n}=\partial^{E}\Omega\cap M$, then we say that
$(M^{n},\\{\partial_{n}M^{n}\\})\subset(\Omega^{n+1},\\{\partial_{n+1}\Omega^{n+1}\\})$
is properly embedded.

In the sequel, we will only consider properly embedded locally wedge-shaped
hypersurfaces.

###  2.2. Variations for locally wedge-shaped hypersurfaces.

We now consider the variational problem for free boundary minimal
hypersurfaces (FBMHs) properly embedded in $\Omega^{n+1}_{\theta}$. Let
$\Sigma^{n}\subset\Omega^{n+1}$ is an embedded locally wedge-shaped
hypersurface of $\Omega^{n+1}$. By Stokes’ theorem, the first variation
formula of the area functional is given by

$\delta\mathcal{A}_{\Sigma}(X)=-\int_{\Sigma}\langle
H_{\Sigma},X\rangle\mathrm{dvol}_{\Sigma}+\int_{\partial^{F}\Sigma}\langle\eta,X\rangle\mathrm{dvol}_{\partial^{F}\Sigma},$

where $\eta$ denotes the unit outward co-normal along $\partial^{F}\Sigma$ and
$X$ is an admissible vector field defined in [MW23, Definition 3.3]. Moreover,
we have the following characterization.

######  Proposition 2.5 ([MW23]).

A properly embedded hypersurface
$(\Sigma^{n},\\{\partial_{n}\Sigma^{n}\\})\subset(\Omega^{n+1},\\{\partial_{n+1}\Omega^{n+1}\\})$
is a FBMH if and only if the following two conditions hold:

  * •

$H_{\Sigma}\equiv 0$ in $\mathring{\Sigma}$,

  * •

$\eta\perp\partial^{F}\Omega$ on $\partial^{F}\Sigma$.

Next, we investigate the stability of locally wedge-shaped hypersurfaces.
Suppose that $\Sigma$ is two-sided so that it admits a continuous choice of
normal vector $\nu$.

######  Proposition 2.6 ([MW23]).

Assume that $\Sigma$ is a FBMH. Suppose that the admissible vector field $X$
restricts to vector field $f\nu$ on $\Sigma$. Then the second variation of the
area functional is given by

(2.1) $\delta^{2}\mathcal{A}_{\Sigma}(X)=\int_{\Sigma}\left(|\nabla
f|^{2}-|A|^{2}f^{2}\right)\mathrm{dvol}_{\Sigma}$

where $A$ denotes the second fundamental form of $\Sigma$ and $\eta$ is the
unit outward co-normal along $\partial\Sigma$.

######  Remark 2.7.

The boundary integral in the original form vanishes due to the properly
embedding and a reflection technique used in [MW23, Theorem 5.3].

Similar to [CM11, Lemma 1.36], the stability of $\Sigma$:

(2.2) $\delta^{2}\mathcal{A}_{\Sigma}(X)=\int_{\Sigma}\left(|\nabla
f|^{2}-|A|^{2}f^{2}\right)\mathrm{dvol}_{\Sigma}\geqslant 0$

is equivalent to the existence of a positive function $w$ on $\Sigma$
satisfying

(2.3)
$\begin{split}\left(\Delta_{\Sigma}+|A|^{2}\right)(w)&=0,~{}~{}\mbox{in}~{}~{}\Sigma,\\\
\frac{\partial
w}{\partial{\eta}}&=0,~{}~{}\mbox{on}~{}~{}\partial\Sigma.\end{split}$

By the well-known construction, the universal cover of $\Sigma$ is also a FBMH
because the canonical projection map is locally isometric. Moreover, if
$\Sigma$ is a stable FBMH, the universal cover $\overline{\Sigma}$ is also
stable because the lifting of $w$ satisfies (2.3) on $\overline{\Sigma}$.
Without loss of generality, in the sequel, we assume that $\Sigma$ is simply
connected.

##  3\. One-endness

Suppose that $\Omega^{n+1}=H^{n+1}_{+}\cap H^{n+1}_{-}$ is a $n+1$-dimensional
wedge domain in $\mathbb{R}^{n+1}$ with wedge angle $\theta$, and
$\left(\Sigma^{n},\left\\{\partial_{n}\Sigma\right\\}\right)\subset\left(\Omega^{n+1},\left\\{\partial_{n+1}\Omega\right\\}\right)$
is a stable $C^{1,1}$-to-edge properly embedded FBMH. We define

$\partial^{F}_{+}\Sigma:=\partial^{F}\Sigma\cap\partial
H^{n+1}_{+},\quad\partial^{F}_{-}\Sigma:=\partial^{F}\Sigma\cap\partial
H^{n+1}_{-}.$

for all $p\in\partial^{F}\Sigma$. We can assume that
$p\in\partial^{F}_{+}\Sigma$. Let $\tilde{\Sigma}$ be the hypersurface
consisting of $\Sigma$ together with its reflection across $\partial
H^{n+1}_{+}$. Then $\tilde{\Sigma}$ is a smooth minimal hypersurface with no
boundary in a neighborhood of $p$ in $\mathbb{R}^{n+1}$.

We claim that $\partial^{F}_{+}\Sigma$ and $\partial^{F}_{-}\Sigma$ are
totally geodesic in $(\Sigma,g)$ with induced metric $g$. We extend $e_{i}\in
T_{p}\partial^{F}_{+}\Sigma$, $i=\\{1,\cdots,n\\}$, to be an orthonormal frame
in a neighborhood of $p$ in $\tilde{\Sigma}$. Let
$\gamma:(-1,1)\to\tilde{\Sigma}$ be a curve that

$\gamma(0)=p,\quad\gamma^{{}^{\prime}}(0)=\eta,\quad\gamma(t)=R(\gamma(-t)),$

where $R$ denotes reflection across $\partial H^{n+1}_{+}$. Note that the
function

$f(t):=\langle\nabla^{g}_{e_{i}}e_{j},\gamma^{{}^{\prime}}\rangle(\gamma(t)),\quad
i,j=\\{1,\cdots,n\\},$

is an odd function, which implies that for $i,j=\\{1,\cdots,n\\}$

$f(0)=\langle\nabla^{g}_{e_{i}}e_{j},\eta\rangle=0.$

Therefore, we conclude that $\partial^{F}_{+}\Sigma$ and
$\partial^{F}_{-}\Sigma$ are totally geodesic.

By analyzing harmonic functions on $\Sigma^{n}$, we will show that
$\Sigma^{n}$ has only one end. In the sequel, without loss of generality, we
may assume that $0\in\partial^{E}\Sigma^{n}$.

