# On the mean curvature type flow for convex capillary hypersurfaces in the
ball

Yingxiang Hu  School of Mathematics, Beihang University, Beijing 100191, P.R.
China<EMAIL_ADDRESS>,  Yong Wei  School of Mathematical Sciences,
University of Science and Technology of China, Hefei 230026, P.R. China
<EMAIL_ADDRESS>,  Bo Yang  Institute of Mathematics, Academy of
Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing,
100190, P. R. China<EMAIL_ADDRESS>and  Tailong Zhou  School of
Mathematical Sciences, University of Science and Technology of China, Hefei
230026, P.R. China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the mean curvature type flow for hypersurfaces in the
unit Euclidean ball with capillary boundary, which was introduced by Wang-Xia
[Wang-Xia2019] and Wang-Weng [WW2020]. We show that if the initial
hypersurface is strictly convex, then the solution of this flow is strictly
convex for $t>0$, exists for all positive time and converges smoothly to a
spherical cap. As an application, we prove a family of new Alexandrov-Fenchel
inequalities for convex hypersurfaces in the unit Euclidean ball with
capillary boundary.

###### Key words and phrases:

Alexandrov-Fenchel inequalities, quermassintegral, capillary hypersurfaces in
a ball, mean curvature type flow

###### 2010 Mathematics Subject Classification:

53C44, 53C21, 35K93, 52A40

###### Contents

  1. 1 Introduction
  2. 2 Convex hypersurfaces with capillary boundary
  3. 3 Evolution equations
  4. 4 Preserving of convexity
  5. 5 Proofs of Theorem 1.1 and Theorem 1.2
  6. A Approximation result

##  1\. Introduction

In this paper, we are interested in the mean curvature type flow for
hypersurfaces in the unit Euclidean ball with capillary boundary, which was
introduced recently by Wang-Xia [Wang-Xia2019] and Wang-Weng [WW2020]. We
first describe some definitions and notations for such hypersurfaces. The
readers are referred to [Weng-Xia2022] for more details. Let
$\Sigma\subset\bar{\mathbb{B}}^{n+1}(n\geq 2)$ be a properly embedded smooth
hypersurface in the unit Euclidean ball $\bar{\mathbb{B}}^{n+1}$, given by an
embedding $x:\bar{\mathbb{B}}^{n}\to\bar{\mathbb{B}}^{n+1}$ satisfying

$\displaystyle\Sigma=x(\mathbb{B}^{n})\subset\mathbb{B}^{n+1},\quad\partial\Sigma=x(\partial\mathbb{B}^{n})\subset\mathbb{S}^{n}.$

Let $\bar{N}$ be the unit outward normal of
$\mathbb{S}^{n}=\partial\mathbb{B}^{n+1}$, and $\nu$ be a smooth choice of the
unit normal of $\Sigma$. For $\theta\in(0,\pi)$, we call that $\Sigma$ has
$\theta$-capillary boundary $\partial\Sigma$ on $\mathbb{S}^{n}$ if $\Sigma$
intersects $\mathbb{S}^{n}$ at the constant contact angle $\theta$, that is,

$\langle\bar{N}\circ x,\nu\rangle=-\cos\theta,\quad\text{along
$\partial\mathbb{B}^{n}$}.$

In particular, if $\theta=\frac{\pi}{2}$, i.e., $\Sigma$ intersects
$\mathbb{S}^{n}$ orthogonally, we call that $\Sigma$ has free boundary. Two
model examples are the spherical cap of radius $r$ around a constant unit
vector $e\in\mathbb{S}^{n}\subset\mathbb{R}^{n+1}$ with $\theta$-capillary
boundary, given by

$\displaystyle
C_{\theta,r}(e):=\left\\{x\in\bar{\mathbb{B}}^{n+1}:\left|x-\sqrt{r^{2}+2r\cos\theta+1}e\right|=r\right\\},$
(1.1)

and the flat ball around $e$ with $\theta$-capillary boundary, given by

$\displaystyle C_{\theta,\infty}(e):=\\{x\in\bar{\mathbb{B}}^{n+1}:\langle
x,e\rangle=\cos\theta\\}.$ (1.2)

Denote the principal curvatures of $\Sigma$ by
$\kappa=(\kappa_{1},\cdots,\kappa_{n})$. For $k=1,\cdots,n$, we denote by
$H_{k}$ the normalized $k$th mean curvature of $\Sigma$, which is defined as
the normalized $k$th elementary symmetric polynomial of $\kappa$:

$H_{k}=\binom{n}{k}^{-1}\sigma_{k}(\kappa),\quad k=1,\cdots,n.$

We also denote the mean curvature as $H=nH_{1}$. We say that $\Sigma$ is
convex (resp. strictly convex) if its principal curvatures $\kappa_{i}\geq 0$
(resp. $\kappa_{i}>0$). In this paper, we always assume that $\Sigma$ is
convex. Denote by $\widehat{\partial\Sigma}$ the convex body in
$\mathbb{S}^{n}$ enclosed by $\partial\Sigma$, and $\widehat{\Sigma}$ the
convex domain in $\overline{\mathbb{B}}^{n+1}$ enclosed by
$\widehat{\partial\Sigma}$ and $\Sigma$. See Figure 1. We choose the unit
normal $\nu$ of $\Sigma$ as the one pointing outward of $\widehat{\Sigma}$.
For each $e\in\mathbb{S}^{n}\subset\mathbb{R}^{n+1}$, the smooth vector field
$X_{e}$ in $\mathbb{R}^{n+1}$ defined by

$X_{e}=\langle x,e\rangle x-\frac{1}{2}(|x|^{2}+1)e$ (1.3)

is a conformal Killing vector field in $\mathbb{R}^{n+1}$ and its restriction
on $\partial\mathbb{B}^{n+1}$ is a tangential vector field on
$\partial\mathbb{B}^{n+1}$ (see [Wang-Xia2019-2, Prop. 3.1]). Using $X_{e}$,
the following Minkowski type formula (see [Weng-Xia2022, Prop. 2.8]) holds:

$\int_{\Sigma}{H_{k-1}\langle
x+\cos\theta\nu,e\rangle}\,dA=\int_{\Sigma}{H_{k}\langle
X_{e},\nu\rangle}\,dA,\quad k=1,\cdots,n.$ (1.4)

Based on the formula (1.4), the following mean curvature type flow with
$\theta$-capillary boundary was introduced by Wang and Xia [Wang-Xia2019] for
the case $\theta=\frac{\pi}{2}$, and later by Wang and Weng [WW2020] for
general case $\theta\in(0,\pi)$:

$\left\\{\begin{aligned} \left(\partial_{t}x\right)^{\bot}&=\Big{(}n\langle
x+\cos\theta\nu,e\rangle-H\langle
X_{e},\nu\rangle\Big{)}\nu&\text{in}\quad\bar{\mathbb{B}}^{n}\times[0,T),\\\
\langle\bar{N}\circ
x,\nu\rangle&=-\cos\theta&\text{on}\quad\partial\bar{\mathbb{B}}^{n}\times[0,T),\\\
x(\cdot,0)&=x_{0}(\cdot)&\text{on}\quad\bar{\mathbb{B}}^{n},\end{aligned}\right.$
(1.5)

where $(\cdot)^{\bot}$ denotes the projection to the normal bundle of
$\Sigma$. Denote $\Sigma_{t}=x(\bar{\mathbb{B}}^{n},t)$ as the solution of the
flow (1.5). The flow (1.5) has a property that the volume
$|\widehat{\Sigma_{t}}|$ is preserved which follows from (1.4) for $k=1$. The
convergence results [Wang-Xia2019, WW2020] of the flow (1.5) for star-shaped
initial hypersurface $\Sigma$, i.e., $\langle X_{e},\nu\rangle>0$ holds
everywhere on $\Sigma$, can be stated as follows.

