# Capillary Surfaces in Manifolds with Nonnegative Scalar Curvature
and Strictly Mean Convex Boundary

Yujie Wu<EMAIL_ADDRESS>
###### Abstract.

In this paper we use stable capillary surfaces (analogous to the $\mu$-bubble
construction) to study manifolds with strictly mean convex boundary and
nonnegative scalar curvature. We give an obstruction to filling 2-manifolds by
such 3-manifolds based on the Urysohn width. We also obtain a bandwidth
estimate and establish other geometric properties of such manifolds.

##  1\. Introduction

In [Gro19], Gromov asked the question of finding sufficient conditions to
allow or disallow fill-in of a given Riemannian manifold $Y^{n}$ as the
boundary of a Riemannian manifold $X^{n+1}$ with nonnegative scalar curvature.
Can the mean curvature of $Y=\partial X$ prove enough influence so that the we
cannot prescribe certain geometry properties on $X$?

In this paper we are interested in the case when $n=2$. If $Y$ is a connected
orientable closed surface with positive Gaussian curvature, then there is an
isometric embedding of $Y$ into $\mathbb{R}^{3}$ as a strictly convex surface,
we denote the mean curvature of this embedding as $H_{0}$ (such embedding is
unique up to isometry of $\mathbb{R}^{3}$). Using this, Shi-Tam [ST02] proved
that if there is some $(X,g)$ with nonnegative scalar curvature filling such a
$Y$ with positive mean curvature $H$, then

$\int_{Y}Hd\sigma\leq\int_{Y}H_{0}d\sigma,$

where $d\sigma$ is the volume form induced from the metric $g$. Moreover,
equality holds if and only if $X$ is a domain in $\mathbb{R}^{3}$. This
results gives a positive answer to the question in the first paragraph.

In this paper, we give another answer in the following theorem using capillary
surfaces. As an example, if a 3-manifold fills $\mathbb{S}^{2}$ with $H\geq 2$
and nonnegative scalar curvature, then the first first Urysohn width of
$\mathbb{S}^{2}$ in the induced metric is no more than $4.5\pi$ (take
$a_{0}=\frac{2}{3},d_{0}=\frac{3\pi}{4}$).

######  Theorem 1.1.

If $(N^{3},\partial N,g)$ is a compact manifold with simply-connected boundary
such that $R_{g}\geq 0$, $H_{\partial N}\geq\frac{\pi}{d_{0}}+a_{0}$, then the
first Urysohn width of $\partial N$ (with respect to the induced metric $g$)
is bounded: $U_{1}(\partial N)\leq 4d_{0}+\frac{\pi}{a_{0}}$.

In another direction, Gromov proved the following bandwidth estimate, here we
denote $T^{n-1}$ as the $n-1$ dimensional torus. The theorem can be proved
using the idea of $\mu$-bubbles [Gro21].

######  Theorem ([Gro18]).

Let $M_{0}=T^{n-1}\times[-1,1],\partial_{\pm}M_{0}=T^{n-1}\times\\{\pm 1\\}$,
if a Riemannian manifold $(M^{n},g),2\leq n\leq 7$ admits a continuous map
$f:(M,\partial_{\pm}M)\rightarrow(M_{0},\partial_{\pm}M_{0})$ with nonzero
degree and scalar curvature bounded from below $R_{g}\geq n(n-1)$, then the
distance of $\partial_{+}M$ and $\partial_{-}M$ is bounded:
$\text{dist}_{g}(\partial_{+}M,\partial_{-}M)\leq\frac{2\pi}{n}$.

Using $\mu$-bubbles, that is, studying stable hypersurfaces with prescribed
mean curvature in a manifold with positive scalar curvature (PSC) has given
fruitful results in recent years (see for example [Li19]
[Zhu21][CL20][CLS22]). In these work, the fact that the scalar curvature of
the manifold has to obtain a strictly positive lower bound is crucial. On the
other hand, for manifolds with boundary, to constrain a minimizer of a
generalized area functional, we can prescribe both mean curvature and the
angle of intersection along the boundary. In particular, mean convexity
assumption is helpful to constrain capillary surfaces. This allows us to relax
the assumption of PSC when studying manifolds with boundary.

Gromov studied a band with PSC of the form $\Sigma^{2}\times[-1,1]$, for a
closed surface with $\chi_{\Sigma}\leq 0$ (for example the torus). Our model
example in the case $\Sigma$ has boundary would be
$M_{0}=(\Sigma,\partial\Sigma)\times[-1,1]$, $R_{M_{0}}\geq 0$, and the
boundary is strictly mean convex. We will use capillary surfaces in $M_{0}$ as
an analogy to the $\mu$-bubbles in Gromov’s bandwidth estimate. To avoid
regularity problem that may occur when the capillary surface touches the
corner $\partial\Sigma\times\\{\pm 1\\}$, we will cap the model example
$M_{0}$ with a top $M_{+}$ and a bottom $M_{-}$. This explains the
decomposition we have in the following theorem.

######  Theorem 1.2 (Bandwidth estimate).

Consider a compact 3-manifold $(M,\partial M)$ with the following
decomposition (see Figure 1),

$M=M_{0}\cup M_{\pm},\,\,\partial
M=\partial_{0}M\cup\partial_{\pm}M,\,\,\partial
M_{\pm}=S_{\pm}\cup\partial_{\pm}M,\,\,\partial M_{0}=\partial_{0}M\cup
S_{\pm}.$

Here $M_{0}$ is homeomorphic to $S_{-}\times[-1,1]$, $S_{-}$ is an orientable
surface with boundary and Euler characteristic $\chi(S_{-})\leq 0$. Assume
$\partial_{+}M\cap\partial_{-}M=\emptyset$, and for any component $\Gamma_{+}$
of $\partial_{+}M$, there is a component $\Gamma_{-}$ of $\partial_{-}M$ such
that, $d_{\partial M}(\Gamma_{+},\Gamma_{-})<\infty$.

We denote the scalar curvature of $M$ by $R_{M}$, mean curvature of
$\partial_{i}M$ by $H_{i}$ for $i\in\\{+,-,0\\}$. If $R_{M}\geq 0,H_{0}\geq
2,H_{\pm}\geq 0$, then $d_{M}(S_{+},S_{-})\leq 4+\pi.$

$M_{+}$
---
$M_{0}$
---
$M_{-}$
---
$S_{+}$
---
$S_{-}$
---
Figure 1. An example of $M$ when $S_{-}$ is an annulus.

In [RS97], Ros and Souam studied constant mean curvature surfaces which
intersect the ambient boundary at a fixed angle, called capillary surfaces. In
the second variation formula for capillary surfaces, instead of fixed angle,
we allow the prescribed angle condition to vary. For manifolds with
nonnegative scalar curvature and strictly mean convex boundary, we are able to
gain rigidity results for 2-manifolds (Theorem 1.3), and exhaustion of the
boundary for 3-manifolds (Theorem 1.4).

######  Theorem 1.3.

Consider a connected Riemannian manifold $\Sigma^{2}$ with boundary, such that
$R_{\Sigma}\geq 0,k_{\partial\Sigma}\geq 1$, then $\partial\Sigma$ is
connected with length no more than $2\pi$; for any $x\in\Sigma$,
$d_{\Sigma}(x,\partial\Sigma)\leq 1,$ so $\Sigma$ is compact. $\Sigma$ is a
topological disk. If $|\partial\Sigma|=2\pi$, then $\Sigma$ is a flat unit
disk in $\mathbb{R}^{2}$.

The fact that every point in this manifold must be at most distance 1 away
from the boundary follows from Li-Nguyen [LN14] and Li [Li14]. Furthermore, Li
[Li14] conjectured the following, a complete Riemannian manifold $M^{n}$ with
nonnegative Ricci curvature and $A_{\partial M}\geq 1$ must be compact ($A$ is
the second fundamental form of the boundary). This theorem solves the
2-dimensional case of this conjecture. We note in Theorem 1.3 we don’t assume
orientability or embeddedness. In the case of $M^{n}\subset\mathbb{R}^{n+1}$,
a strictly convex hypersurface bounding a region with pinched second
fundamental form, Hamilton [Ham94] showed that $M$ must be compact.

Contrary to the 2-dimensional case, for a connected 3-manifold with
nonnegative scalar curvature and $H_{\partial M}\geq 2$ ($H_{\partial M}$ is
mean curvature of the boundary), its boundary might not be connected or
compact. But we can obtain an exhaustion of the boundary as below, and show
that the boundary has linear volume growth under further topological
assumptions (see Corollary 1.5).

######  Theorem 1.4.

Let $(M^{3},\partial M)$ be a complete connected Riemannian manifold with
nonnegative scalar curvature, $\partial M$ is connected non-compact with
$H_{\partial M}\geq 2$.