######  Lemma 3.1.

Every end of $\Sigma^{n}$ has infinite volume.

###### Proof.

We let $r(x)=\mathrm{dist}_{\mathbb{R}^{n+1}}(0,x)$ be the distance function
of $\mathbb{R}^{n+1}$, and $\bar{r}(x)=\mathrm{dist}_{\Sigma}(0,x)$ be the
distance function on $\Sigma$ with respect to the induced metric. Choosing
$s_{0}$ large enough so that

$\Sigma\backslash\mathbb{B}_{\mathbb{R}^{n+1}}(s_{0})=\bigcup_{j=1}^{k}E_{j},\quad
k\geqslant 1$

is the disjoint union of noncompact connected components. By Sard’s theorem,
we can assume that $\mathbb{B}_{\mathbb{R}^{n+1}}(s_{0})$ intersects with each
$E_{j}$ transversally. Without loss of generality, we take $E=E_{1}$ as an
example. There are three mutually exclusive possibilities: $E$ admits the
boundary which

  1. (1)

doesn’t intersect with $\partial H^{n+1}_{+}\cup\partial H^{n+1}_{-}$;

  2. (2)

intersects with $\partial H^{n+1}_{+}$ or $\partial H^{n+1}_{-}$ merely;

  3. (3)

intersects with both $\partial H^{n+1}_{+}$ and $\partial H^{n+1}_{-}$.

In the second case, without loss of generality, we can assume that $E$
intersects with $\partial H^{n+1}_{+}$. Let $\tilde{E}$ be the hypersurface
consisting of $E$ together with its reflection across $\partial H^{n+1}_{+}$.
For convenience, in the first and third case, let $\tilde{E}$ be $E$.

Let $\mathbb{B}_{\Sigma}(s)$ be the geodesic ball contained in $\tilde{E}$, of
radius $s$ centered at the origin. By Lemma 1 in [CSZ97], we know that
$s^{-n}\mathrm{Vol}(B_{\Sigma}(s))$ is non-decreasing. Therefore

$\frac{\mathrm{Vol}(B_{\Sigma}(s))}{s^{n}}\geqslant\lim_{s\to
0}\frac{\mathrm{Vol}(B_{\Sigma}(s))}{s^{n}}=\omega(n),$

where $\omega(n)$ is the volume of unit ball in $\mathbb{R}^{n}$. If $E$ has
finite volume, it implies that $\tilde{E}$ has finite volume as well. In the
first and second cases, we can choose $R$ big enough such that

$\mathrm{Vol}(\tilde{E})\geqslant\mathrm{Vol}(B_{p}(s))\geqslant\omega(n)R^{n}>\mathrm{Vol}(\tilde{E}),$

a contradiction.

We follow the argument in [CSZ97, Lemma 1] to deal with the third case. As
shown in [CSZ97], we have

(3.1) $\frac{\partial r}{\partial\bar{r}}\leqslant 1.$

By a direct computation and using the fact that $\Sigma$ is minimal, we can
show that in the interior of $\Sigma$,

(3.2) $\Delta_{\Sigma}r^{2}(x)=2n.$

Note that the radial function $r^{2}(x)$ can be extended to be an even
function on $\tilde{\Sigma}$ around any point $p$ on $\partial^{F}\Sigma$.
Similarly, we can conclude that

(3.3) $\frac{\partial
r^{2}}{\partial\eta_{\pm}}(p)=0~{}~{}\mbox{on}~{}~{}\partial H^{n+1}_{\pm},$

where $\eta_{\pm}$ are the unit outward co-normal along $\partial
H^{n+1}_{\pm}$.

Due to (3.3), integrating (3.2) over $\mathbb{B}_{\Sigma}(s)\cap E$ yields

(3.4) $2n\mathrm{vol}(\mathbb{B}_{\Sigma}(s)\cap
E)=\int_{\partial\mathbb{B}_{\Sigma}(s)\cap\mathring{E}}\frac{\partial
r^{2}}{\partial\eta_{1}}\mathrm{dvol}+\int_{\partial
E\cap\overline{\mathbb{B}}_{\mathbb{R}^{n+1}}(s_{0})}\frac{\partial
r^{2}}{\partial\eta_{2}}\mathrm{dvol},$

where $\eta_{1}$ and $\eta_{2}$ are the unit outward co-normal vector fields
along $\partial\mathbb{B}_{\Sigma}(s)\cap\mathring{E}$ and $\partial
E\cap\overline{\mathbb{B}}_{\mathbb{R}^{n+1}}(s_{0})$, respectively.

By our assumption, $\mathbb{B}_{\mathbb{R}^{n+1}}(s_{0})$ intersects with $E$
transversally. It implies that $\eta_{2}$ points inward the ball
$\mathbb{B}_{\mathbb{R}^{n+1}}(s_{0})$ along $\partial
E\cap\overline{\mathbb{B}}_{\mathbb{R}^{n+1}}(s_{0})$. Therefore,

(3.5) $\int_{\partial
E\cap\overline{\mathbb{B}}_{\mathbb{R}^{n+1}}(s_{0})}\frac{\partial
r^{2}}{\partial\eta_{2}}\mathrm{dvol}\leqslant 0.$

Similarly, we have

$2n\mathrm{vol}(\mathbb{B}_{\Sigma}(s)\cap E)\leqslant
2s\mathrm{vol}(\partial\mathbb{B}_{\Sigma}(s)\cap\mathring{E}).$

Note that

$\mathrm{vol}(\partial\mathbb{B}_{\Sigma}(s)\cap\mathring{E})=\frac{\partial}{\partial
t}\Big{|}_{t=s}\mathrm{vol}(\mathbb{B}_{\Sigma}(t)\cap E).$

In the same spirit as [CSZ97, Lemma 1], we conclude that the quantity
$s^{-n}\mathrm{vol}(\mathbb{B}_{\Sigma}(s)\cap\mathring{E})$ is nondecreasing.
Therefore,

$\frac{\mathrm{vol}(\mathbb{B}_{\Sigma}(s)\cap\mathring{E})}{s^{n}}\geqslant\frac{\mathrm{vol}(\mathbb{B}_{\Sigma}(s_{0})\cap\mathring{E})}{s_{0}^{n}}>0,$

for fixed value $s_{0}$. If $E$ has finite volume, similar to the above, we
obtain a contradiction. ∎

######  Lemma 3.2.

Suppose that $\Sigma^{n}$ has at lease two ends, then here exists a non-
constant bounded harmonic function with finite energy on $\Sigma^{n}$.