######  Theorem A.

Let $\Sigma\subset\bar{\mathbb{B}}^{n+1}(n\geq 2)$ be a properly embedded
smooth hypersurface with $\theta$-capillary boundary, given by an embedding:
$x:\bar{\mathbb{B}}^{n}\to\Sigma\subset\bar{\mathbb{B}}^{n+1}$. Assume that
$\Sigma$ is star-shaped with respect to $e\in\mathbb{S}^{n}$, i.e., $\langle
X_{e},\nu\rangle>0$ holds everywhere in $\Sigma$.

  1. $(i)$

[Wang-Xia2019] If $\theta=\frac{\pi}{2}$, then the solution $\Sigma_{t}$ of
the flow (1.5) exists for all positive time and converges smoothly to a
spherical cap with free boundary as $t\rightarrow\infty$, whose enclosed
domain has the same volume as $|\widehat{\Sigma}|$.

  2. $(ii)$

[WW2020] If $\theta$ satisfies $|\cos\theta|<\frac{3n+1}{5n-1}$, then the
solution $\Sigma_{t}$ of the flow (1.5) exists for all positive time and
subsequently converges smoothly to a spherical cap with $\theta$-capillary
boundary as $t\rightarrow\infty$, whose enclosed domain has the same volume as
$|\widehat{\Sigma}|$.

The first aim of this paper is to prove the preservation of the strict
convexity along the flow (1.5). The main tool is the tensor maximum principle
in Theorem 4.1, which was developed by the authors in [HWYZ2022]. With help of
the convexity preserving, we establish the convergence result of the flow
(1.5) for convex hypersurfaces with $\theta$-capillary boundary in the unit
Euclidean ball.

######  Theorem 1.1.

Let $\Sigma\subset\bar{\mathbb{B}}^{n+1}(n\geq 2)$ be a properly embedded,
strictly convex smooth hypersurface with $\theta$-capillary boundary
($\theta\in(0,\frac{\pi}{2}]$), given by an embedding:
$x:\bar{\mathbb{B}}^{n}\to\Sigma\subset\bar{\mathbb{B}}^{n+1}$. Then there
exists a constant unit vector $e\in\mathbb{S}^{n}$ such that $\langle
X_{e},\nu\rangle>0$ holds everywhere on $\Sigma$, and the solution
$\Sigma_{t}$ of the flow (1.5) starting from $\Sigma$ is strictly convex and
exists for all time $t>0$. Moreover, $\Sigma_{t}$ converges smoothly to a
spherical cap $C_{\theta,r_{\infty}}(e)$ as $t\rightarrow\infty$, where
$r_{\infty}$ is determined by
$|\widehat{C_{\theta,r_{\infty}}}(e)|=|\widehat{\Sigma}|$.

The mean curvature type flow (1.5) is a locally constrained curvature flow,
which was previously considered by Guan and Li [GL-2015] for closed
hypersurfaces in space forms. Other kinds of locally constrained curvature
flows have been investigated by many authors, see [BGL, GLW-2019, Lambert-
Scheuer2021, Sch21, SX-2019, HLW2020, GL-2021, Wei-X2022a, Wei-X2022b], etc. A
main motivation of studying these locally constrained curvature flows is their
powerful applications in proving geometric inequalities including the
Alexandrov-Fenchel type inequalities for quermassintegrals.

The quermassintegrals for convex hypersurfaces with $\theta$-capillary
boundary in $\bar{\mathbb{B}}^{n+1}$ were first introduced by Scheuer, Wang
and Xia [Scheuer-Wang-Xia2018] for $\theta=\frac{\pi}{2}$, and later by Weng
and Xia [Weng-Xia2022] for $\theta\in(0,\frac{\pi}{2}]$. Let
$\Sigma\subset\bar{\mathbb{B}}^{n+1}(n\geq 2)$ be a properly embedded, convex
smooth hypersurface with $\theta$-capillary boundary
$\partial\Sigma\subset\mathbb{S}^{n}$, where $\theta\in(0,\frac{\pi}{2}]$.
Then the quermassintegrals $W_{k,\theta}(\widehat{\Sigma})$ for
$\widehat{\Sigma}$ are defined by (2.2)–(2.4). In particular,

$\displaystyle W_{0,\theta}(\widehat{\Sigma})=$
$\displaystyle|\widehat{\Sigma}|,\qquad
W_{1,\theta}(\widehat{\Sigma})=\frac{1}{n+1}\left(|\Sigma|-\cos\theta|\widehat{\partial\Sigma}|\right).$

For strictly convex hypersurfaces in $\bar{\mathbb{B}}^{n+1}$ with
$\theta$-capillary boundary, the following flow

$\left\\{\begin{aligned}
\left(\partial_{t}x\right)^{\bot}&=\left(\frac{\langle
x,e\rangle+\cos\theta\langle\nu,e\rangle}{H_{n}/H_{n-1}}-\langle
X_{e},\nu\rangle\right)\nu&\text{in}\quad\bar{\mathbb{B}}^{n}\times[0,T),\\\
\langle\bar{N}\circ
x,\nu\rangle&=-\cos\theta&\text{on}\quad\partial\bar{\mathbb{B}}^{n}\times[0,T),\\\
x(\cdot,0)&=x_{0}(\cdot)&\text{on}\quad\bar{\mathbb{B}}^{n},\end{aligned}\right.$
(1.6)

was studied in [Scheuer-Wang-Xia2018, Weng-Xia2022], and as an application
they proved that for any $\ell=0,1,\cdots,n-1$, there holds

$W_{n,\theta}(\widehat{\Sigma})\geq(f_{n}\circ
f_{\ell}^{-1})(W_{\ell,\theta}(\widehat{\Sigma})),$ (1.7)

where $f_{\ell}(r)$ is the strictly increasing function given by
$f_{\ell}(r)=W_{\ell,\theta}(\widehat{C_{\theta,r}})$. Equality holds in (1.7)
if and only if $\Sigma$ is a spherical cap or a flat ball with
$\theta$-capillary boundary.