We can find an exhaustion of the boundary $\partial M=\cup_{k}Z_{k}$, with
$\partial Z_{k}=\partial\Sigma^{\prime}_{k}$, where $\Sigma^{\prime}_{k}$ is a
union of finitely many capillary surface of disk type in $M$. Each component
$\Sigma_{k}$ of $\Sigma^{\prime}_{k}$ has bounded boundary length:
$|\partial\Sigma_{k}|\leq 2\pi$, and $d_{\Sigma_{k}}(x,\partial\Sigma_{k})\leq
2$ for any $x\in\Sigma_{k}$ .

Furthermore, if $\pi_{1}(\partial M)=0$, let $(E_{k})_{k=1}^{\infty}$ be an
end of $\partial M$, then $\overline{E_{k+1}}\setminus E_{k}$ is connected and
$\sup_{k}\text{diam}_{\partial M}(\overline{E_{k+1}}\setminus E_{k})\leq
5\pi$.

Using Theorem 1.4 when $\pi_{1}(\partial M)=0$ and $\partial M$ has weakly
bounded geometry we can apply section 3 in [Wu23] to get the following
corollary.

######  Corollary 1.5.

Let $M^{3}$ be a complete connected Riemannian manifold with nonnegative
scalar curvature, whose non-compact boundary $\partial M$ is simply connected,
uniformly mean convex and has weakly bounded geometry, then each end of
$\partial M$ has linear volume growth. In particular, if $\partial M$ has
finitely many ends, then $\partial M$ has linear volume growth.

Acknowledgements. The author wants to thank Otis Chodosh for his continuous
support and encouragement, and for many helpful discussions. The author thanks
Chao Li for discussions over the fill-in problem and encouragements through
improvements of the first draft. The author thanks Shuli Chen and Jared Marx-
Kuo for interest in this work.

##  2\. 2D results

Throughout this paper, if $(N^{k+1},\partial N),k\geq 1$ is a manifold with
boundary, we denote the second fundamental form of $\partial N\subset N$ with
respect to the outward pointing unit normal $\nu$ as
$\mathrm{I\\!I}(X,Y)=-\langle\nabla_{X}Y,\nu\rangle$ for vector fields $X,Y$
tangent to $\partial N$. Then the scalar-valued mean curvature is written as
$H=\sum_{i=1}^{k}-\langle\nabla_{e_{i}}e_{i},\nu\rangle$ for an orthonormal
basis $(e_{i})_{i}^{k}$ of $T_{p}\partial N$ at some point $p\in\partial N$.
In this convention, boundary of the unit ball $\mathbb{B}^{3}$ in
$\mathbb{R}^{3}$ has positive mean curvature, $H_{\partial\mathbb{B}^{3}}=2$.

###### Proof of Theorem 1.3.

The $d_{\Sigma}(x,\partial\Sigma)\leq 1$ part follows directly from [LN14]
Proposition 2.1 and [Li14] Theorem 1.1. Note that from this result we know
that $\Sigma$ is compact if and only if $\partial\Sigma$ is compact.

Step 1. If we know that $\Sigma$ is compact, and $\partial\Sigma$ is
connected, we can show the length of $\partial\Sigma$ is no more than $2\pi$.

We apply Gauss Bonnet to get, denoting $\frac{1}{2}R_{\Sigma}=K$:

(2.1) $2\pi\geq
2\pi\chi=\int_{\Sigma}K+\int_{\partial\Sigma}k\geq|\partial\Sigma|>0.$

Here we used $\Sigma$ has only one boundary component, so $\chi_{\Sigma}\leq
1$. This gives us $|\partial\Sigma|\leq 2\pi$, and $\chi(\Sigma)>0$, so
$\chi_{\Sigma}=1$, and $\Sigma$ is a topological disk.

Step 2. We now look at the rigidity statement under the assumption that
$\Sigma$ is compact and $\partial\Sigma$ is connected.

If $K\geq 0,k\geq 1$ and $|\partial\Sigma|=2\pi$, then from (2.1) we must have
$K=0,k=1$ everywhere. So $\Sigma$ is locally isometric to the flat unit disk
$\mathbb{D}=\\{(x_{1},x_{2}),x_{1}^{2}+x_{2}^{2}\leq 1\\}$ , we show that the
local isometry can be extended globally.

We denote the unit speed loop of $\partial\Sigma$ as
$\gamma:[0,2\pi]\rightarrow\partial\mathbb{D}$, and write
$\gamma(0)=\gamma(2\pi)=p$. Then by choosing $\epsilon$ small enough, we can
find an isometry for each $t\in[0,2\pi]$ with
$\Psi_{t}:B_{\epsilon}(\gamma(t))\rightarrow\mathbb{D}$. Combining $\Psi_{t}$
with a rotation of $\mathbb{D}$ around origin, we can patch up these isometry
$\Psi_{t}$ to get a global isometry $\Psi_{0}$ from
$B_{\epsilon}(\partial\Sigma)$ to a neighborhood of $\partial\mathbb{D}$ in
$\mathbb{D}$.

We now define a global isometry $\Psi$ from $\Sigma$ to $\mathbb{D}$ as
follows, for $x\in B_{\epsilon}(\partial\Sigma)$, $\Psi(x)=\Psi_{0}(x)$. Now
fix $p\in\partial\Sigma$ and for any $q\in\Sigma\setminus
B_{\epsilon}(\partial\Sigma)$, for any path
$s(t):(0,1]\rightarrow\Sigma^{\circ}$ with $s(0)=p,s(1)=q$. We cover $s(t)$
with interior balls each of which can be mapped isometrically into an interior
ball in $\mathbb{D}$, in a way that agrees on the overlap (see [Lee18] Theorem
12.4). Then applying Theorem 12.2 in [Lee18], we have obtained a global
isometry from $\Sigma$ to $\mathbb{D}$.

Step 3. We now show $\partial\Sigma$ is connected ($\Sigma$ might be non-
compact).

If not, then there are at least two components
$\partial_{1}\Sigma,\partial_{2}\Sigma$ of $\partial\Sigma$. Let’s first
assume that
$\alpha=\inf\left\\{d(x,y),x\in\partial_{1}\Sigma,y\in\partial_{2}\Sigma\right\\}$
can be obtained by a stable free boundary geodesic $\beta_{0}(t)$. Let $T$ be
a unit normal vector field of $\beta_{0}$, notice that along $\partial\Sigma$,
$T$ is tangent to $\partial\Sigma$, this vector field generate a local
variation of $\beta_{0}$ called $\beta_{s}(t)$. Then the second variation
formula of length $L(\beta_{s})$ implies,

$0\leq\frac{d^{2}}{ds^{2}}\Bigg{\rvert}_{s=0}L(\beta_{s})=\int_{\beta}-\frac{R_{\Sigma}}{2}+\int_{\partial\beta}\langle\nabla_{T}T,\beta^{\prime}\rangle\leq\int_{\partial\beta}\langle\nabla_{T}T,\beta^{\prime}\rangle\leq-2<0,$

where we used convexity at the end points, i.e.
$\langle\nabla_{T}T,\beta^{\prime}(0)\rangle=k_{\partial\Sigma}(\beta(0))=-1$,
this is a contradiction.

Now we look at the case $\alpha$ can’t be realized by a stable free boundary
geodesic. In this case, at least one of the component is not compact. Assume
$\partial_{1}\Sigma$ is non-compact, then we can try to find a minimizing
geodesic line via points on $\partial_{1}\Sigma$. To elaborate, consider the
universal cover of $\Sigma$ named $\tilde{\Sigma}$, then we have that again
(at least) two components of boundary $\partial\tilde{\Sigma}$, denoted
$\partial_{1}\tilde{\Sigma}\cup\partial_{2}\tilde{\Sigma}$ and
$\partial_{1}\tilde{\Sigma}$ is not compact. We denote
$\partial_{1}\tilde{\Sigma}\cong\mathbb{R}$ as $\gamma(t)$, with
$|\gamma^{\prime}(t)|=1$, $\gamma(0)=q$, $\gamma(\pm n)=\pm q_{n}$. Consider
the minimizing geodesic in $\tilde{\Sigma}$ from $+q_{n}$ to $-q_{n}$ denoted
$l_{n}(t)$ , which cannot touch the boundary except at end points due to
strict convexity.

We fix a point $p\in\partial_{2}\tilde{\Sigma}$ and fix a minimizing path from
$p$ to $q$, denoted $\overline{pq}(t)$. Now any minimizing geodesic in
$\tilde{\Sigma}$ between $\pm q_{n}$ must intersect $\overline{pq}$. Indeed,
if there is a minimizing geodesic $l$ between $\pm q_{n}$ and
$l\cap\overline{pq}=\emptyset$, we can minimize distance of $+q_{n}$
(similarly for $-q_{n}$) to $\overline{pq}$ and then use this to build a path
transversal to $\overline{pq}$ with intersection number 1. Concatenating with
$l$ , we get a loop in $\tilde{\Sigma}$ whose intersection number with
$\overline{pq}$ is equal to 1. But this loop is homotopic to the constant loop
at e.g. $+q_{n}$ by simply-connectedness, a contradiction (see section 2.4 in
[GP74]).