###### Proof.

As constructed in Lemma 2 [CSZ97], there is an exhausion
$\left\\{D_{i}\right\\}$ of $\Sigma$ by compact submanifolds with boundary.
For $i\geqslant i_{0}$ and $i_{0}$ sufficiently large, let

$\Sigma\backslash D_{i}=\bigcup_{j=1}^{n}E^{(i)}_{j},\quad n\geqslant 1$

be the disjoint union of connected components. By Lemma 3.1 and the
assumption, $\Sigma$ has at least two components with infinite volume. Let
$E^{(i_{0})}_{1}$ and $E^{(i_{0})}_{2}$ be two such components. On each
compact domain $D_{i}$, we consider the following Dirichlet–Neumann problem

(3.6)
$\left\\{\begin{array}[]{ll}\Delta_{\Sigma}u&=0,~{}~{}\mbox{in}~{}~{}\mathring{D_{i}}\\\
u&=1,~{}~{}\mbox{on}~{}~{}\partial E^{(i)}_{1}\\\
u&=0,~{}~{}\mbox{on}~{}~{}\partial E^{(i)}_{j},j\neq 1\\\ \frac{\partial
u}{\partial\eta}&=0,~{}~{}\mbox{on}~{}~{}\partial
D_{i}\cap\partial^{F}\Sigma,\end{array}\right.$

where $\eta$ is the unit conormal to $\partial^{F}\Sigma$. Let $u_{i}$ be the
unique solution of (3.6). By the maximum principle and Hopf lemma, we have
$0\leqslant u_{i}\leqslant 1$ on $D_{i}$. Moreover, it’s easy to see that for
$i>j$

$\int_{D_{i}}|\nabla
u_{i}|^{2}\mathrm{dvol}_{\Sigma}\leqslant\int_{D_{j}}|\nabla
u_{j}|^{2}\mathrm{dvol}_{\Sigma}.$

For simplicity, in the following arguments, let $C>0$ be a varying uniform
constant. Hence, there is a constant $C$ such that

$\int_{D_{i}}|\nabla u_{i}|^{2}\mathrm{dvol}_{\Sigma}<C.$

Therefore by passing to a subsequence, still denoted by $u_{i}$, we can find a
harmonic function $u$ on $\Sigma$ satisfying

$\lim_{i\to\infty}u_{i}=u,\quad\int_{\Sigma}|\nabla
u|^{2}\mathrm{dvol}_{\Sigma}<C,\quad 0\leqslant u\leqslant 1.$

In the following we prove that the limiting harmonic function $u$ is not a
constant function. We will prove this by contradiction.

Setting $\phi=u_{i}(1-u_{i})$. Note that from the construction of $u_{i}$,
$\phi$ vanishes on $\partial E^{(i)}_{1}\cup\partial E^{(i)}_{2}$ and has
finite energy as well. Besides, $\phi$ satisfies the Neumann condition on
$\partial D_{i}\cap\partial^{F}\Sigma$. We define the Sobolev trace quotient
$\mathcal{Q}_{\partial}(\Sigma)$ by

$\mathcal{Q}_{\partial}(\Sigma)=\inf\left\\{\mathcal{Q}^{g}_{\partial}(\varphi):\varphi\in
C^{1}(\Sigma),\varphi\neq
0,\frac{\partial\varphi}{\partial\eta_{\pm}}=0\right\\},$

where

$\mathcal{Q}^{g}_{\partial}(\varphi):=\frac{\int_{\Sigma}\left(|\nabla\varphi|^{2}_{g}+\frac{n-2}{4(n-1)}R_{g}\varphi^{2}\right)\mathrm{dvol}_{\Sigma}+\int_{\partial\Sigma}\left(\varphi\frac{\partial\varphi}{\partial\eta}+\frac{n-2}{2}H_{\partial\Sigma}\varphi^{2}\right)\mathrm{dvol}_{\partial\Sigma}}{\left(\int_{\partial\Sigma}|\varphi|^{\frac{2(n-1)}{n-2}}\mathrm{dvol}_{\partial\Sigma}\right)^{\frac{n-2}{n-1}}}.$

The boundary integral in the numerator of
$\mathcal{Q}^{g}_{\partial}(\varphi)$ vanishes because $\varphi$ satisfies the
Neumann condition and $\partial^{F}\Sigma$ is totally geodesic. Moreover, by
the stability of $\Sigma^{n}$ and the Gauss equation, we have

$\int_{\Sigma}\left(|\nabla\varphi|^{2}_{g}+\frac{n-2}{4(n-1)}R_{g}\varphi^{2}\right)\mathrm{dvol}_{\Sigma}>\int_{\Sigma}\left(|\nabla\varphi|^{2}_{g}-|A|^{2}\varphi^{2}\right)\mathrm{dvol}_{\Sigma}\geqslant
0,$

which implies that $\mathcal{Q}_{\partial}(\Sigma)>0$.

Therefore, we have

(3.7) $\left(\int_{\partial
D_{i}}\phi^{\frac{2(n-1)}{n-2}}\mathrm{dvol}_{\partial
D_{i}}\right)^{\frac{n-2}{2(n-1)}}\leqslant
C\left(\int_{D_{i}}|\nabla\phi|^{2}\mathrm{dvol}_{D_{i}}\right)^{\frac{1}{2}}.$

Combining this with [Bre21, Theorem 1.1], similar to arguments in [CL23,
Section 3.3], we obtain that

$\left(\int_{D_{i}}\phi^{\frac{2n}{n-2}}\mathrm{dvol}_{\Sigma}\right)^{\frac{n-2}{n}}\leqslant
C\int_{D_{i}}|\nabla\phi|^{2}\mathrm{dvol}_{\Sigma}\leqslant C.$

Since $\mathrm{Vol}(D_{i})$ goes to infinity, it follows that if $u$ is a
constant function, then $u\equiv 0$ or $u\equiv 1$. As argued in [CSZ97, Lemma
2], we may assume that $u\equiv 1$. We choose a smooth function $\psi$ such
that

(3.8)
$\psi=\left\\{\begin{array}[]{cl}&1,~{}~{}\mbox{in}~{}~{}E^{(i_{0})}_{2}\\\
&0,~{}~{}\mbox{in}~{}~{}E^{(i_{0})}_{j},j\neq 2\\\ \end{array}\right.$

and

$|\nabla\psi|<C,\quad 0\leqslant\psi\leqslant 1$

for some constant $C$ independent of $i$ and $u_{i}$. Note that $|\nabla\psi|$
vanishes outside a compact set.