As an application of our Theorem 1.1, we prove new sharp Alexandrov-Fenchel
type inequalities relating $W_{k,\theta}(\widehat{\Sigma})$ and
$W_{0,\theta}(\widehat{\Sigma})$ for $k\geq 1$.

######  Theorem 1.2.

Let $\Sigma\subset\bar{\mathbb{B}}^{n+1}(n\geq 2)$ be a properly embedded,
convex smooth hypersurface in the unit ball with $\theta$-capillary boundary,
where $\theta\in(0,\frac{\pi}{2}]$. Then for any $k=1,\cdots,n-1$, there holds

$W_{k,\theta}(\widehat{\Sigma})\geq(f_{k}\circ
f_{0}^{-1})(W_{0,\theta}(\widehat{\Sigma})).$ (1.8)

Equality holds in (1.8) if and only if $\Sigma$ is a spherical cap or a flat
ball with $\theta$-capillary boundary.

######  Remark 1.3.

The inequality (1.8) for $k=n$ is the inequality (1.7) for $\ell=0$ and has
been proved in [Scheuer-Wang-Xia2018, Weng-Xia2022].

The paper is organized as follows: In §2, we give some preliminaries for
convex capillary hypersurfaces in the unit ball, including some estimates on
the geometric quantities on the hypersurfaces. We also recall the definition
of quermassintegrals for such hypersurfaces. In §3, the evolution equations
along the flow (1.5) will be deduced. In §4, we apply the tensor maximum
principle to prove the preservation of convexity along the flow (1.5). In §5,
we give the proofs of Theorem 1.1 and Theorem 1.2.

######  Acknowledgments.

The authors would like to thank Professor Chao Xia for his helpful
suggestions. The research was surpported by National Key Research and
Development Program of China 2021YFA1001800 and 2020YFA0713100, National
Natural Science Foundation of China NSFC11721101 and NSFC12101027, and
Research grant KY0010000052 from University of Science and Technology of
China.

##  2\. Convex hypersurfaces with capillary boundary

In this section, we collect some preliminaries on smooth hypersurfaces in the
unit ball with capillary boundary, the quermassintegrals, and some estimates
on convex hypersurfaces with capillary boundary.

###  2.1. Hypersurfaces in the ball with capillary boundary

Let $\theta\in(0,\frac{\pi}{2}]$. Suppose that
$\Sigma\subset\bar{\mathbb{B}}^{n+1}$ is a smooth, properly embedded, convex
hypersurface with $\theta$-capillary boundary, which is given by an embedding
$x:\bar{\mathbb{B}}^{n}\rightarrow\Sigma$ such that

$\Sigma=x(\mathbb{B}^{n})\subset\mathbb{B}^{n+1},\quad\partial\Sigma=x(\partial\mathbb{B}^{n})\subset\mathbb{S}^{n}.$

Then $\partial\Sigma\subset\mathbb{S}^{n}$ is a convex hypersurface of the
unit sphere and bounds a convex body in $\mathbb{S}^{n}$, which we denote by
$\widehat{\partial\Sigma}$, cf. [CW1970, Theorem 1.1]. Denote by
$\widehat{\Sigma}$ the bounded domain in $\bar{\mathbb{B}}^{n+1}$ enclosed by
$\Sigma$ and $\widehat{\partial\Sigma}$. We denote by $\nu$ the unit normal
field of $\Sigma$ and $\bar{N}$ the position vector of $\mathbb{S}^{n}$. We
identify the outward pointing conormal of $\partial\Sigma\subset\Sigma$ with
$\mu$ and the unit normal of $\partial\Sigma$ in $\mathbb{S}^{n}$ with
$\bar{\nu}$ such that $\\{\nu,\mu\\}$ and $\\{\bar{\nu},\bar{N}\circ x\\}$
have the same orientation in the normal bundle of
$\partial\Sigma\subset\mathbb{S}^{n}$, see Figure 1.

o$\Sigma$$\widehat{\partial\Sigma}$$\widehat{\Sigma}$$\partial\Sigma$$\theta$$\theta$$\mu$$\nu$$\bar{N}$$\bar{\nu}\circ
x$ Figure 1. A convex $\theta$-capillary hypersurface in the unit ball

It follows from $\langle\bar{N}\circ x,\nu\rangle=-\cos\theta$ that

$\left\\{\begin{aligned} \bar{N}\circ x=&\sin\theta\mu-\cos\theta\nu,\\\
\bar{\nu}=&\cos\theta\mu+\sin\theta\nu.\end{aligned}\right.$ (2.1)

We denote by $D$ the Levi-Civita connection of $\mathbb{R}^{n+1}$ with respect
to the Euclidean metric $\delta$, and $\nabla$ the Levi-Civita connection on
$\Sigma$ with respect to the induced metric from the embedding
$x:\bar{\mathbb{B}}^{n}\rightarrow\Sigma\subset\mathbb{R}^{n+1}$,
respectively. The second fundamental of $\Sigma$ in $\mathbb{R}^{n+1}$ is
given by

$h(X,Y):=-\langle D_{X}Y,\nu\rangle,\ \ X,Y\in T(\Sigma).$

Note that $\partial\Sigma$ can be viewed as a smooth closed hypersurface in
$\mathbb{S}^{n}$ and $\Sigma$, respectively. The second fundamental form of
$\partial\Sigma$ in $\mathbb{S}^{n}$ is given by

$\widehat{h}(X,Y):=-\langle\nabla^{\mathbb{S}^{n}}_{X}Y,\bar{\nu}\rangle=-\langle
D_{X}Y,\bar{\nu}\rangle,\ \ X,Y\in T(\partial\Sigma).$

The second fundamental form of $\partial\Sigma$ in $\Sigma$ is given by

$\tilde{h}(X,Y):=-\langle\nabla_{X}Y,\mu\rangle=-\langle D_{X}Y,\mu\rangle,\ \
X,Y\in T(\partial\Sigma).$

######  Proposition 2.1 ([Weng-Xia2022]).

Let $\Sigma\subset\bar{\mathbb{B}}^{n+1}$ be a $\theta$-capillary
hypersurface. Let $\\{e_{\alpha}\\}_{\alpha=1}^{n-1}$ be an orthonormal frame
of $\partial\Sigma$. Then the following relations hold on $\partial\Sigma$:

  1. $(1)$

$\mu$ is a principal direction of $\Sigma$, i.e., $h(\mu,e_{\alpha})=0$.

  2. $(2)$

$h_{\alpha\beta}=\sin\theta\widehat{h}_{\alpha\beta}-\cos\theta\delta_{\alpha\beta}$.

  3. $(3)$

$\tilde{h}_{\alpha\beta}=\cos\theta\widehat{h}_{\alpha\beta}+\sin\theta\delta_{\alpha\beta}=\cot\theta
h_{\alpha\beta}+\frac{1}{\sin\theta}\delta_{\alpha\beta}$.