Now consider a minimizing geodesics $\gamma_{n}(t)$ connecting $\pm q_{n}$.
First, $|\gamma_{n}|\rightarrow\infty$, since we can pick $\pm q_{n}$ to be in
$\tilde{\Sigma}\setminus B_{n}(\overline{pq})$ for large $n$. We look at one
of the intersection points of $\gamma_{n}$ and $\overline{pq}$ named $s_{n}$
and the velocity vector of $\gamma_{n}$ at $s_{n}$ named $v_{n}$, then
$\\{(s_{n},v_{n}),n\in\mathbb{N}\\}$ is contained in a compact set, and we
have a subsequence converge to some point $(s,v)$. The geodesic ray from $s$
with velocity $v$ (respectively $-v$) must have infinite length because we can
get $C^{1}_{\text{loc}}$-convergence of these geodesics when
$(s_{n},v_{n})\rightarrow(s,v)$. This means that we have found a geodesic
line.

Now we can use the Toponogov’s splitting theorem generalized to manifolds with
convex boundary ([BZ77] Theorem 5.2.2) to conclude that
$\tilde{\Sigma}=\mathbb{R}\times I$ where $I$ is a connected one manifold with
boundary, this is a contradiction to the boundary being strictly convex.

Step 4. We assume $\Sigma$ is not compact and get a contradiction. In
particular, we assume the diameter of $\Sigma$ is unbounded.

By the result of Step 3, we can assume that $\partial\Sigma$ is connected. The
idea is that if the diameter of the boundary $\partial\Sigma$ is large, then
we can prescribe a “capillary” minimizing geodesic, and use stability
inequality to get a contradiction.

Now we minimize the functional in (2.2) among open sets with finite perimeter,
containing $S:=B_{2}(x)\cap\partial\Sigma$ (for some fixed
$x\in\partial\Sigma$). Consider a small neighborhood $U_{0}$ of
$\partial\Sigma\cap\left(B_{2\pi}(x)\setminus B_{\pi}(x)\right)$ in
$\partial\Sigma$, and $\rho(y)$ a mollification of the function
$\pi-d(x,\cdot)$ with $|\nabla\rho|<1$. Let $w$ be a smooth function such that
$w=-\cos(\rho(y))$ when $y\in U_{0}$ and $|w|>1$ otherwise.Take such a
Caccioppoli set $\Omega$ and let,

(2.2)
$A(\Omega)=\mathcal{H}^{1}(\partial\Omega)+\int_{\Omega\cap\partial\Sigma}w,$

where $\mathcal{H}^{1}$ is the perimeter of $\partial\Omega$ in $\Sigma$.

We note that if a smooth minimizer $\partial\Omega=\gamma\neq\emptyset$ exists
and $\partial\gamma\neq\emptyset$ then using the second variation formula
[RS97], we have over the minimizing geodesic $\gamma$,

$\displaystyle A^{\prime\prime}(\gamma)=$
$\displaystyle\int_{\gamma}-\phi\Delta_{\gamma}\phi-(\text{Ric}(N,N)+|\mathrm{I\\!I}_{\gamma}|^{2})\phi^{2}$
$\displaystyle+\int_{\partial\gamma}\frac{\phi^{2}}{\sin\rho}\langle\nabla_{\bar{\nu}}\bar{\nu},\bar{N}\rangle+\frac{\phi^{2}}{\sin^{2}\rho}\nabla_{\bar{\nu}}w+\phi\nabla_{\nu}\phi+\phi^{2}\cot\rho\langle
N,\nabla_{\nu}\nu\rangle,$

where $\overline{N}$ (respectively $N$) is the outward unit normal of
$\partial\Sigma\subset\Sigma$ (respectively $\partial\Omega\subset\Omega$),
and $\overline{\nu}$ (respectively $\nu$) is the outward unit normal of
$\partial\Omega\cap\partial\Sigma\subset\Omega\cap\partial\Sigma$
(respectively $\partial\gamma\subset\gamma$).

We can plug in $\phi=1$, use $\mathrm{I\\!I}_{\gamma}=0=\langle
N,\nabla_{\nu}\nu\rangle$, and use Gauss Equation,

$\displaystyle 0$ $\displaystyle\leq
A^{\prime\prime}(\gamma)=\int_{\gamma}-\frac{1}{2}R_{\Sigma}+\int_{\partial\gamma}\frac{1}{\sin\rho}(\langle\nabla_{\bar{\nu}}\bar{\nu},\bar{N}\rangle-\nabla_{\nu}\rho)$
$\displaystyle<\int_{\partial\gamma}\frac{1}{\sin\rho}\left(-1+1\right)=0,$

using $|\nabla\rho|<1$ and
$-k_{\partial\Sigma}=\langle\nabla_{\overline{\nu}}\overline{\nu},\overline{N}\rangle\leq-1$,
a contradiction.

Step 5. In this step we take care of some technical details needed to show
that a smooth minimizer exists (used in Step 4.)

If $\Omega$ is a candidate in a minimizing sequence of $\mathcal{A}$, by
assumption we have
$\partial\Omega\cap\left(B_{2}(x)\cap\partial\Sigma\right)=\emptyset$. In fact
we can find an open neighborhood $\Omega^{\prime}$ around
$B_{1}(x)\cap\partial\Sigma$, so that any minimizing sequence must contain
$\Omega^{\prime}$.

To elaborate, we assume without loss of generality each $\Omega$ in the
minimizing sequence has smooth boundary, now we show that for some choice of
$\Omega^{\prime}$,

$\delta(\Omega):=\mathcal{H}^{1}(\partial\Omega^{\prime}\setminus\Omega)-\mathcal{H}^{1}(\partial\Omega\cap\Omega^{\prime})<0.$

We consider the following family $\Phi_{t}:=\exp(-t\phi(x)\overline{N}(x))$;
here again $\overline{N}$ is the outward pointing unit normal of
$\partial\Sigma\subset\Sigma$ , $\phi(x)$ is a compactly supported nonnegative
function on $S=B_{2}(x)\cap\partial\Sigma$ that is equal to 1 on
$B_{1}(x)\cap\partial\Sigma$.

Let $S_{t}:=(\Phi_{t}(S))$. We show that $\Phi_{t}:S\rightarrow S_{t}$ is a
diffeomorphism: since $d\Phi_{0}$ is bijective and $\Phi_{t}$ is continuous in
$t$, this is true for small $t$. By continuity (see formula A.13 in [CM20]),
we get that $H_{S_{t}}\geq 1/2$ for small $t$ ($H_{S_{0}}\geq 1$ by
assumption). Let $\Omega^{\prime}:=\cup_{t\in[0,\delta]}S_{t}$ be the union of
these “foliation”, since $\phi=1$ on $B_{1}(x)\cap\partial\Sigma$, we have
$\Omega^{\prime}$ contains a tubular neighborhood of height $\delta$ of
$B_{1}(x)\cap\Sigma$.

By [Mag12] Theorem 16.3 with $E=\Omega^{\prime}$ and $F=\Omega$,

$\delta(\Omega)=|E|(F^{0})+|F|(E^{0})+\mathcal{H}^{1}(\nu_{E}=\nu_{F})-|F|=|E|(F^{0})-|F|(E^{1}).$

So by divergence theorem for Lipschitz domain we have,

$\delta(\Omega)=|\partial\Omega^{\prime}\setminus\Omega|-|\partial\Omega\setminus\Omega^{\prime}|\leq\int_{\partial\Omega^{\prime}\setminus\Omega}\nu_{t}\cdot\nu_{\partial\Omega^{\prime}}-\int_{\partial\Omega\setminus\Omega^{\prime}}\nu_{t}\cdot\nu_{\partial\Omega}=\int_{\Omega^{\prime}\setminus\Omega}\text{div}^{\Sigma}(\nu_{t})<0,$

since $\text{div}_{S_{t}}(\nu_{t})=\text{div}^{\Sigma}\nu_{t}\leq-1/2$
($\nu_{0}=-\overline{Nf}$). We get that any minimizing sequence must
(eventually) contain $\Omega^{\prime}$.