By (3.6) and (3.8), the function $\psi u_{i}$ vanishes on on $\partial
E^{(i)}_{1}\cup\partial E^{(i)}_{2}$. Similar to above, we can claim that

$\left(\int_{D_{i}}(\psi
u_{i})^{\frac{2n}{n-2}}\mathrm{dvol}_{\Sigma}\right)^{\frac{n-2}{n}}\leqslant
C.$

In particular,

$\left(\int_{D_{i}\cap E^{(i_{0})}_{2}}(\psi
u_{i})^{\frac{2n}{n-2}}\mathrm{dvol}_{\Sigma}\right)^{\frac{n-2}{n}}\leqslant
C,\quad~{}~{}\mbox{for all}~{}~{}i\geqslant i_{0}.$

Letting $i\to\infty$, we get $\mathrm{Vol}(E^{(i_{0})}_{2})\leqslant C$, a
contradiction with Lemma 3.1. Therefore, the limiting hamronic function is not
a constant function. ∎

######  Lemma 3.3.

Suppose that
$\left(\Sigma^{3},\left\\{\partial_{3}\Sigma\right\\}\right)\subset\left(\Omega^{4},\left\\{\partial_{4}\Omega\right\\}\right)$
is a stable $C^{1,1}$-to-edge properly embedded FBMH and u is a harmonic
function on $\Sigma$. Then

$\left(1-\frac{1}{\sqrt{2}}\right)\int_{\Sigma}\varphi^{2}|A|^{2}|\nabla
u|^{2}\mathrm{dvol}_{\Sigma}+\frac{1}{2}\int_{\Sigma}\varphi^{2}|\nabla|\nabla
u||^{2}\mathrm{dvol}_{\Sigma}\leqslant\int_{\Sigma}|\nabla\varphi|^{2}|\nabla
u|^{2}\mathrm{dvol}_{\Sigma},$

for any $\varphi\in C^{1}(\Sigma)$.

###### Proof.

Detecting the proof of [CL23, Lemma 21], we find that there is an extra
boundary integral in (12) when integrating by parts; i.e, we need to estimate

$\int_{\partial\Sigma}|\nabla u|\langle\nabla|\nabla
u|,\eta\rangle\varphi^{2}\mathrm{dvol}_{\partial\Sigma}.$

Since $\mathcal{H}^{n-1}(\partial^{E}\Sigma)=0$, it suffices to show that
$\langle\nabla|\nabla u|,\eta\rangle(p)=0$ for all $p\in\partial^{F}\Sigma$.
Note that the harmonic function $u$ satisfies the Neumann condition
$\frac{\partial u}{\partial\eta}=0$ on $\partial^{F}\Sigma$. Let $\tilde{u}$
be the even reflection of $u$ across $\partial H_{+}$. Then $\tilde{u}$ is a
harmonic function in a neighborhood of $p$. Now let
$\gamma:(-1,1)\to\tilde{\Sigma}$ be a curve that

$\gamma(0)=p,\quad\gamma^{{}^{\prime}}(0)=\eta,\quad\gamma(t)=R(\gamma(-t)),$

where $R$ denotes reflection across $\partial H_{+}$. Then
$|\nabla\tilde{u}|(\gamma(t))$ is an even function of $t$ and so

(3.9)
$0=\frac{d}{dt}\bigg{|}_{t=0}|\nabla\tilde{u}|(\gamma(t))=\langle\nabla|\nabla
u|,\eta\rangle(p).$

Since $p$ is arbitrary, the boundary integral therefore vanishes. The
remainder of the proof can now be completely exactly as in [CL23]. ∎

Now we are ready to prove the one-endness.

######  Proposition 3.4.

Suppose that
$\left(\Sigma^{3},\left\\{\partial_{3}\Sigma\right\\}\right)\subset\left(\Omega^{4},\left\\{\partial_{4}\Omega\right\\}\right)$
is a stable $C^{1,1}$-to-edge properly embedded FBMH. $\Sigma$ has only one
end.

###### Proof.

Suppose the contrary, that $\Sigma$ has at least two ends. Then Lemma 3.2
implies that $\Sigma$ admits a nontrivial harmonic function $u$ with finite
energy. Take an arbitrary point $p\in\mathring{\Sigma}$. For $\rho>0$, take
$\varphi\in C^{1}(\Sigma)$ such that

(3.10)
$\varphi=\left\\{\begin{array}[]{cl}&1,~{}~{}\mbox{in}~{}~{}B_{\Sigma}(p,\rho)\\\
&0,~{}~{}\mbox{in}~{}~{}B^{c}_{\Sigma}(p,2\rho),\\\ \end{array}\right.$

and $|\nabla\varphi|\leqslant\frac{2}{\rho}$. Note that the constructed
$\varphi$ are not needed to be compact supported in $\mathring{\Sigma}$. Then
Lemma 3.3 implies that

$\left(1-\frac{1}{\sqrt{2}}\right)\int_{B_{\Sigma}(p,\rho)}\varphi^{2}|A|^{2}|\nabla
u|^{2}\mathrm{dvol}_{\Sigma}+\frac{1}{2}\int_{B_{\Sigma}(p,\rho)}\varphi^{2}|\nabla|\nabla
u||^{2}\mathrm{dvol}_{\Sigma}\leqslant\frac{4C}{\rho^{2}}.$

Sending $\rho\to\infty$, we conclude that $|\nabla|\nabla u||^{2}\equiv 0$,
which implies that $|\nabla u|$ is a constant. Since $u$ is nonconstant, we
have that $|\nabla u|>0$. However, this implies that

$\mathrm{Vol}(\Sigma)=\frac{1}{|\nabla u|^{2}}\int_{\Sigma}|\nabla
u|^{2}<\infty,$

a contradiction. ∎

##  4\. A conformal deformation of metrics

Suppose that
$\left(\Sigma^{n},\left\\{\partial_{n}\Sigma\right\\}\right)\subset\left(\Omega^{n+1},\left\\{\partial_{n+1}\Omega\right\\}\right)$
is a stable $C^{1,1}$-to-edge properly embedded FBMH. Without loss of
generality, we may assume that $0\in\partial^{E}\Sigma^{n}$. In this section
we carry out the conformal deformation technique used by Chodosh–Li [CL23].