  4. $(4)$

$\nabla_{\mu}h_{\alpha\beta}=\tilde{h}_{\beta\gamma}\left(h_{\mu\mu}\delta_{\alpha\gamma}-h_{\alpha\gamma}\right)$.

###  2.2. Quermassintegrals

The quermassintegrals for $\theta$-capillary hypersurfaces in the unit ball
were first introduced for $\theta=\frac{\pi}{2}$ by Scheuer, Wang and Xia
[Scheuer-Wang-Xia2018], and later generalized by Weng and Xia [Weng-Xia2022]
to $\theta\in(0,\frac{\pi}{2}]$. Let $\Sigma\subset\bar{\mathbb{B}}^{n+1}$ be
a smooth convex, embedded hypersurface with $\theta$-capillary boundary
$\partial\Sigma\subset\mathbb{S}^{n}$. The quermassintegrals for
$\widehat{\Sigma}$ are defined by

$\displaystyle W_{0,\theta}(\widehat{\Sigma})=$
$\displaystyle|\widehat{\Sigma}|,$ (2.2) $\displaystyle
W_{1,\theta}(\widehat{\Sigma})=$
$\displaystyle\frac{1}{n+1}\left(|\Sigma|-\cos\theta
W_{0}^{\mathbb{S}^{n}}(\widehat{\partial\Sigma})\right),$ (2.3)

and for $1\leq k\leq n-1$,

$\displaystyle\begin{aligned}
&W_{k+1,\theta}(\widehat{\Sigma})=\frac{1}{n+1}\left\\{\int_{\Sigma}H_{k}dA-\cos\theta\sin^{k}\theta
W_{k}^{\mathbb{S}^{n}}(\widehat{\partial\Sigma})\right.\\\
&\quad\quad\quad\left.-\cos^{k-1}\theta\sum_{\ell=0}^{k-1}\frac{(-1)^{k+\ell}}{n-\ell}\binom{k}{\ell}\left[(n-k)\cos^{2}\theta+k-\ell\right]\tan^{\ell}\theta
W_{\ell}^{\mathbb{S}^{n}}(\widehat{\partial\Sigma})\right\\}.\end{aligned}$
(2.4)

For the free boundary case $\theta=\pi/2$, the definition (2.4) simplifies as

$\displaystyle
W_{k+1,\frac{\pi}{2}}(\widehat{\Sigma})=\frac{1}{n+1}\left\\{\int_{\Sigma}H_{k}dA-\frac{k}{n-k+1}W_{k-1}^{\mathbb{S}^{n}}(\widehat{\partial\Sigma})\right).$
(2.5)

Here $dA$ is the area element and $H_{k}$ is the $k$th normalized mean
curvature of the hypersurface $\Sigma$ in $\mathbb{R}^{n+1}$ respectively, and
for a $k$-dimensional submanifold $M\subset\mathbb{R}^{n+1}$ (with or without
boundary), $|M|$ denotes the $k$-dimensional Hausdorff measure of $M$. The
quermassintegrals $W_{k}^{\mathbb{S}^{n}}(\widehat{\partial\Sigma})$ in
$\mathbb{S}^{n}$ are defined by (see [Sol06, ChenGLS22])

$\displaystyle W_{0}^{\mathbb{S}^{n}}(\widehat{\partial\Sigma})=$
$\displaystyle|\widehat{\partial\Sigma}|,$ $\displaystyle
W_{1}^{\mathbb{S}^{n}}(\widehat{\partial\Sigma})=$
$\displaystyle\frac{1}{n}|\partial\Sigma|,$ $\displaystyle
W_{k+1}^{\mathbb{S}^{n}}(\widehat{\partial\Sigma})=$
$\displaystyle\frac{1}{n}\int_{\partial\Sigma}H_{k}^{\mathbb{S}^{n}}ds+\frac{k}{n-k+1}W^{\mathbb{S}^{n}}_{k-1}(\widehat{\partial\Sigma}),\quad
1\leq k\leq n-1,$

where $ds$ is the area element and $H_{k}^{\mathbb{S}^{n}}$ is the $k$th
normalized mean curvature of the hypersurface $\partial\Sigma$ in
$\mathbb{S}^{n}$ respectively.

The quermassintegrals $W_{k,\theta}(\widehat{\Sigma})$ defined in (2.2) -
(2.4) are viewed as a natural counterparts to the quermassintegrals for smooth
closed hypersurfaces in $\mathbb{R}^{n+1}$, as evidenced by the following nice
variational formula:

######  Proposition 2.2 ([Weng-Xia2022]).

Let $\Sigma_{t}\subset\bar{\mathbb{B}}^{n+1}$ be a family of smooth, properly
embedded hypersurfaces with $\theta$-capillary boundary on $\mathbb{S}^{n}$,
given by
$x(\cdot,t):\bar{\mathbb{B}}^{n}\rightarrow\Sigma_{t}\subset\bar{\mathbb{B}}^{n+1}$,
such that

$\left(\partial_{t}x\right)^{\perp}=f\nu$

for some normal speed $f$. Then there holds

$\frac{d}{dt}W_{k,\theta}(\widehat{\Sigma}_{t})=\frac{n+1-k}{n+1}\int_{\Sigma_{t}}H_{k}fdA,\quad
0\leq k\leq n.$ (2.6)

Using the Minkowski formula (1.4) and the variational formula (2.6), we have
the following monotonicity of quermassintegrals along the flow (1.5).

######  Lemma 2.3.

Let $\Sigma_{t}$ be a smooth strictly convex solution of the flow (1.5). Then
along the flow, $W_{0,\theta}(\widehat{\Sigma_{t}})$ is preserved and
$W_{k,\theta}(\widehat{\Sigma_{t}})$ is non-increasing in time for all $1\leq
k\leq n-1$. Moreover, $\frac{d}{dt}W_{k,\theta}(\widehat{\Sigma_{t}})=0$ if
and only if $\Sigma_{t}$ is a spherical cap.

###### Proof.

Firstly, by the variational formula (2.6) and the Minkowski formula (1.4) for
$k=1$, we have

$\displaystyle\frac{d}{dt}W_{0,\theta}(\widehat{\Sigma_{t}})=\frac{n}{n+1}\int_{\Sigma_{t}}\Big{(}\langle
x+\cos\theta\nu,e\rangle-H_{1}\langle X_{e},\nu\rangle\Big{)}dA=0.$

As the solution $\Sigma_{t}$ is strictly convex for $t>0$, using the Newton’s
inequality $H_{k+1}\leq H_{k}H_{1}$ and the Minkowski formula (1.4) again, we
obtain

$\displaystyle\frac{d}{dt}W_{k,\theta}(\widehat{\Sigma_{t}})=$
$\displaystyle\frac{n(n+1-k)}{n+1}\int_{\Sigma_{t}}\Big{(}H_{k}\langle
x+\cos\theta\nu,e\rangle-\langle X_{e},\nu\rangle H_{1}H_{k}\Big{)}dA$
$\displaystyle=$
$\displaystyle\frac{n(n+1-k)}{n+1}\int_{\Sigma_{t}}\left(H_{k+1}-H_{1}H_{k}\right)\langle
X_{e},\nu\rangle dA$ $\displaystyle\leq$ $\displaystyle 0,$ (2.7)

for $k=1,\cdots,n-1$. If equality holds in (2.2), then $\Sigma_{t}$ are
umbilic and hence it is a spherical cap with $\theta$-capillary boundary. ∎

###  2.3. Estimates on convex capillary hypersurfaces

In this subsection, we collect some estimates on strictly convex hypersurfaces
with $\theta$-capillary boundary in the unit ball, which were obtained by Weng
and Xia [Weng-Xia2022].