A similar argument using $|w|>1$ outside of $U_{0}$, a small open neighborhood
of $B_{2\pi}(x)\setminus B_{\pi}(x)$, shows that given a minimizing sequence
$\Omega_{i}$, $\partial\Omega_{i}=\gamma_{i}$, we can modify $\Omega_{i}$ so
that
$\Omega_{i}\cap\partial\Sigma\subset\left(\overline{B_{2\pi}(x)}\cup\overline{U_{0}}\right)$
and $\partial\gamma_{i}\subset\overline{B_{2\pi}(x)\setminus B_{\pi}(x)}$,
without increasing $A(\Omega_{i})$.

Now because
$\Omega_{i}\cap\partial\Sigma\subset\left(\overline{B_{2\pi}(x)}\cup\overline{U_{0}}\right)$,
the term $\int_{\partial\Sigma\cap\Omega_{i}}w$ is uniformly bounded for a
minimizing sequence, so the length of $\partial\Omega_{i}$ is also uniformly
bounded. Together with
$\partial\gamma_{i}\subset\overline{B_{2\pi}(x)\setminus B_{\pi}(x)}$, we get
that the minimizing sequence must be contained in a bounded set from the point
$x$.

We can continue the minimization process with standard BV compactness and
regularity theory.

We now check that the smooth minimizer $\gamma=\partial\Omega$ must have
nonempty boundary. Note
$\left(B_{1}(x)\cap\partial\Sigma\right)\subset\Omega^{\prime}\subset\Omega$,
and $\Omega$ is disjoint from $U_{0}$. So if $\Omega$ is a minimizer, then
$\Omega\cap\partial\Sigma$ and $\partial\Sigma\setminus{\Omega}$ is two
disjoint nonempty open sets, whose union is $\partial\Sigma$ in case
$\partial\gamma$ is empty, a contradiction to $\partial\Sigma$ being
connected. ∎

##  3\. 3D results

Contrary to the 2-dimensional case, a 3-manifold with nonnegative scalar
curvature and uniformly mean convex boundary, might not have connected
boundary. We also might not have that any point is at bounded distance away
from the boundary (using the same method to prove either would require non-
negative Ricci curvature).

Before we start the proof of Theorem 1.4, we need the following lemma which is
analogous to Lemma 16 in [CL20]. The inequality (3.1), when compared to the
requirement of [CL20] in the PSC setting, is suitable for nonnegative scalar
curvature. And we added the assumption (3.2), which is suitable for mean-
convexity.

######  Lemma 3.1.

If there is a smooth function $u>0$ over a surface $(\Sigma,\partial\Sigma)$
with nonempty boundary, such that,

(3.1)
$\displaystyle\Delta_{\Sigma}u\leq\frac{R_{\Sigma}}{2}u+\frac{|\nabla_{\Sigma}u|^{2}}{2u}\quad$
$\displaystyle\text{over }\Sigma,$ (3.2)
$\displaystyle\frac{\nabla_{\nu}u}{u}\geq a_{0}-k_{\partial\Sigma}\quad$
$\displaystyle\text{over }\partial\Sigma,$

then $d(x,\partial\Sigma)\leq\frac{2}{a_{0}}$ for any $x\in\Sigma$. Here we
denote the outward pointing unit normal along $\partial\Sigma$ as $\nu$.

###### Proof.

If not, assume $d(z,\partial\Sigma)\geq\frac{2}{a_{0}}+2\delta$ for some
$z\in\Sigma$ and $0<\delta<1$. We then find a minimizer of the following
functional $\mathcal{F}(\Gamma)$ for sets with finite perimeters
$\Gamma\subset\Sigma$ containing a neighborhood $U_{0}$ of $z$ (will be chosen
later) and disjoint from $\partial\Sigma$. Denote $\partial\Gamma=\gamma$, and
$\nu_{\gamma}$ outward unit normal of $\gamma\subset\Gamma$,

$\displaystyle\mathcal{F}(\Gamma)=\int_{\gamma}u-\int_{\Sigma}hu(\chi_{\Gamma}-\chi_{\Gamma_{0}}),$

here $h(y)$ is a mollification of the function $\hat{h}(y)=\frac{2}{\alpha-
d_{\Sigma}(y,\partial\Sigma)}>0$ when $x\in U$ with
$U:=\\{y\in\Sigma,d(y,\partial\Sigma)<\alpha\\}$, for
$\alpha=\frac{2}{a_{0}}+\delta$. We require
$h\rvert_{\partial\Sigma}<a_{0},h\rvert_{\Sigma\setminus U}=\infty$, and
$\frac{1}{2}h^{2}+\nabla_{\nu_{\gamma}}h>0$ everywhere on $\Sigma$.

Here $\Gamma_{0}$ is a fixed Caccioppoli set to be chosen.

By the proof of Proposition 12 in [CL20], there is a neighborhood $U_{0}$ of
$z$ with smooth boundary, $h\rvert_{\Sigma\setminus U_{0}}\in L^{\infty}$, we
take $\Gamma_{0}=U_{0}$. From Proposition 12 in [CL20] we also have
$\mathcal{F}(\Gamma^{\prime}\cup U_{0})<\mathcal{F}(\Gamma^{\prime})$ for any
$\Gamma^{\prime}$ with smooth boundary if $U_{0}\nsubseteq\Gamma^{\prime}$.
This implies $\inf\mathcal{F}>-\infty$.

We now check that any minimizing sequence must be disjoint from a fixed open
neighborhood of $\partial\Sigma$. So by interior regularity we have that a
smooth minimizer exist [Tam84] [CL20].

By first variation, if a smooth minimizer exists then its geodesic curvature
is $k_{\gamma}=h-\frac{\nabla_{\nu}u}{u}$, so if $x\in\partial\Sigma$,

$\displaystyle k_{\gamma}(x)$
$\displaystyle=h\rvert_{\partial\Sigma}(x)-\frac{\nabla_{\nu}u}{u}(x)$ (3.3)
$\displaystyle\leq
h\rvert_{\partial\Sigma}(x)-a_{0}+k_{\partial\Sigma}(x)<k_{\partial\Sigma},$

where we used 3.2 and the assumptions of $h\rvert_{\partial\Sigma}$. Using
this observation, similar to Step 5 in Theorem 1.3, we use a foliation along
$\partial\Sigma$ to modify our minimizing sequence so that it is disjoint from
a fixed open neighborhood of $\partial\Sigma$.

To be precise, given any smooth minimizing sequence
$\gamma_{i}=\partial\Gamma_{i}$ , we show that
$\mathcal{F}(\Gamma_{i}\setminus T^{\prime})<\mathcal{F}(\Gamma_{i})$ if
$T^{\prime}\setminus\Gamma_{i}\neq\emptyset$, where
$T^{\prime}=\cup_{t\in[0,\epsilon],z\in\partial\Sigma}\exp_{z}(-t\nu)$ for
some small $\epsilon$ to be decided. If we fix $t\in[0,\epsilon]$, then we get
$T^{\prime}_{t}=\cup_{z\in\partial\Sigma}\exp_{z}(-t\nu)$ a hypersurface with
unit normal $\nu_{t}$, chosen so that as $t\rightarrow
0,\nu_{t}\rightarrow-\nu$. Then writing $T_{i}=\Gamma_{i}\setminus
T^{\prime}$, we have

$\displaystyle\mathcal{F}(T_{i})-\mathcal{F}(\Gamma_{i})$
$\displaystyle\leq\int_{\partial
T_{i}\cap\Gamma_{i}}u(-\nu_{t})\cdot\nu_{\partial
T_{i}}+\int_{\partial\Gamma_{i}\cap
T_{i}}u\nu_{t}\cdot\nu_{\partial\Gamma_{i}}+\int_{\Gamma_{i}\cap
T^{\prime}}hu$
$\displaystyle=\int_{T^{\prime}\setminus\Gamma_{i}}\text{div}^{\Sigma}(u\nu_{t})+hu$
$\displaystyle=\int_{T^{\prime}\setminus\Gamma_{i}}u\text{div}^{\Sigma}(\nu_{t})+\nabla_{\nu_{t}}u+hu$
$\displaystyle=\int_{T^{\prime}\setminus\Gamma_{i}}u(\text{div}^{\Sigma}(\nu_{t})+\frac{\nabla_{\nu_{t}}u}{u}+h)<0$

in the final inequality we used as $t\rightarrow 0$, $\nu_{t}\rightarrow-\nu$,
so $\text{div}(\nu_{t})\rightarrow-k_{\partial\Sigma}$, this implies by (3),

$h-\frac{\nabla_{-\nu_{t}}u}{u}+\text{div}(\nu_{t})<0.$

So any minimizing sequence must be disjoint from $T^{\prime}$ provided we
choose $\epsilon$ small enough.