Consider the function $r(x)=\mathrm{dist}_{\mathbb{R}^{n+1}}(0,x)$ on
$\Sigma$. As shown in [CL23, Section 5], we find that

$\Delta r=\frac{n}{r}-\frac{|\nabla r|^{2}}{r}.$

Suppose that $w>0$ is a smooth function on $\Sigma\backslash\\{0\\}$. On
$\Sigma\backslash\\{0\\}$, define $\tilde{g}=w^{2}g$. For
$\lambda\in\mathbb{R}$, we consider the quadratic form

$\mathcal{Q}_{w}(\varphi)=\int_{\Sigma}\left(|\tilde{\nabla}\varphi|^{2}+\left(\frac{1}{2}\tilde{R}-\lambda\right)\varphi^{2}\right)dV_{\tilde{g}},$

where $\varphi\in C^{1}_{c}(\Sigma\backslash\\{0\\})$ satisfies

(4.1)
$\frac{\partial\varphi}{\partial\tilde{\eta}}=0,\quad~{}~{}\mbox{on}~{}~{}\partial\left(\Sigma\backslash\\{0\\}\right),$

and $\tilde{\nabla}$, $\tilde{R}$, $dV_{\tilde{g}}$, $\tilde{\eta}$ are the
gradient, the scalar curvature, the volume form and the outward normal
derivative with respect to $\tilde{g}$, respectively. In the sequel, we choose
$w=r^{-1}$ on $\Sigma\backslash\\{0\\}$, and the dimension of $\Sigma^{n}$ is
equal to $3$.

One relates the geometric quantities in $g$ and $\tilde{g}$ as follows:

(4.2)
$|\nabla\varphi|_{g}^{2}=w^{2}|\tilde{\nabla}\varphi|_{\tilde{g}}^{2},\quad
dV_{\tilde{g}}=w^{-n}dV_{g},\quad\tilde{\eta}=w^{-1}\eta.$

Moreover, we have

$w^{2}\tilde{R}=R-2(n-1)\Delta\log w-(n-1)(n-2)|\nabla\log w|^{2}.$

For $\varphi\in C^{1}_{c}(\Sigma\backslash\\{0\\})$ satisfying (4.1), by
(4.2), we know that

(4.3) $\frac{\partial\varphi}{\partial\eta}=0.$

Note that the radial function $w$ can be extended to be an even function on
the reflection of $\Sigma$. Combining it with the argument used in Section 3,
we have

(4.4)
$\frac{\partial(w^{\frac{2-n}{2}}\varphi)}{\partial\tilde{\eta}}=w^{\frac{2-n}{2}}\frac{\partial\varphi}{\partial\tilde{\eta}}+\varphi\frac{\partial
w^{\frac{2-n}{2}}}{\partial\tilde{\eta}}\\\ =\varphi\frac{\partial
w^{\frac{2-n}{2}}}{\partial\tilde{\eta}}=\varphi w^{-1}\frac{\partial
w^{\frac{2-n}{2}}}{\partial{\eta}}=0.$

Denote by
$\tilde{\mathcal{Q}}_{w}(\varphi):=\mathcal{Q}_{w}(w^{\frac{2-n}{2}}\varphi)$.
Following the same computation in [CL23, Section 5], we have

$\begin{split}&\tilde{\mathcal{Q}}_{w}(\varphi)\\\
=&\int_{\Sigma}\left(w^{-2}|{\nabla}(w^{\frac{2-n}{2}}\varphi)|_{g}^{2}+\left(\frac{1}{2}\tilde{R}-\lambda\right)w^{2-n}\varphi^{2}\right)w^{n}dV_{{g}}\\\
=&\int_{\Sigma}\left(|{\nabla}\varphi-\frac{n-2}{2}\varphi\nabla\log
w|_{g}^{2}+\left(\frac{1}{2}\tilde{R}-\lambda\right)w^{2}\varphi^{2}\right)dV_{{g}}\\\
=&\int_{\Sigma}\left(|{\nabla}\varphi|_{g}^{2}-\frac{n-2}{2}\langle{\nabla}\varphi^{2},\nabla\log
w\rangle_{g}+\left(\frac{n-2}{2}\right)^{2}|\nabla\log
w|^{2}_{g}\varphi^{2}+\left(\frac{1}{2}\tilde{R}-\lambda\right)w^{2}\varphi^{2}\right)dV_{{g}}\\\
=&\int_{\Sigma}\left(|{\nabla}\varphi|_{g}^{2}+\left(\frac{n-2}{2}\Delta\log
w+\left(\frac{n-2}{2}\right)^{2}|\nabla\log
w|^{2}_{g}+\left(\frac{1}{2}\tilde{R}-\lambda\right)w^{2}\right)\varphi^{2}\right)dV_{{g}}\\\
&-\int_{\partial\Sigma}\frac{n-2}{2}\varphi^{2}\frac{\partial\log
w}{\partial\eta}dA_{g}\\\
=&\int_{\Sigma}\left(|{\nabla}\varphi|_{g}^{2}+\frac{1}{2}{R}\varphi^{2}-\left(\frac{n}{2}\left(\Delta\log
w+\frac{n-2}{2}|\nabla\log w|^{2}_{g}\right)+\lambda
w^{2}\right)\varphi^{2}\right)dV_{{g}}\end{split}$

where the boundary term vanishes due to (4.4). Note that

$\Delta\log w+\frac{n-2}{2}|\nabla\log
w|^{2}_{g}=-\frac{n}{r^{2}}+\frac{n+2}{2}\frac{|\nabla r|^{2}}{r^{2}}.$

Therefore,

(4.5)
$\begin{split}\tilde{\mathcal{Q}}_{w}(\varphi)&=\int_{\Sigma}\left(|{\nabla}\varphi|_{g}^{2}+\frac{1}{2}{R}\varphi^{2}+\left(\frac{n}{2}\left(n-\frac{n+2}{2}|\nabla
r|^{2}_{g}\right)-\lambda\right)r^{-2}\varphi^{2}\right)dV_{{g}}\\\
&\geqslant\int_{\Sigma}\left(|{\nabla}\varphi|_{g}^{2}+\frac{1}{2}{R}\varphi^{2}+\left(\frac{n(n-2)}{4}-\lambda\right)r^{-2}\varphi^{2}\right)dV_{{g}}.\end{split}$

By the Gauss equation, we have $|A|_{g}^{2}+R_{g}=0$. On the other hand, by
[MW23, Theorem 5.1], we know that for any $\varphi\in
C^{1}_{c}(\Sigma\backslash\\{0\\})$,

$\int_{\Sigma}\left(|{\nabla}\varphi|_{g}^{2}+\frac{1}{2}{R}\varphi^{2}\right)dV_{g}\geqslant\int_{\Sigma}\left(|{\nabla}\varphi|_{g}^{2}-|A|^{2}_{g}\varphi^{2}\right)dV_{g}\geqslant
0.$

We summarize these in the following Proposition.