######  Proposition 2.4.

[Weng-Xia2022, Prop. 2.15] Let $\Sigma\subset\bar{\mathbb{B}}^{n+1}$ be a
strictly convex hypersurface with $\theta$-capillary boundary for
$\theta\in(0,\frac{\pi}{2}]$. Then for any $p\in\Sigma$, $\Sigma$ lies on one
side of $T_{p}\Sigma$.

Let $\Sigma\subset\bar{\mathbb{B}}^{n+1}$ be a strictly convex hypersurface
with $\theta$-capillary boundary. Then it is proved in [Weng-Xia2022, Prop.
2.16] that there exists a point
$e\in\operatorname{int}(\widehat{\partial\Sigma})$ and a constant
$0<\delta_{0}<1-\cos\theta$ depending only on $\Sigma$ such that

$\displaystyle\langle x,e\rangle\geq\cos\theta+\delta_{0},$ (2.8)
$\displaystyle\langle x,\nu\rangle\leq-\cos\theta.$ (2.9)

Recall that the foliations $C_{\theta,s}(e)$, $s\geq 0$ are spherical caps of
radius $s$ around $e$ with $\theta$-capillary boundary such that
$C_{\theta,\infty}(e)$ is the flat ball around $e$ with $\theta$-capillary
boundary. Then there exists some $0<R_{1}<R_{2}<\infty$ such that

$\displaystyle\Sigma\subset\widehat{C_{\theta,R_{2}}(e)}\backslash\widehat{C_{\theta,R_{1}}(e)}.$

######  Proposition 2.5 (cf. Prop.2.16 of [Weng-Xia2022]).

Let $\Sigma\subset\bar{\mathbb{B}}^{n+1}$ be a strictly convex hypersurface
with $\theta$-capillary boundary such that
$\Sigma\subset\widehat{C_{\theta,R_{2}}(e)}\backslash\widehat{C_{\theta,R_{1}}(e)}$.
Then the following estimates hold:

  1. $(1)$

$\langle x-e,\nu\rangle\geq\delta_{1}>0$;

  2. $(2)$

$\langle X_{e},\nu\rangle\geq\delta_{2}>0$;

  3. $(3)$

$\cos\theta+\delta_{0}\leq\langle x,e\rangle\leq 1-\delta_{3}$;

  4. $(4)$

$\langle e,\nu\rangle\leq-\delta_{4}$.

Here $\delta_{i}$, $i=1,2,3,4$ are positive constants depending only on
$R_{1}$, $\theta$, and $\delta_{0}$ is a positive constant depending only on
$R_{2}$, $\theta$ such that $\cos\theta+\delta_{0}\leq 1-\delta_{3}$.

###### Proof.

The estimates are proved in Prop.2.16 of [Weng-Xia2022]. Here we provide a
proof by clarifying that the constants $\delta_{i},i=0,1,\cdots,4$ can be
chosen explicitly depending only on $\theta,R_{1}$ and $R_{2}$. These
estimates will be used in the proof of preserving of the strict convexity of
$\Sigma_{t}$ in Theorem 4.3. Let $S$ be a flat ball in
$\bar{\mathbb{B}}^{n+1}$ such that $\partial S=\partial C_{\theta,R_{1}}(e)$.
Since $S\subset\widehat{C_{\theta,R_{1}}(e)}\subset\widehat{\Sigma}$, we know
that $S$ lies in the upper half-space determined by $T_{x}\Sigma$ in the
direction of $-\nu$, see Figure 2.

$e$$S$$C_{\theta,R_{1}}(e)$$C_{\theta,R_{2}}(e)$$O$$T_{x}\Sigma$$x$$\nu$$\Sigma$
Figure 2.  Capillary hypersurface between two spherical caps

Let $d^{\mathbb{R}}$ be the distance in $\mathbb{R}^{n+1}$. Since $\langle
x-e,\nu\rangle$ measures the distance from $e$ to $T_{x}\Sigma$, which can be
bounded from below by the distance from $e$ to $S$, i.e., $\langle
x-e,\nu\rangle\geq d^{\mathbb{R}}(e,S)$. We calculate $d^{\mathbb{R}}(e,S)$
explicitly as follows:

$\displaystyle
1-(1-d^{\mathbb{R}}(e,S))^{2}=R_{1}^{2}-(\sqrt{R_{1}^{2}+2R_{1}\cos\theta+1}-1+d^{\mathbb{R}}(e,S))^{2},$

which gives

$d^{\mathbb{R}}(e,S)=1-\frac{1+R_{1}\cos\theta}{\sqrt{R_{1}^{2}+2R_{1}\cos\theta+1}}=:\delta_{1}(R_{1},\theta)>0.$

Then we get $\langle x-e,\nu\rangle\geq\delta_{1}$. It follows that

$\displaystyle\langle X_{e},\nu\rangle=\frac{1+|x|^{2}}{2}\langle
x-e,\nu\rangle-\langle
x,\nu\rangle\frac{|e-x|^{2}}{2}\geq\frac{1}{2}\delta_{1}=:\delta_{2}(R_{1},\theta),$

where we used $\langle x,\nu\rangle\leq-\cos\theta\leq 0$ by (2.9).

Note that $\Sigma\subset\widehat{C_{\theta,R_{2}}(e)}$, i.e., $\Sigma$ lies
above $C_{\theta,R_{2}}(e)$ in the direction of $e$. Then we get

$\displaystyle\min_{\Sigma}\langle x,e\rangle\geq$
$\displaystyle\min_{C_{\theta,R_{2}}(e)}\langle
x,e\rangle=\sqrt{R_{2}^{2}+2R_{2}\cos\theta+1}-R_{2}$ $\displaystyle=$
$\displaystyle\cos\theta+\frac{\sin^{2}\theta}{\sqrt{R_{2}^{2}+2R_{2}\cos\theta+1}+R_{2}+\cos\theta}$
$\displaystyle=:$ $\displaystyle\cos\theta+\delta_{0}(R_{2},\theta).$

Similarly,

$\displaystyle\max_{\Sigma}\langle x,e\rangle\leq$
$\displaystyle\max_{C_{\theta,R_{1}}(e)}\langle
x,e\rangle=\frac{1+R_{1}\cos\theta}{\sqrt{R_{1}^{2}+2R_{1}\cos\theta+1}}$
$\displaystyle=:$ $\displaystyle 1-\delta_{3}(R_{1},\theta).$