We now write out the second variation formula for a minimizer of
$\mathcal{F}(\Gamma)$ this was derived in Theorem 6.3 of [Wu23] (see also
[CL20] Lemma 16),

$\displaystyle 0$
$\displaystyle\leq\int_{\gamma}|\nabla\psi|^{2}u-K_{\Sigma}\psi^{2}u-\frac{1}{2}k^{2}_{\gamma}\psi^{2}u+(\Delta_{\Sigma}u-\Delta_{\gamma}u)\psi^{2}-\langle\nabla_{\Sigma}u,\nu\rangle\psi^{2}h-\langle\nabla_{\Sigma}h,\nu\rangle\psi^{2}u$
$\displaystyle\leq\int_{\gamma}|\nabla\psi|^{2}u-(\Delta_{\gamma}u)\psi^{2}-(\frac{h^{2}}{2}+\nabla_{\nu}h)\psi^{2}u$
$\displaystyle<\int_{\gamma}|\nabla_{\gamma}\psi|^{2}u+\nabla_{\gamma}\psi^{2}\nabla_{\gamma}u\quad(\star)$

where we used (3.1) in the second inequality and also
$\frac{k_{\gamma}^{2}}{2}=\frac{h^{2}}{2}+\frac{(\nabla_{\nu}u)^{2}}{2u}-\frac{h\nabla_{\nu}u}{u}$.
To get the strict inequality in ($\star$ ) we used
$\frac{1}{2}h^{2}+\nabla_{\nu_{\gamma}}h>0$. Then we can plug
$\psi=u^{-\frac{1}{2}}$ into ($\star$) to get
$0<\int_{\gamma}\frac{-3}{4u^{2}}|\nabla_{\gamma}u|^{2}$, a contradiction. So
$d_{\Sigma}(x,\partial\Sigma)\leq\frac{2}{a_{0}}$ as claimed. ∎

######  Remark 3.2.

One can check that in the above proof, it’s sufficient to have equation (3.1):
$\Delta_{\Sigma}u\leq\frac{R_{\Sigma}}{2}u+\frac{|\nabla_{\Sigma}u|^{2}}{2u}$
over the set $U:=\\{y\in\Sigma,d(y,\partial\Sigma)<\frac{2}{a_{0}}+3\delta\\}$
instead of requiring it everywhere in $\Sigma$, because (3.1) is only used in
the step before ($\star$) over the minimizer $\gamma$, which must lie in $U$
by construction.

Before proving Theorem 1.4, we start with the simple case which allows us to
find capillary surfaces in a compact manifold with sufficiently large boundary
diameter.

######  Lemma 3.3.

If $(V,\partial V,g)$ is a connected compact 3-manifold with $R_{g}\geq 0$,
the mean curvature of the boundary has $H_{\partial
V}\geq\frac{\pi}{d_{0}}+a_{0}$ for some $a_{0}>0$ , and $\partial V$ is
connected with intrinsic $\text{diam}(\partial V)>d_{0}>0$. Then there are
finitely many capillary surfaces $(\Sigma_{i})_{i=1}^{k}$, whose boundary has
finite length $|\partial\Sigma_{i}|\leq\frac{2\pi}{a_{0}}$, and
$d_{\Sigma_{i}}(x,\partial\Sigma_{i})\leq\frac{2}{a_{0}}$ for any
$x\in\Sigma_{i}$. Furthermore, $\cup_{i=1}^{k}\Sigma_{i}$ separates $\partial
V$.

###### Proof.

Let $\text{diam}(\partial V)=d_{\partial V}(p,q)>d_{0}+5\delta$, for some
$\delta>0$. Then we can build a smooth function $w:\partial
V\rightarrow\mathbb{R}$ such that $w(x)<-1$ if $d_{\partial V}(p,x)\leq
2\delta$ and $w(x)>1$ if $d_{\partial V}(p,x)\geq d_{0}+3\delta$, and if
$-1<w(x)<1$ then $w(x)=\cos\rho(x)$ with $|\nabla^{\partial
V}\rho|<\frac{\pi}{d_{0}}$.

We consider a minimizer of the following functional among open sets $\Omega$
with finite perimeters, containing $S:=B^{\partial V}_{2\delta}(p)$ and
disjoint from the set $S^{\prime}:=\partial V\setminus B^{\partial
V}_{d_{0}+3\delta}(p)$,

$\displaystyle\mathcal{A}(\Omega)=|\partial\Omega|+\int_{\partial
V\cap\Omega}w.$

Then any minimizing sequence $\Omega_{i}$ must (eventually) contain a fixed
open neighborhood around $B_{\delta}^{\partial V}(p)$ and be disjoint from
some fixed open neighborhood of $\partial V\setminus B^{\partial
V}_{d_{0}+4\delta}(p)$, also the boundary $\partial\Omega_{i}\cap\partial V$
of any minimizing sequence $\Omega_{i}$ must lie in $\\{x\in\partial
V,|w|(x)\leq 1\\}$ (the proof of these two claims is analogous to Step 5 of
Theorem 1.3). So we find a minimizing Caccioppoli set $\Omega$ with smooth
boundary $\tilde{\Sigma}=\partial\Omega\neq\emptyset$ since $V$ is connected,
and $\partial\tilde{\Sigma}\neq\emptyset$ since $\partial V$ is connected (see
([DM14],[CEL24]) for regularity of capillary surfaces).

We first show that each component $\Sigma$ of $\tilde{\Sigma}$ has length no
more than $\frac{2\pi}{a_{0}}$. We note that we have strict inequality
$-1<w\rvert_{\partial\Sigma}<1$, because if not,
$\langle\nu,\overline{\nu}\rangle=\pm 1$ at some point of $\partial\Sigma$,
meaning $\Sigma$ is tangent to $\partial V$ at a boundary point from the
inside, a contradiction to $H_{\Sigma}=0$ and $H_{\partial V}\geq 2$. So we
write $\sqrt{1-w^{2}}=\sin\rho\neq 0$ in the calculations below.

We have the first and second variation formulas over $\Sigma$ (computed
analogous to [RS97] when $w$ is a constant function), here we also used the
Gauss-Codazzi equation
$R_{g}=R_{\Sigma}+2\text{Ric}(N,N)+|\mathrm{I\\!I}_{\Sigma}|^{2}-H_{\Sigma}^{2}$
in the second variation:

$\displaystyle H_{\Sigma}$
$\displaystyle=0,\quad\langle\nu,\overline{\nu}\rangle=-w;$ $\displaystyle 0$
$\displaystyle\leq\int_{\Sigma}-\phi\Delta_{\Sigma}\phi-\frac{1}{2}({R_{g}}-R_{\Sigma}+|\mathrm{I\\!I}_{\Sigma}|^{2}+H_{\Sigma}^{2})\phi^{2}$
(3.4)
$\displaystyle+\int_{\partial\Sigma}\frac{\phi^{2}}{\sin\rho}(\langle\nabla_{\bar{\nu}}\bar{\nu},\bar{N}\rangle+\nabla_{\bar{\nu}}\rho+\cos(\rho)\langle
N,\nabla_{\nu}\nu\rangle)+\phi\nabla_{\nu}\phi.$

where $\overline{\nu}$ is the outward unit normal of
$\partial\Sigma\subset\left(\overline{\Omega}\cap\partial V\right)$, $\nu$ is
the outward unit normal of $\partial\Sigma\subset\Sigma$, $\overline{N}$ is
the outward unit normal of $\partial V\subset V$ and $N$ is the outward unit
normal of $\Sigma\subset\overline{\Omega}$.

Now by (3.4) we can choose $u>0$ over $\Sigma$ with,

(3.5)
$\displaystyle\Delta_{\Sigma}u+\frac{1}{2}(R_{g}-R_{\Sigma}+|\mathrm{I\\!I}|+H^{2})u$
$\displaystyle\leq 0,$ (3.6)
$\displaystyle\nabla_{\nu}u+\frac{u}{\sin\rho}[\langle\nabla_{\bar{\nu}}\bar{\nu},\bar{N}\rangle+\nabla_{\bar{\nu}}\rho+\cos\rho\langle\nabla_{\nu}\nu,N\rangle]$
$\displaystyle=0.$

The following computation appeared in equation (3.8) in Li’s paper [Li19] is
very helpful, here $\gamma^{\prime}$ is a unit tangent vector along
$\partial\Sigma$,

(3.7) $-H_{\partial
V}=\langle\nabla_{\bar{\nu}}\bar{\nu},\bar{N}\rangle+\cos\rho\langle
N,\nabla_{\nu}\nu\rangle+\sin\rho\langle\nabla_{\gamma^{\prime}}\gamma^{\prime},\nu\rangle.$

Combining equation (3.5),(3.6),(3.7) and $R_{g}\geq 0$, we have a smooth
$u>0$,

(3.8) $\displaystyle\Delta_{\Sigma}u$
$\displaystyle\leq\frac{1}{2}R_{\Sigma}u$
$\displaystyle\frac{\nabla_{\nu}u}{u}$
$\displaystyle=\frac{1}{\sin\rho}(H_{\partial
V}-\nabla_{\overline{\nu}}\rho)+\langle\nabla_{\gamma^{\prime}}\gamma^{\prime},\nu\rangle\geq\frac{\frac{\pi}{d_{0}}+a_{0}}{\sin\rho}-\frac{\pi}{d_{0}}-k_{\partial\Sigma}\geq
a_{0}-k_{\partial\Sigma}.$

So we can now apply Lemma 3.1 to get that
$d_{\Sigma}(x,\partial\Sigma)\leq\frac{2}{a_{0}}$ for all $x\in\Sigma$.