######  Proposition 4.1.

Suppose that
$\left(\Sigma^{n},\left\\{\partial_{n}\Sigma\right\\}\right)\subset\left(\Omega^{n+1},\left\\{\partial_{n}\Omega\right\\}\right)$
is a stable $C^{1,1}$-to-edge properly embedded FBMH. Then the conformally
deformed manifold $(\Sigma\backslash\\{0\\},\tilde{g}=r^{-2}g)$ satisfies

(4.6)
$\lambda^{N}_{1}\left(-\tilde{\Delta}+\frac{1}{2}\tilde{R}\right)\geqslant\lambda,$

where $\lambda^{N}_{1}$ is the first Neumann eigenvalue and
$\lambda=\frac{n(n-2)}{4}$.

##  5\. Volume Estimate

We first prove a diameter bound for free boundary warped $\mu-$bubbles in
$3-$manifolds with boundary satisfying
$\lambda^{N}_{1}\left(-\tilde{\Delta}+\frac{1}{2}\tilde{R}\right)\geqslant\lambda>0$.

Recall that $r(x)=\mathrm{dist}_{\mathbb{R}^{4}}(0,x)$ and
$\bar{r}(x)=\mathrm{dist}_{(\Sigma,g)}(x,0)$, and we consider
$\tilde{g}=r^{-2}g$. Combining the reflection technique used above and the
fact that the conformal factor is symmetric with respect to the reflection, we
conclude that $\partial^{F}_{+}\Sigma$ and $\partial^{F}_{-}\Sigma$ are
totally geodesic with respect to $\tilde{g}$ as well.

Fix $\rho>0$, and consider the ball
$B_{\mathbb{R}^{4}}(0,e^{\frac{5\pi}{\sqrt{\lambda}}}\rho)$. By Proposition
3.4, $\Sigma\backslash
B_{\mathbb{R}^{4}}(0,e^{\frac{5\pi}{\sqrt{\lambda}}}\rho)$ has only one
unbounded connected component $E$. Denote by
$\Sigma^{{}^{\prime}}=\Sigma\backslash E$. We claim that
$\partial\Sigma^{{}^{\prime}}\cap\mathring{\Sigma}=\partial
E\cap\mathring{\Sigma}$ is connected. Indeed, since $\Sigma^{{}^{\prime}}$ and
$E$ are both connected, if $\partial\Sigma^{{}^{\prime}}\cap\mathring{\Sigma}$
has more than one connected components, one can find a loop in
$\mathring{\Sigma}$ intersecting one component of
$\partial\Sigma^{{}^{\prime}}\cap\mathring{\Sigma}$ exactly once,
contradicting that $\Sigma$ is simply connected. For convenience, we denote
$\partial\Sigma^{{}^{\prime}}\cap\mathring{\Sigma}$,
$\partial\Sigma^{{}^{\prime}}\cap\partial H^{4}_{+}$ and
$\partial\Sigma^{{}^{\prime}}\cap\partial H^{4}_{-}$ by
$\partial\mathring{\Sigma}^{{}^{\prime}}$,
$\partial^{F}_{+}{\Sigma}^{{}^{\prime}}$ and
$\partial^{F}_{-}{\Sigma}^{{}^{\prime}}$, respectively.

######  Lemma 5.1.

Let $(\Sigma^{{}^{\prime}},\tilde{g})$ be constructed as above satisfying

(5.1)
$\lambda^{N}_{1}\left(-\tilde{\Delta}+\frac{1}{2}\tilde{R}\right)\geqslant\lambda>0.$

Then there exists a connected proper embedded open set $\Pi$ containing
$\partial\mathring{\Sigma}^{{}^{\prime}}$, $\Pi\subset
B_{\Sigma}(\partial\mathring{\Sigma}^{{}^{\prime}},\frac{5\pi}{\sqrt{\lambda}})$,
such that each connected component of
$\partial\Pi\backslash\partial\mathring{\Sigma}^{{}^{\prime}}$ is a $2$-sphere
with area at most $\frac{8\pi}{\lambda}$ and intrinsic diameter at most
$\frac{2\pi}{\sqrt{\lambda}}$.

###### Proof.

This is an application of estimates for the free boundary warped $\mu-$bubbles
(see, e.g. [CL23, Section 4]). Since $\Sigma^{{}^{\prime}}$ satisfies (5.1),
there exists $u\in C^{\infty}(\mathring{\Sigma}^{{}^{\prime}})$ satisfying
(4.1), $u>0$ in $\mathring{\Sigma}^{{}^{\prime}}$ and $C^{1,\alpha}$-to-edge,
such that

(5.2)
$\tilde{\Delta}_{\Sigma^{{}^{\prime}}}u\leqslant-\frac{1}{2}\left(2\lambda-\tilde{R}_{\Sigma^{{}^{\prime}}}\right)u.$

Take $\varphi_{0}\in C^{\infty}(\Sigma^{{}^{\prime}})$ to be a smoothing of
$\mathrm{dist}_{\Sigma^{{}^{\prime}}}(\cdot,\partial\mathring{\Sigma}^{{}^{\prime}})$
such that $|\mathrm{Lip}(\varphi_{0})|\leqslant 2$, and $\varphi_{0}=0$ on
$\partial\mathring{\Sigma}^{{}^{\prime}}$. Choose $\epsilon\in(0,\frac{1}{2})$
such that $\epsilon$, $\frac{4\pi}{\sqrt{\lambda}}+2\epsilon$ are regular
values of $\varphi_{0}$. Define

$\varphi=\frac{\varphi_{0}-\epsilon}{\frac{4}{\sqrt{\lambda}}+\frac{\epsilon}{\pi}}-\frac{\pi}{2},$

$\Pi_{1}=\\{x\in\Sigma^{{}^{\prime}}:-\frac{\pi}{2}<\varphi<\frac{\pi}{2}\\}$,
and $\Pi_{0}=\\{x\in\Sigma^{{}^{\prime}}:-\frac{\pi}{2}<\varphi\leqslant
0\\}$. We have that
$|\mathrm{Lip}(\varphi)|\leqslant\frac{\sqrt{\lambda}}{2}$. In $\Pi_{1}$,
define $h(x)=-\frac{1}{2}\tan(\varphi(x))$. By a direct computation, we have

(5.3) $\lambda+h^{2}-2|\tilde{\nabla}h|\geqslant 0.$

Note that
$\partial\mathring{\Sigma}^{{}^{\prime}}\cap\partial^{F}_{\pm}{\Sigma}^{{}^{\prime}}$
consisting of smooth $1$-dimensional closed submanifolds. Moreover, we may
assume that $\partial^{F}_{\pm}{\Sigma}^{{}^{\prime}}$ meets
$\partial\mathring{\Sigma}^{{}^{\prime}}$ orthogonally.