Since $\partial C_{\theta,R_{1}}(e)$ is enclosed by $\partial\Sigma$ in
$\mathbb{S}^{n}$, we know that $\nu|_{\partial\Sigma}$ is strictly contained
in $B_{\frac{\pi}{2}-\beta}(-e)$ in $\mathbb{S}^{n}$, where $\beta$ is given
by

$\cos\beta=\frac{1+R_{1}\cos\theta}{\sqrt{R_{1}^{2}+2R_{1}\cos\theta+1}}.$

The convexity of $\Sigma$ implies that $\nu|_{\Sigma}$ also lies in
$B_{\frac{\pi}{2}-\beta}(-e)$ and hence

$\langle\nu,e\rangle\leq-\sin\beta=-\frac{R_{1}\sin\theta}{\sqrt{R_{1}^{2}+2R_{1}\cos\theta+1}}=-\delta_{4}(R_{1},\theta).$

∎

##  3\. Evolution equations

In this section, we derive the evolution equations along the flow (1.5).
Denote the normal speed of the flow by

$f=n\langle x+\cos\theta\nu,e\rangle-H\langle X_{e},\nu\rangle,$ (3.1)

where we choose $e\in\text{int}(\widehat{\partial\Sigma_{0}})$ as in Lemma
A.1. As in [Weng-Xia2022, §2.4], the $\theta$-capillary boundary condition
along the flow requires the tangential component $V=(\partial_{t}x)^{\top}\in
T\Sigma_{t}$ must satisfy
$V\big{|}_{\partial\Sigma_{t}}=f\cot\theta\mu+\tilde{V}$, where $\tilde{V}\in
T\left(\partial\Sigma_{t}\right)$. Hence up to a time-dependent diffeomorphism
of $\partial\mathbb{B}^{n}$, we can assume that $\tilde{V}=0$ and hence

$V\big{|}_{\partial\Sigma_{t}}=f\cot\theta\mu.$ (3.2)

The $\theta$-capillary boundary condition also ensures that (see [Weng-
Xia2022, Prop. 2.12])

$\nabla_{\mu}f=\left(\frac{1}{\sin\theta}+\cot\theta h_{\mu\mu}\right)f\ \
\text{along}\ \partial\Sigma_{t}.$ (3.3)

So we rewrite the flow (1.5) equivalently as the following form:

$\left\\{\begin{aligned} \partial_{t}x&=f\nu+V\qquad&\text{in
$\bar{\mathbb{B}}^{n}\times[0,T)$},\\\ \langle\bar{N}\circ
x,\nu\rangle&=-\cos\theta&\text{on
$\partial\bar{\mathbb{B}}^{n}\times[0,T)$},\\\
x(\cdot,0)&=x_{0}(\cdot)&\text{on
$\bar{\mathbb{B}}^{n}$},\end{aligned}\right.$ (3.4)

where $f$ is given by (3.1) and the tangential component
$V=(\partial_{t}x)^{\top}\in T\Sigma_{t}$ satisfies (3.2) on the boundary
$\partial\Sigma_{t}$. By Proposition 2.5, we have $\langle
X_{e},\nu\rangle\geq\delta_{2}>0$ on $\Sigma_{0}$. Then the short time
existence of the flow (3.4) follows from rewriting it as a strictly parabolic
scalar equation with oblique boundary condition, using Möbius coordinate
transformation (see [WW2020, §3] for details).

It is easy to check that the spherical cap $C_{\theta,r}(e)$ of radius $r$
satisfies

$\displaystyle\langle
x,e\rangle+\cos\theta\langle\nu,e\rangle=\frac{1}{r}\langle X_{e},\nu\rangle,$

and hence it is stationary along the flow (3.4). Using the spherical caps as
barriers gives the following $C^{0}$ estimate.

######  Lemma 3.1.

Assume that the initial hypersurface $\Sigma$ satisfies

$\Sigma\subset\widehat{C_{\theta,R_{2}}(e)}\backslash\widehat{C_{\theta,R_{1}}(e)},$

for some $0<R_{1}<R_{2}<\infty$. Then this property also holds for the
solution $\Sigma_{t}$ along the flow (3.4) for $t>0$.

Let $g_{ij}$, $h_{ij}$ and $H=g^{ij}h_{ij}$ be the induced metric, the second
fundamental form and the mean curvature of the flow hypersurface $\Sigma_{t}$.
We recall the following general evolution equations:

######  Proposition 3.2.

[Weng-Xia2022, Proposition 2.11] Along a general flow satisfying

$\partial_{t}x=f\nu+V,$

where $V\in T\Sigma_{t}$, we have

$\displaystyle\partial_{t}g_{ij}=$ $\displaystyle
2fh_{ij}+\nabla_{i}V_{j}+\nabla_{j}V_{i},$ (3.5)
$\displaystyle\partial_{t}h_{ij}=$
$\displaystyle-\nabla_{i}\nabla_{j}f+f\left(h^{2}\right)_{ij}+\nabla_{V}h_{ij}+h_{j}^{k}\nabla_{i}V_{k}+h_{i}^{k}\nabla_{j}V_{k},$
(3.6) $\displaystyle\partial_{t}H=$ $\displaystyle-\Delta
f-|A|^{2}f+\nabla_{V}H,$ (3.7)

where $(h^{2})_{ij}=h_{i}^{k}h_{kj}$ and $|A|^{2}=g^{ij}h_{i}^{k}h_{kj}$.

We also need the following lemma.

######  Lemma 3.3 ([Scheuer-Wang-Xia2018]).

There hold

$\nabla_{i}\langle X_{e},\nu\rangle=\langle e_{i},e\rangle\langle
x,\nu\rangle-\langle e_{i},x\rangle\langle e,\nu\rangle+h_{i}^{k}\langle
X_{e},e_{k}\rangle$ (3.8)

and

$\displaystyle\nabla_{i}\nabla_{j}\langle X_{e},\nu\rangle=$
$\displaystyle\langle X_{e},\nabla h_{ij}\rangle+\langle x,e\rangle
h_{ij}-\left(h\right)^{2}_{ij}\langle X_{e},\nu\rangle-g_{ij}\langle
e,\nu\rangle$ $\displaystyle+h_{j}^{k}\left(\langle e_{i},e\rangle\langle
x,e_{k}\rangle-\langle e_{i},x\rangle\langle e,e_{k}\rangle\right)$
$\displaystyle+h_{i}^{k}\left(\langle e_{j},e\rangle\langle
x,e_{k}\rangle-\langle e_{j},x\rangle\langle e,e_{k}\rangle\right).$ (3.9)

Now, we calculate the evolution equations along the flow (3.4).