Now recall Gauss-Bonnet (for a surface with nonempty boundary
$\chi_{\Sigma}\leq 1$),

(3.9) $\displaystyle 2\pi$ $\displaystyle\geq
2\pi\chi_{\Sigma}=\int_{\Sigma}K_{\Sigma}+\int_{\partial\Sigma}k_{g}=\int_{\Sigma}K_{\Sigma}-\int_{\partial\Sigma}\langle\nabla_{\gamma^{\prime}}\gamma^{\prime},\nu\rangle,$

Combining (3.7), (3.9) and the second variation (3.4), we have that for
$\phi=1$,

(3.10) $\displaystyle 0$
$\displaystyle\leq\int_{\Sigma}-\frac{1}{2}(R_{g}+|\mathrm{I\\!I}_{\Sigma}|^{2})+2\pi\chi_{\Sigma}+\int_{\partial\Sigma}\frac{1}{\sin\rho}(\nabla_{\bar{\nu}}\rho-
H_{\partial V})$
$\displaystyle<2\pi-|\partial\Sigma|\cdot(-\frac{\pi}{d_{0}}+\frac{\pi}{d_{0}}+a_{0})$
$\displaystyle=2\pi-|\partial\Sigma|\cdot a_{0},$

note if $\chi_{\Sigma}\leq 0$, we would get a contradiction in (3.10). So any
$\Sigma$ with nonempty boundary, must be a topological disk. We now get
$|\partial\Sigma|\leq\frac{2\pi}{a_{0}}$ as desired. ∎

We can now continue to the proof of Theorem 1.4.

###### Proof of Theorem 1.4.

We first build one capillary surface $\Sigma=\Sigma_{1}$ using the same idea
in Lemma 3.3 adapted to non-compact manifolds, then we build
$\Sigma_{2},\Sigma_{3},\Sigma_{4}...$ one by one.

We consider a minimizer of the following functional among open sets $\Omega$
with finite perimeters, containing $S:=B^{\partial M}_{2}(x)\cap$ (for some
fixed $x\in\partial M$) and contained in $B^{M}_{20}(x)$,

$\displaystyle\mathcal{A}(\Omega)=|\partial\Omega|+\int_{\partial
M\cap\Omega}w,$

here $w:\partial M\rightarrow\mathbb{R}$ with $|\nabla^{\partial M}w|\leq 1$
is a mollification of the Lipschitz function $\tilde{w}(y)$,

$\displaystyle\tilde{w}(y)$ $\displaystyle=-\cos(\pi-d_{\partial
M}(x,y))\text{ if }y\in B^{\partial M}_{2\pi}(x)\setminus B^{\partial
M}_{\pi}(x);$ $\displaystyle\tilde{w}(y)$ $\displaystyle=\frac{2d_{\partial
M}(x,y)}{\pi}-3\text{ otherwise.}$

We want to apply the same proof of Lemma 3.3 to $B^{M}_{20}(x)$ so we need to
perturb the metric of $M$ near $\partial B^{M}_{20}(x)$ to get mean convexity.
In particular, in a local coordinates near $\partial B^{M}_{20}(x)$ one can
change the Christoffel symbols (which are first derivatives of the metric)
while maintaining the coordinates being orthonormal (which is a condition only
depending on the zero-th order derivatives of the metric). So we can perturb
the metric near $\partial B^{M}_{20}(x)$ so that the mean curvature is at
least 1.5, and the perturbation happens within $M\setminus B_{19}(x)$. We
denote the scalar curvature after perturbation as $\hat{R}_{M}$, and
$\hat{R}_{M}\rvert_{B_{19}(x)}=R_{M}\rvert_{B_{19}(x)}$.

Then we apply the same minimizing scheme as in Lemma 3.3, note we again have
that the boundary of a minimizer $\partial\Sigma$ must lie in $\\{y\in\partial
M,-1<w(y)<1\\}$ and in this region $w(y)$ can be written as $w=\cos(\rho(y))$
for some smooth function $\rho$ with $|\nabla^{\partial M}\rho|\leq 1$. Now we
have a smooth capillary surface (with finitely many components)
$\Sigma=\partial\Omega$, $\Sigma\neq\emptyset$ since $M$ is connected, and
$\partial\Sigma\neq\emptyset$ since $\partial M$ is connected (see
([DM14],[CEL24]) for regularity of capillary surfaces).

Similar to 3.8, using second variation of each component $\Sigma$ of the
capillary surface, we have a smooth $u>0$,

$\displaystyle\Delta_{\Sigma}u$
$\displaystyle\leq\frac{1}{2}(R_{\Sigma}-\hat{R}_{M})u$
$\displaystyle\frac{\nabla_{\nu}u}{u}$
$\displaystyle=\frac{1}{\sin\rho}(H_{\partial
M}-\nabla_{\overline{\nu}}\rho)+\langle\nabla_{\gamma^{\prime}}\gamma^{\prime},\nu\rangle\geq
1-k_{\partial\Sigma},$

where $\overline{\nu}$ is the outward unit normal of
$\partial\Sigma\subset\left(\overline{\Omega}\cap\partial M\right)$, $\nu$ is
the outward unit normal of $\partial\Sigma\subset\Sigma$.

Using $\hat{R}_{M}\rvert_{B_{19}(x)}=R_{M}\rvert_{B_{19}(x)}\geq 0$ we have,

$\Delta_{\Sigma}u\leq\frac{1}{2}(R_{\Sigma}-\hat{R}_{M})u\leq\frac{1}{2}R_{\Sigma}u\quad\text{over
the set }B_{19}(x).$

Using Remark 3.2 to the set
$U:=\\{y\in\Sigma,d(y,\partial\Sigma)<\frac{2}{1}+3\\}\subset B_{19}(x)$, we
can now apply Lemma 3.1 to get that $d(x,\partial\Sigma)\leq 2$ for all
$x\in\Sigma$. So $\Sigma\subset B_{19}(x)$,
$\hat{R}_{M}\rvert_{\Sigma}=R_{M}\rvert_{\Sigma}$, and in the analysis below
we just write $R_{M}$.

Similar to (3.10) we have (put $a_{0}=1,d_{0}=\pi,H_{\partial M}\geq
2,|\nabla^{\partial M}\rho|\leq 1$),

$\displaystyle
0\leq\int_{\Sigma}-\frac{1}{2}(R_{M}+|\mathrm{I\\!I}_{\Sigma}|^{2})+2\pi\chi_{\Sigma}+\int_{\partial\Sigma}\frac{1}{\sin\rho}(\nabla_{\bar{\nu}}\rho-
H_{\partial M})\leq 2\pi-|\partial\Sigma|,$

here again $\chi_{\Sigma}\leq 1$ since there’s at least one boundary
component; if $\chi_{\Sigma}\leq 0$, we would get a contradiction in (3.10).
So any $\Sigma$ with nonempty boundary, must be a topological disk, and we
have $|\partial\Sigma|\leq 2\pi$ as desired.