Consider the functional

$\mathcal{A}(\Pi)=\int_{\partial\Pi}ud\mathcal{H}^{2}-\int_{\Pi_{1}}\left(\chi_{\Pi}-\chi_{\Pi_{0}}\right)hud\mathcal{H}^{3}$

among Caccioppoli sets $\Pi$ in $\Pi_{1}$ with $\Pi\Delta\Pi_{0}$ compactly
contained in $\Pi_{1}$. By [CL23, Proposition 15], there exists
$\overline{\Pi}$ with
$\partial\overline{\Pi}\subset\Pi_{1}\cup\partial\mathring{\Sigma}^{{}^{\prime}}$
minimizing $\mathcal{A}$ among such regions.
$\partial\overline{\Pi}\cap\mathring{\Sigma}^{{}^{\prime}}$ is smooth and
along it,

(5.4)
$H=-u^{-1}\langle\tilde{\nabla}_{\Sigma^{{}^{\prime}}}u,\nu_{\partial\Pi}\rangle+h.$

Moreover, $\partial\overline{\Pi}$ meets
$\partial^{F}_{\pm}{\Sigma}^{{}^{\prime}}$ orthogonally and it may have
nonempty intersection with $\partial^{E}\Sigma^{{}^{\prime}}$.

We take $\Pi$ to be the connected component of
$\\{x\in\Sigma^{{}^{\prime}}:0\leqslant\varphi_{0}\leqslant\epsilon\\}\cup\overline{\Pi}$
that contains $\partial\mathring{\Sigma}^{{}^{\prime}}$. We claim that each
connected component $\Xi$ of $\partial\Pi\cap\Pi_{1}$ is properly embedded in
$\Sigma^{{}^{\prime}}$. Without loss of generality, we may assume that
$\partial\Xi$ is not contained in $\mathring{\Sigma}^{{}^{\prime}}$. First, by
(4.1) and the fact that $\partial^{F}_{\pm}{\Sigma}^{{}^{\prime}}$ are totally
geodesic, we know that $\mathring{\Xi}$ can’t touch
$\partial^{F}_{\pm}{\Sigma}^{{}^{\prime}}$. Besides, at any
$p\in\partial^{F}\Xi\cap\partial^{E}\Sigma^{{}^{\prime}}$, without loss of
generality, we may assume that the unit outward co-normal $\varsigma$ to
$\partial^{F}\Xi$ is orthogonal to $\partial^{F}_{+}{\Sigma}^{{}^{\prime}}$.
Denote $X\in T_{p}\partial^{F}_{-}{\Sigma}^{{}^{\prime}}$ the unit vector
orthogonal to $T_{p}\partial^{F}\Xi$. Note that by our assumption, the inner
product $\langle\varsigma,X\rangle$ at the point $p$ is negative. We can
extend $X$ to be a variation around $p$ which preserves the negativity of
$\langle\varsigma,X\rangle$. Combining this with (5.4) and the first variation
formula, we have

$\delta\mathcal{A}_{\Pi}(X)=\int_{\partial^{F}\Xi}\langle\varsigma,X\rangle<0,$

which is contradicted with the minimizing property of $\Pi$.

Due to the $C^{1,\alpha}$-to-edge regularity, we shall make use of the log-
cutoff trick. Specifically, we define cutoff functions $\eta_{r}$ on $\Pi$
which vanish in a neighborhood of $\partial^{E}\Pi$ (possibly empty) and which
convergence pointwisely to $1$ on $\Pi\backslash\partial^{E}\Pi$ as $r\to 0$.
Since $\partial^{E}\Pi$ is of codimension 2 in $\Pi$, we can arrange that

$\int_{\Pi}|\nabla\eta_{r}|^{2}d\mathcal{H}^{2}\to 0,~{}~{}\mbox{as}~{}~{}r\to
0.$

Now, we verify that $\Pi$ satisfies the conclusions of this Lemma. Given a
test function $\psi$ defined on $\Pi$, composing with the cutoff function
$\eta_{r}$ if necessary. For convenience, we still denote it by $\psi$.
Indeed, for any connected component $\Xi$ of $\partial\Pi\cap\Pi_{1}$, the
stability of $\mathcal{A}$ [CL23, Proposition] implies

$\begin{split}0\leqslant&\int_{\Xi}\left(|\nabla\psi|^{2}u-\frac{1}{2}\left(\tilde{R}_{\Sigma^{{}^{\prime}}}-\lambda-\tilde{R}_{\Xi}+|\mathring{\tilde{A}}_{\Xi}|^{2}\right)\psi^{2}u+\left(\tilde{\Delta}_{\Sigma^{{}^{\prime}}}u-\tilde{\Delta}_{\Xi}u\right)\psi^{2}\right)d\mathcal{H}^{2}\\\
&\int_{\Xi}\left(-\frac{1}{2}u^{-1}\langle\tilde{\nabla}_{\Sigma^{{}^{\prime}}}u,\nu_{\partial\Pi}\rangle^{2}\psi^{2}-\frac{1}{2}\left(\lambda+h^{2}+2\langle\tilde{\nabla}_{\Sigma^{{}^{\prime}}}h,\nu_{\partial\Pi}\right)\psi^{2}u\right)d\mathcal{H}^{2}\\\
&-\int_{\partial\Xi}\tilde{A}_{\partial^{F}_{\pm}{\Sigma}^{{}^{\prime}}}(\nu_{\partial\Pi},\nu_{\partial\Pi})\psi^{2}ud\mathcal{H}^{1}.\end{split}$

Using (5.2) and (5.3), we have

$\begin{split}0\leqslant&\int_{\Xi}\left(|\nabla\psi|^{2}u+\frac{1}{2}\left(\tilde{R}_{\Xi}-\lambda\right)\psi^{2}u-\tilde{\Delta}_{\Xi}u\psi^{2}\right)d\mathcal{H}^{2}-\int_{\partial\Xi}\tilde{A}_{\partial^{F}_{\pm}{\Sigma}^{{}^{\prime}}}(\nu_{\partial\Pi},\nu_{\partial\Pi})\psi^{2}ud\mathcal{H}^{1}.\end{split}$