######  Lemma 3.4.

Along the flow (3.4) with $f$ given by (3.1), we have

  1. (i)

The induced metric $g_{ij}$ evolves by

$\displaystyle\frac{\partial}{\partial t}g_{ij}=2\left(n\langle
x+\cos\theta\nu,e\rangle-H\langle
X_{e},\nu\rangle\right)h_{ij}+\nabla_{i}V_{j}+\nabla_{j}V_{i}.$ (3.10)

  2. (ii)

The mean curvature $H$ evolves by

$\displaystyle\frac{\partial}{\partial t}H=$ $\displaystyle\langle
X_{e},\nu\rangle\Delta H+\Big{\langle}HX_{e}-n\cos\theta e+V,\nabla
H\Big{\rangle}$ $\displaystyle+2\langle\nabla\langle X_{e},\nu\rangle,\nabla
H\rangle+(H^{2}-n|A|^{2})\langle x,e\rangle.$ (3.11)

and

$\displaystyle\nabla_{\mu}H=0,\quad\text{on}~{}\partial\Sigma_{t}.$ (3.12)

  3. (iii)

The second fundamental form $h=(h_{ij})$ evolves by

$\displaystyle\frac{\partial}{\partial t}h_{ij}=$ $\displaystyle\langle
X_{e},\nu\rangle\Delta h_{ij}+\Big{\langle}HX_{e}-n\cos\theta e+V,\nabla
h_{ij}\Big{\rangle}$ $\displaystyle+\nabla_{i}H\left(\langle
e_{j},e\rangle\langle x,\nu\rangle-\langle e_{j},x\rangle\langle
e,\nu\rangle+h_{j}^{k}\langle X_{e},e_{k}\rangle\right)$
$\displaystyle+\nabla_{j}H\left(\langle e_{i},e\rangle\langle
x,\nu\rangle-\langle e_{i},x\rangle\langle e,\nu\rangle+h_{i}^{k}\langle
X_{e},e_{k}\rangle\right)$ $\displaystyle+(h^{2})_{ij}\Big{(}n\langle
x,e\rangle+2n\cos\theta\langle e,\nu\rangle-3\langle X_{e},\nu\rangle
H\Big{)}$ $\displaystyle+h_{ij}\Big{(}n\langle e,\nu\rangle+\langle
X_{e},\nu\rangle|A|^{2}+H\langle x,e\rangle\Big{)}$
$\displaystyle+h_{j}^{k}\nabla_{i}V_{k}+h_{i}^{k}\nabla_{j}V_{k}-g_{ij}H\langle
e,\nu\rangle.$ (3.13)

###### Proof.

(1) Equation (3.10) follows from (3.1) and (3.5).

(2) Substituting (3.1) into (3.7), we get

$\displaystyle\partial_{t}H=$ $\displaystyle-n\Delta\langle
x,e\rangle-n\cos\theta\Delta\langle\nu,e\rangle$ $\displaystyle+\Delta
H\langle X_{e},\nu\rangle+2\langle\nabla H,\nabla\langle
X_{e},\nu\rangle\rangle+H\Delta\langle X_{e},\nu\rangle$
$\displaystyle-|A|^{2}(n\langle x+\cos\theta\nu,e\rangle-H\langle
X_{e},\nu\rangle)+\nabla_{V}H.$ (3.14)

Recall that

$\displaystyle-\Delta\langle x,e\rangle=$ $\displaystyle
H\langle\nu,e\rangle,$ (3.15) $\displaystyle\Delta\langle\nu,e\rangle=$
$\displaystyle\langle\nabla^{i}(h_{i}^{k}e_{k}),e\rangle=\langle\nabla
H,e\rangle-|A|^{2}\langle\nu,e\rangle.$ (3.16)

Substituting (3.15), (3.16) and (3.9) into (3), we obtain the equation (ii).

For the boundary condition (3.12), by [Wang-Xia2019-2, Proposition 3.3], along
$\partial\Sigma_{t}$ we have

$\displaystyle\nabla_{\mu}\langle
X_{e},\nu\rangle=\left(\frac{1}{\sin\theta}+\cot\theta
h_{\mu\mu}\right)\langle X_{e},\nu\rangle,$

and

$\displaystyle\nabla_{\mu}\langle
x+\cos\theta\nu,e\rangle=\left(\frac{1}{\sin\theta}+\cot\theta
h_{\mu\mu}\right)\langle x+\cos\theta\nu,e\rangle.$

Substituting the above two equalities into (3.3), we obtain $\nabla_{\mu}H=0$.

(3) Substituting (3.1) into (3.6), we have

$\displaystyle\frac{\partial}{\partial t}h_{ij}=$
$\displaystyle-n\nabla_{i}\nabla_{j}\langle
x,e\rangle-n\cos\theta\nabla_{i}\nabla_{j}\langle\nu,e\rangle$
$\displaystyle+\nabla_{i}\nabla_{j}H\langle
X_{e},\nu\rangle+H\nabla_{i}\nabla_{j}\langle
X_{e},\nu\rangle+\nabla_{i}H\nabla_{j}\langle
X_{e},\nu\rangle+\nabla_{j}H\nabla_{i}\langle X_{e},\nu\rangle$
$\displaystyle+\Big{(}n\langle
x,e\rangle+n\cos\theta\langle\nu,e\rangle-H\langle
X_{e},\nu\rangle\Big{)}(h^{2})_{ij}$
$\displaystyle+\nabla_{V}h_{ij}+h_{j}^{k}\nabla_{i}V_{k}+h_{i}^{k}\nabla_{j}V_{k}.$

For the first two terms, we have

$\displaystyle-\nabla_{i}\nabla_{j}\langle x,e\rangle=$ $\displaystyle
h_{ij}\langle\nu,e\rangle,$
$\displaystyle\nabla_{i}\nabla_{j}\langle\nu,e\rangle=$
$\displaystyle\nabla_{i}\langle h_{j}^{k}e_{k},e\rangle=\langle\nabla
h_{ij},e\rangle-(h^{2})_{ij}\langle\nu,e\rangle.$