In total, we have been able to construct finitely many capillary disks
$\Sigma^{\prime}=\partial\Omega$, denoting
$\Sigma^{\prime}_{1}=\Sigma,Z_{1}=\Omega_{1}\cap\partial M$, then each
component $\Sigma_{1}$ of $\Sigma^{\prime}_{1}$ has

$\displaystyle B^{\partial M}_{2}(x)\subset Z_{1},\quad$
$\displaystyle\partial Z_{1}=\partial\Sigma^{\prime}_{1}\subset B^{\partial
M}_{2\pi}(x)\setminus B^{\partial M}_{\pi}(x),$
$\displaystyle|\partial\Sigma_{1}|\leq 2\pi,\quad$ $\displaystyle
d(y,\partial\Sigma_{1})\leq 2\text{ for all }y\in\Sigma_{1}.$

For the non-compact boundary $\partial M$, to get our desired exhaustion, we
can replace the $w$ function above by $w_{k}$, such that $w_{k}$ is a
mollification of $\tilde{w_{k}}=-\cos\rho_{k}$ for $\rho_{k}=k\pi-d_{\partial
M}(x,\cdot)$ on $B^{\partial M}_{(k+1)\pi}(x)\setminus B^{\partial
M}_{k\pi}(x)$, and $|\tilde{w_{k}}|>1$ elsewhere. For the corresponding
$\mathcal{A}_{k}(\Omega)$, we can obtain a minimizer $\Omega_{k}$,
$\Sigma^{\prime}_{k}=\partial\Omega_{k}$ has finitely many components. We
write $Z_{k}=\Omega_{k}\cap\partial M$. For each component
$\partial\Sigma_{k}$ of $\partial\Sigma^{\prime}_{k}$, we have

$\displaystyle B^{\partial M}_{(k-1)\pi+2}(x)\subset Z_{k},\quad$
$\displaystyle\partial Z_{k}=\partial\Sigma^{\prime}_{k}\subset B^{\partial
M}_{(k+1)\pi}(x)\setminus B^{\partial M}_{k\pi}(x),$
$\displaystyle|\partial\Sigma_{k}|\leq 2\pi,\quad$ $\displaystyle
d(y,\partial\Sigma_{k})\leq 2\text{ for all }y\in\Sigma_{k}.$

Since $B^{\partial M}_{(k-1)\pi+2}(x)\subset Z_{k}$, we have
$\cup_{k}Z_{k}=\partial M$. For any $z\in\partial M\setminus\Omega_{k}$, any
path from $z$ to $x$ must contain a point in $\partial\Sigma_{k}$ by
connectedness of $\partial M$. We therefore obtained an exhaustion of
$\partial M$ via boundary of capillary surfaces of disk type, whose boundary
is at most $2\pi$ in length.

If we know $\pi_{1}(\partial M)=0$, consider $E_{k}$ an unbounded component of
$\partial M\setminus Z_{k}$ such that $E_{k+1}\subset E_{k}$. Then as in
[CLS22] Proposition 3.2, for each $k$, $\partial E_{k}$ must be connected,
$\overline{E_{k+1}}\setminus E_{k}$ is also connected. We have
$\sup_{k}\text{diam}_{\partial M}(E_{k+1}\setminus E_{k})\leq 5\pi$. Indeed,
consider any two points $z_{1},z_{2}\in E_{k+1}\setminus E_{k}$, then for each
$z_{i}$ there’s some $y_{i}\in\partial\Sigma_{k}$, such that $d_{\partial
M}(z_{i},y_{i})\leq 2\pi$, now $|\partial\Sigma_{k}|\leq 2\pi$ implies
$d_{\partial M}(y_{1},y_{2})\leq\pi$, adding up the length we get the claim. ∎

######  Remark 3.4.

As a concrete example, take any surface $\Sigma$ satisfies Theorem 1.3, i.e.
with nonnegative scalar curvature and strictly convex boundary, then Theorem
1.4 applies to $\Sigma\times\mathbb{R}$. For example if $\Sigma^{2}$ is a
strictly convex spherical cap (see [RS97]), then Theorem 1.4 applies to any
small perturbation of $\Sigma^{2}\times\mathbb{R}$. We note that if $M^{3}$
has nonnegative Ricci curvature and strictly mean convex boundary, it might
not split as $\Sigma\times\mathbb{R}$. For example consider capping off one
end of a solid cylinder by a half-ball. If $M^{3}$ has nonnegative Ricci
curvature, and strictly positive second fundamental form, then [Li14] has
conjectured that it must be compact.

######  Remark 3.5.

In the proof above the bound $d(x,\partial\Sigma)\leq 2$ for all $x\in\Sigma$
will depend on mean curvature of $\partial M\subset M$ and also on the angle
function $w$ prescribed on the boundary. In [RS97], the authors showed the
disks and spherical caps are capillary stable surfaces of constant mean
curvature in $\mathbb{B}^{3}$ when $w=\cos\theta$ is a constant function over
$\partial\mathbb{B}^{3}$.

######  Remark 3.6.

Theorem 1.4 only describes the behavior of $\partial M$, and has no control of
the interior of $M$. As an example, if $B=\mathbb{B}^{3}_{r}(z)$ is a small
interior ball in $M^{3}$ with $R_{M}\rvert_{B}>0$, then one can concatenate
$M\setminus B$ along $\partial B\approx\mathbb{S}^{2}$ with
$\mathbb{S}^{2}\times[0,\infty)$ or to any complete (non-compact) manifold
$(N,\partial N)$ with $\partial N$ homeomorphic to $\mathbb{S}^{2}$, and
$R_{N}\rvert_{\partial N}>0$.

We can now apply the same method to prove the bound for 1-Urysohn width if a
simply connected surface can be filled in by a 3-manifold with a metric with
strict mean convexity and nonnegative scalar curvature.

###### Proof of Theorem 1.1.

There is some $n\in\mathbb{N}$ so that $nd_{0}<\text{diam}(\partial
N)\leq(n+1)d_{0}$. If $n\leq 1$ then we are done, otherwise let $d_{\partial
N}(p,q)=\text{diam}(\partial N)=nd_{0}+5\delta$, then we can obtain functions
$(w_{l})_{l=1}^{n}:\partial N\rightarrow\mathbb{R}$, $w_{l}(y)<-1$ if
$d_{\partial N}(p,y)\leq 2\delta+(l-1)d_{0}$ and $w_{l}(y)>1$ if $d_{\partial
N}(p,y)\geq ld_{0}+3\delta$. When $-1<w_{l}(y)<1$ then $w_{l}=\cos\rho_{l}(x)$
for $|\nabla^{\partial N}\rho_{l}(y)|<\frac{\pi}{d_{0}}$ for any $y\in\partial
N$.

We consider a minimizer of the following functional among open sets $\Omega$
with finite perimeters, containing $S_{l}:=B^{\partial
N}_{2\delta+(l-1)d_{0}}(p)$ and disjoint from the set
$S_{l}^{\prime}:=\partial N\setminus B^{\partial N}_{ld_{0}+3\delta}(p)$,

$\displaystyle\mathcal{A}(\Omega)=|\partial\Omega|+\int_{\partial
N\cap\Omega}w_{l}.$

Then we get (using proof of Lemma 3.3) $\Sigma_{l}=\partial\Omega_{l}$
capillary surfaces with finitely many disk components
$(\Sigma_{l}^{k})_{k=1}^{K}$ and the boundary $\partial\Sigma_{l}^{k}$ of each
component $\Sigma_{l}^{k}$ must separate $\partial N$ into two components
using simply connectedness of $\partial N$. We also have
$|\partial\Sigma_{l}^{k}|\leq\frac{2\pi}{a_{0}}$.

Now $\partial M\setminus\left(\cup_{l=1}^{n}\Sigma_{l}\right)$ is a union of
finitely many 2-manifolds
$\left(\Gamma_{l}^{j}\right)_{l=0}^{n}\subset\left(\overline{\Omega_{l+1}}\setminus\Omega_{l}\right)$,
if $l=0$ think of $\Omega_{l}$ as the empty set, if $l=n$ think of
$\Omega_{l+1}$ as $N$. Each component $\Gamma_{l}^{j}$ has piecewise smooth
boundary, such that $\partial\Gamma_{l}^{j}\cap\partial\Sigma_{l}$ has at most
one component (exactly 1 if $l\geq 1$) by simply connectedness. Indeed, if not
then assume that there are $\partial\Sigma_{l}^{1},\partial\Sigma_{l}^{2}$ two
distinct boundary components in $\Gamma_{l}^{j}$, then take a fixed point
$z\in\Gamma_{l}^{j}$ and distance-minimizing paths $l_{s}$ to
$\partial\Sigma_{l}^{s}$ for $s\in\\{1,2\\}$, concatenate $l_{1},l_{2}$
together with a path in $\Omega_{l}$ transversal to both $l_{1},l_{2}$ (for
example by minimize distance to the point $p$), we get a contradiction in
terms of intersection number with respect to each of $l_{1},l_{2}$ using
simply connectedness.

We show that each $\Gamma_{l}^{j}$ has diameter bounded by
$4d_{0}+\frac{\pi}{a_{0}}$. If $l=0$ or $l=n$ this is true by triangle
inequality. We now look at $0<l<n$, for any $z_{1},z_{2}\in\Gamma_{l}^{j}$,
take the distance minimizing path to
$\partial\Gamma_{l}^{j}\cap\partial\Sigma_{l}$ called $t_{1},t_{2}$
respectively. The length of each $t_{1},t_{2}$ is no more than $2d_{0}$ by
construction. We have just shown that
$\partial\Gamma_{l}^{j}\cap\partial\Sigma_{l}$ is a connected curve with
length no more than $\frac{2\pi}{a_{0}}$, so the distance between any two
points on the curve is no more than $\frac{\pi}{a_{0}}$. Adding these together
we get the bound of diameter.