Taking $\psi=u^{-\frac{1}{2}}$ and integrating by parts, we have

(5.5) $\begin{split}0&\leqslant\int_{\Xi}\left(\frac{1}{4}|\nabla
u|^{2}u^{-2}+\tilde{K}_{\Xi}-\frac{1}{2}\lambda-u^{-1}\tilde{\Delta}_{\Xi}u\right)d\mathcal{H}^{2}-\int_{\partial\Xi}\tilde{A}_{\partial^{F}_{\pm}{\Sigma}^{{}^{\prime}}}(\nu_{\partial\Pi},\nu_{\partial\Pi})\psi^{2}ud\mathcal{H}^{1}\\\
&=\int_{\Xi}\left(-\frac{3}{4}|\nabla
u|^{2}u^{-2}+\tilde{K}_{\Xi}-\frac{1}{2}\lambda\right)d\mathcal{H}^{2}-\int_{\partial\Xi}\left(\tilde{A}_{\partial^{F}_{\pm}{\Sigma}^{{}^{\prime}}}(\nu_{\partial\Pi},\nu_{\partial\Pi})+u^{-1}\frac{\partial
u}{\partial\tilde{\eta}}\right)d\mathcal{H}^{1}.\end{split}$

Recall that $\partial^{F}_{+}\Sigma$ and $\partial^{F}_{-}\Sigma$ are totally
geodesic with respect to $\tilde{g}$ and $u$ satisfies the Neumann condition
(4.1). Boundary integrals in above vanish, and we conclude that

$\lambda|\Xi|\leqslant 2\int_{\Xi}\tilde{K}_{\Xi}d\mathcal{H}^{2}\leqslant
8\pi\Rightarrow|\Xi|\leqslant\frac{8\pi}{\lambda}.$

Note that we have used Gauss–Bonnet formula, which also implies that $\Xi$ is
a $2$-sphere. The diameter upper bound follows from [CL23, Lemma 17]. ∎

###### Proof of Theorem 1.1.

First of all, we consider $\lceil\frac{\pi}{\theta}\rceil$ times reflections
of $\Sigma$ with respect to $\partial H^{4}_{+}$ or $\partial H^{4}_{-}$. The
union of these manifolds is still a stable $C^{1,1}$-to-edge properly embedded
free boundary minimal hypersurface in a wedge domain with angle
$\lceil\frac{\pi}{\theta}\rceil\cdot\theta$. By the assumption on $\theta$, we
know that $\pi<\lceil\frac{\pi}{\theta}\rceil\cdot\theta<2\pi$. For
convenience, we denote this union still by $\Sigma$.

Applying Lemma 5.1 to $(\Sigma^{{}^{\prime}}\backslash\\{0\\},\tilde{g})$, we
find a connected open set $\Pi$ in the $\frac{5\pi}{\sqrt{\lambda}}$
neighborhood of $\partial\mathring{\Sigma}^{{}^{\prime}}$, such that each
connected component of
$\partial\Pi\backslash\partial\mathring{\Sigma}^{{}^{\prime}}$ is a $2-$sphere
with area at most $\frac{8\pi}{\lambda}$ and intrinsic diameter at most
$\frac{2\pi}{\sqrt{\lambda}}$. Let $\Sigma_{0}$ be the connected component of
$\Sigma^{{}^{\prime}}\backslash\Pi$ that contains $0$.

We make a few observations about $\Sigma_{0}$. First, we claim that
$\Sigma\backslash\Sigma_{0}$ is connected. To see this, let $\Sigma_{1}$ be
the union of connected components of $\Sigma^{{}^{\prime}}\backslash\Pi$ other
than $\Sigma_{0}$. Then $\Sigma\backslash\Sigma_{0}=\Sigma_{1}\cup\Pi\cup E$.
Note that each connected component of $\Sigma_{1}$ shares a common boundary
with $\Pi$. Since $\Pi$ is connected, so is $\Sigma_{1}\cup\Pi$. Next, we
claim that $\Sigma_{0}\cap\mathring{\Sigma}$ has only one boundary component:
otherwise, since both $\Sigma_{0}$ and $\Sigma\backslash\Sigma_{0}$ are
connected, as before we can find a loop in $\mathring{\Sigma}$ intersecting
one component of $\partial\Sigma_{0}\cap\mathring{\Sigma}$ exactly once,
contradicting that $\Sigma$ is simply connected.

Denote by $\Xi=\partial\Sigma_{0}\cap\mathring{\Sigma}$. By (1) in [CL23,
Lemma 25], we know that $\min_{x\in\Xi}r(x)\geqslant\rho$. Since $\Sigma\cap
B_{\mathbb{R}^{4}}(0,\rho)$ is connected, this implies that $\left(\Sigma\cap
B_{\mathbb{R}^{4}}(0,\rho)\right)\subset\Sigma_{0}$. Obviously,
$\max_{x\in\Xi}r(x)\leqslant e^{\frac{5\pi}{\sqrt{\lambda}}}\rho$. Therefore,
we have

$|\Xi|_{g}=\int_{\Xi}dA_{g}=\int_{\Xi}r^{2}dA_{\tilde{g}}\leqslant
e^{\frac{10\pi}{\sqrt{\lambda}}}\rho^{2}|\Xi|_{\tilde{g}}\leqslant\frac{8\pi}{\lambda}e^{\frac{10\pi}{\sqrt{\lambda}}}\rho^{2}.$

By our construction, the complement of the wedge domain with angle
$\lceil\frac{\pi}{\theta}\rceil\cdot\theta$ is a convex body in
$\mathbb{R}^{4}$. Note that $\Xi$ is the relative boundary of $\Sigma_{0}$,
and $\partial^{F}_{+}{\Sigma}_{0}:=\partial\Sigma_{0}\cap\partial H^{4}_{+}$,
$\partial^{F}_{-}{\Sigma}_{0}:=\partial\Sigma_{0}\cap\partial H^{4}_{-}$ are
free boundaries of $\Sigma_{0}$. By [LWW23, Theorem 1.2], we have

$|\Sigma\cap
B_{\mathbb{R}^{4}}(0,\rho)|_{g}\leqslant|\Sigma_{0}|_{g}\leqslant\frac{4|\mathbb{B}^{3}|_{g_{c}}}{|\partial\mathbb{B}^{3}|^{\frac{3}{2}}_{g_{c}}}|\Xi|_{g}^{\frac{3}{2}},$

where $\mathbb{B}^{3}$ is the unit round ball in $(\mathbb{R}^{3},g_{c})$ with
canonical metric $g_{c}$. Noting that the union contains the original free
boundary minimal hypersurface, we complete the proof. ∎

## References

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