Using $\text{diam}(\partial\Sigma_{l}^{k})\leq\frac{\pi}{a_{0}}$ for each
$l,k$, we can find some tubular neighborhood $U_{l}^{k}$ of
$\partial\Sigma_{l}^{k}$ in $\partial M$ so that $\text{diam}(U_{l}^{k})\leq
4d_{0}+\frac{\pi}{a_{0}}$.

We now define a graph $G$ and a continuous map $l:\partial N\rightarrow G$.
The graph $G$ has vertices $v_{l}^{j}$ and $l(x)=v_{l}^{j}$ if
$x\in\Gamma_{l}^{j}\setminus\left(\cup_{lk}U_{l}^{k}\right)$. We connect two
vertices $v_{l}^{j}$ and $v_{l+1}^{j^{\prime}}$ with an edge
$E_{lj^{\prime}}=[0,1]$ if $\Gamma_{l}^{j}$ and $\Gamma_{l+1}^{j^{\prime}}$
are separated by some $\partial\Sigma_{l+1}^{k}$. For a point $z$ in the
tubular neighborhood $U_{l+1}^{k}$ homeomorphic to

$\partial\Sigma_{l+1}^{k}\times[0,1]=\\{(y,t),y\in\partial\Sigma_{l+1}^{k},t\in[0,1]\\},$

we map by $l(z)=l(y,t)=t\in E_{lj^{\prime}}$.

One can check that this gives us a continuous map of $\partial N$ to a
connected graph, the preimage of every point has diameter bounded by
$4d_{0}+\frac{\pi}{a_{0}}$. Using the definition of Urysohn width in [Gut17]
we have shown that the 1-Urysohn width of $\partial N$ is bounded by
$4d_{0}+\frac{\pi}{a_{0}}$. ∎

######  Remark 3.7.

In Theorem 1.1 these quantities $R_{g}\geq 0,H_{\partial
N}\geq\frac{\pi}{d_{0}}+a_{0},U_{1}(\partial N)\leq 4d_{0}+\frac{\pi}{a_{0}}$
scale accordingly. One can check that the minimum of

$(\frac{\pi}{d_{0}}+a_{0})(4d_{0}+\frac{\pi}{a_{0}})=5\pi+\frac{\pi^{2}}{a_{0}d_{0}}+4a_{0}d_{0},$

is obtained when $a_{0}d_{0}=\frac{\pi}{2}$. So we can restate the theorem
with $R_{g}\geq 0,H_{\partial N}\geq 3a_{0}$, and $U_{1}(\partial N)\leq
3\pi/a_{0}$.

##  4\. Band Width Estimate

We now continue with the proof of the band width estimate.

###### Proof of Theorem 1.2.

See Figure 1 (in introduction) as an example. We argue by contradiction. If
$d_{M}(S_{+},S_{-})>4+\pi+2\delta$ for some $\delta>0$, then there is a smooth
function $w:\partial_{0}M\rightarrow\mathbb{R}$, such that $|w(x)|>1$ if
$x\in\partial_{0}M\setminus K$, $w(x)=\cos\rho(x)$ if $x\in K$, for a compact
set $K$ with boundary in $M_{0}$:

$(K,\partial_{\pm}K)\Subset
K_{0}:=\left\\{x\in\partial_{0}M,2+\delta<d(x,S_{-})<2+\pi+2\delta\right\\},$

and $\rho$ is a smooth function with

$\rho\rvert_{\partial_{-}K}=\pi,\rho\rvert_{\partial_{+}K}=0,|\nabla\rho|<1.$

We may further assume $|w|$ is a constant greater than 1 on $\partial
M\setminus K_{0}$, in our choice we have
$w\rvert_{S_{-}}<-1<0,w\rvert_{S_{+}}>1>0$.

Now we minimize the following functional over open Caccioppoli sets $\Omega$,
with $\partial_{-}M\subset\Omega,\partial_{+}M\cap\Omega=\emptyset$:

(4.1) $\mathcal{A}(\Omega)=|\partial\Omega|+\int_{\partial M\cap\Omega}w.$

Using the same argument in section 3, we know any minimizing sequence has its
boundary contained in the compact set $K$. By our convexity assumption, we
note that if the minimizing Caccioppoli set has an interior point on
$\partial_{\pm}M$, then by the maximum principle ([SW89]), it must be equal to
a component of $\partial M$ that is minimal in $M$, a contradiction to our
assumption.

By regularity of capillary problem [DM14] [CEL24], we have a smooth minimizer
$\Sigma^{\prime}=\partial\Omega^{\prime}$.

By the same proof as in Lemma 3.3 (plug in $d_{0}=\pi,a_{0}=1$), for each
component $\Sigma$ of $\Sigma^{\prime}$, we have
$d_{\Sigma}(x,\partial\Sigma)\leq 2$ for $x\in\Sigma$, this gives us that
$d_{M}(\Sigma,S_{\pm})>0$. So we have that $M_{-}\subset\Omega^{\prime}$,
$\Omega^{\prime}\cap M_{+}=\emptyset$,
$\partial\Omega^{\prime}=\Sigma^{\prime}\subset M_{0}$ and we can find a map
of degree 1 (mod 2) from a component $\Sigma$ of $\Sigma^{\prime}$ onto
$S_{-}$ as follows.

Consider a diffeomorphism $f:M_{0}\rightarrow S_{-}\times[-1,1]$, then for any
$x=f^{-1}(z_{0},t_{0})\in\Sigma^{\prime}$, we define the projection map
$P(x)=z_{0}$. Consider a regular value $z\in S_{-}$ and the path
$I_{z}:=f^{-1}(z,t)$ in $M_{0}$ , we may write $I_{z}$ just as $(z,t)$ for
$t\in[-1,1]$. We look at the characteristic function of $\Omega^{\prime}$,
then $\chi_{\Omega^{\prime}}(z,-1)=1$ and $\chi_{\Omega^{\prime}}(z,1)=0$, we
note that $z$ can be in $\partial S_{-}$, and $P^{-1}(\partial
S_{-})=\partial\Omega^{\prime}$, so there is some component $\Sigma$ of
$\Sigma^{\prime}$ such that the degree of $P$ is equal to 1 (mod 2). We also
note that this means for any $x\in\partial S_{-}$,
$P^{-1}(x)\cap\Sigma\neq\emptyset$. If $\partial S_{-}$ has at least two
components, then $\chi(\Sigma)\leq 0$, otherwise we have a map of degree 1
(mod 2) from $(\Sigma,\partial\Sigma)\rightarrow(S_{-},\partial S_{-})$

by Kneser’s theorem [Kne30] (see [Edm79] Proposition 4.4 for the case for
manifold with boundary), we have $\chi(\Sigma)\leq 0$. Therefore, using the
stability inequality (see also 3.10) we get,

$\displaystyle 0\leq$
$\displaystyle\int_{\Sigma}-\frac{1}{2}(R_{M}+|\mathrm{I\\!I}_{\Sigma}|^{2})+\int_{\partial\Sigma}\frac{1}{\sin\rho}(\nabla_{\bar{\nu}}\rho-
H_{\partial M})+2\pi\chi_{\Sigma}$ $\displaystyle<$
$\displaystyle\int_{\partial\Sigma}\frac{1}{\sin\rho}(1-2)+2\pi\chi_{\Sigma}<0$

a contradiction. ∎

We first give some examples relating to Theorem 1.2.

######  Example 4.1.

We give an example when $M_{0}$ is homeomorphic to $S_{-}\times[-1,1]$ with
$\chi(S_{-})>0$, to show that in Theorem 1.2, the assumption of
$\chi(S_{-})\leq 0$ is crucial. Take the cylinder
$\mathbb{D}^{2}\times[-L,L]\subset\mathbb{R}^{3}$, with $\mathbb{D}^{2}$ unit
disk in $\mathbb{R}^{2}$, and cap the top (or bottom) slice
$\mathbb{D}^{2}\times\\{\pm L\\}$ with an upper (or lower) hemisphere, then
the length of the cylinder is unbounded when we let $L\rightarrow\infty$.

######  Example 4.2.

If $(\Sigma,\partial\Sigma),\partial\Sigma\neq\emptyset$ is a surface with
strictly convex boundary, nonnegative Euler characteristic, and nonnegative
scalar curvature, then the band width estimate applies to
$\Sigma\times\mathbb{S}^{1}$.

######  Remark 4.3.

For $(M,\partial M)$ as in Theorem 1.2, following the proof, we can replace
the statement $R_{M}\geq 0,H_{0}\geq 2,H_{\pm}\geq 0,d_{\partial
M}(\partial_{+}M,\partial_{-}M)\leq 4+\pi$, with the following: if $R_{M}\geq
0,H_{0}\geq\frac{\pi}{d_{0}}+a_{0},H_{\pm}\geq 0$, then $d_{\partial
M}(\partial_{+}M,\partial_{-}M)\leq d_{0}+\frac{4}{a_{0}}$.

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