By the Simons’ identity

$\displaystyle\nabla_{i}\nabla_{j}H=\Delta
h_{ij}+|A|^{2}h_{ij}-H(h^{2})_{ij},$

and the equations (3.8) - (3.9), we get

$\displaystyle\frac{\partial}{\partial t}h_{ij}=$ $\displaystyle
nh_{ij}\langle e,\nu\rangle+n\cos\theta(h^{2})_{ij}\langle
e,\nu\rangle-n\cos\theta\langle\nabla h_{ij},e\rangle$
$\displaystyle+\nabla_{i}H(\langle e_{j},e\rangle\langle x,\nu\rangle-\langle
e_{j},x\rangle\langle e,\nu\rangle+h_{j}^{k}\langle X_{e},e_{k}\rangle)$
$\displaystyle+\nabla_{j}H(\langle e_{i},e\rangle\langle x,\nu\rangle-\langle
e_{i},x\rangle\langle e,\nu\rangle+h_{i}^{k}\langle X_{e},e_{k}\rangle)$
$\displaystyle+\langle X_{e},\nu\rangle(\Delta
h_{ij}+|A|^{2}h_{ij}-H(h^{2})_{ij})$ $\displaystyle+H\left(\langle
X_{e},\nabla h_{ij}\rangle+h_{ij}\langle x,e\rangle-(h^{2})_{ij}\langle
X_{e},\nu\rangle-g_{ij}\langle e,\nu\rangle\right)$
$\displaystyle+Hh_{j}^{k}\left(\langle e_{i},e\rangle\langle
x,e_{k}\rangle-\langle e_{i},x\rangle\langle e,e_{k}\rangle\right)$
$\displaystyle+Hh_{i}^{k}\left(\langle e_{j},e\rangle\langle
x,e_{k}\rangle-\langle e_{j},x\rangle\langle e,e_{k}\rangle\right)$
$\displaystyle+(n\langle x,e\rangle+n\cos\theta\langle\nu,e\rangle-H\langle
X_{e},\nu\rangle)(h^{2})_{ij}$
$\displaystyle+\nabla_{V}h_{ij}+h_{j}^{k}\nabla_{i}V_{k}+h_{i}^{k}\nabla_{j}V_{k}.$

Then (iii) follows by rearranging the terms. ∎

##  4\. Preserving of convexity

We first recall the following tensor maximum principle which was recently
developed by the authors [HWYZ2022] and can be viewed as a generalization of
the tensor maximum principles of Hamilton [Hamilton1982], Stahl [Stahl1996-2]
and Andrews [And2007].

######  Theorem 4.1 ([HWYZ2022]).

Let $\Sigma$ be a smooth compact manifold with boundary $\partial\Sigma$ and
$\mu$ be the outward pointing unit normal vector field of $\partial\Sigma$ in
$\Sigma$. Assume that $S_{ij}$ is a smooth time-varying symmetric tensor field
on $\Sigma$ satisfying

$\frac{\partial}{\partial
t}S_{ij}=a^{k\ell}\nabla_{k}\nabla_{\ell}S_{ij}+b^{k}\nabla_{k}S_{ij}+N_{ij}$

on $\Sigma\times[0,T]$, where the coefficients $a^{k\ell}$ and $b^{k}$ are
smooth, $\nabla$ is a (possibly time-dependent) smooth symmetric connection,
and $(a^{k\ell})$ is positively definite everywhere. Suppose that

$N_{ij}\xi^{i}\xi^{j}+\sup_{\Gamma}2a^{k\ell}\left(2\Gamma^{p}_{k}\nabla_{\ell}S_{ip}\xi^{i}-\Gamma_{k}^{p}\Gamma_{\ell}^{q}S_{pq}\right)\geq
0\quad\mathrm{on}~{}\Sigma\times(0,T],$ (4.1)
$\left(\nabla_{\mu}S_{ij}\right)\xi^{i}\xi^{j}\geq
0\qquad\mathrm{on}~{}\partial\Sigma\times(0,T]$ (4.2)

whenever $S_{ij}\geq 0$ and $S_{ij}\xi^{i}=0$. If $S_{ij}\geq 0$ everywhere on
$\Sigma\times\\{0\\}$, then it remains so on $\Sigma\times[0,T]$.

We also need the following useful lemma.

######  Lemma 4.2 ([HWYZ2022]).

Let $\Sigma$ be a smooth compact manifold with boundary $\partial\Sigma$ and
$\mu$ be the outward pointing unit normal vector field of $\partial\Sigma$ in
$\Sigma$. Suppose that $S_{ij}$ is a smooth symmetric tensor satisfying
$S_{ij}\geq 0$ on $\Sigma$ and

$(\nabla_{\mu}S_{ij})\xi^{i}\xi^{j}\geq 0,\qquad\mathrm{on}~{}\partial\Sigma$
(4.3)

whenever $S_{ij}\xi^{j}=0$ for some tangent vector $\xi$. Define a function

$Z(p,\xi)=S(p)(\xi,\xi)$

where $p\in\Sigma$ and $\xi\in T_{p}\Sigma$. If $Z$ attains its minimum at
some $(p,\xi)\in T\Sigma$, then $DZ(p,\xi)=0$ and $D^{2}Z(p,\xi)$ is non-
negative definite.

Theorem 4.1 has been used in [HWYZ2022] to prove the preservation of convexity
along the locally constrained inverse curvature flows for capillary
hypersurfaces in the half-space. In the following ,we shall apply Theorem 4.1
to show that the convexity is preserved along the flow (3.4).

######  Theorem 4.3.

Assume that the initial hypersurface $\Sigma\subset\bar{\mathbb{B}}^{n+1}$ is
strictly convex with $\theta$-capillary boundary
$\partial\Sigma\subset\mathbb{S}^{n}$ such that
$\Sigma\subset\widehat{C_{\theta,R_{2}}(e)}\backslash\widehat{C_{\theta,R_{1}}(e)}$.
Let $\Sigma_{t},t\in[0,T),$ be the smooth solution of the flow (3.4) starting
from $\Sigma$. Then $\Sigma_{t}$ is strictly convex and there exists a
constant $\varepsilon>0$ depending only on $\Sigma$ such that the principal
curvatures $\kappa_{i}$ of $\Sigma_{t}$ satisfy

$\kappa_{i}\geq\varepsilon,\qquad i=1,\cdots,n$

for all $t\in[0,T)$.

###### Proof.

We consider the tensor

$S_{ij}=h_{ij}-\varepsilon g_{ij},$

where $\varepsilon>0$ is a small constant to be determined later. Since the
initial hypersurface $\Sigma$ is strictly convex, we can first choose
$\varepsilon>0$ small enough such that $S_{ij}\geq 0$ on $\Sigma$. Combining
the equations (3.10), (iii), we deduce that

$\displaystyle\frac{\partial}{\partial t}S_{ij}=$
$\displaystyle\frac{\partial}{\partial
t}h_{ij}-\varepsilon\frac{\partial}{\partial t}g_{ij}$ $\displaystyle=$
$\displaystyle\langle X_{e},\nu\rangle\Delta S_{ij}+\langle HX_{e}+n\cos\theta
e+V,\nabla S_{ij}\rangle+S_{j}^{k}\nabla_{i}V_{k}+S_{i}^{k}\nabla_{j}V_{k}$
$\displaystyle+\nabla_{i}H\left(\langle e_{j},e\rangle\langle
x,\nu\rangle-\langle e_{j},x\rangle\langle e,\nu\rangle+h_{j}^{k}\langle
X_{e},e_{k}\rangle\right)$ $\displaystyle+\nabla_{j}H\left(\langle
e_{i},e\rangle\langle x,\nu\rangle-\langle e_{i},x\rangle\langle
e,\nu\rangle+h_{i}^{k}\langle X_{e},e_{k}\rangle\right)$
$\displaystyle+((S^{2})_{ij}+2\varepsilon
S_{ij}+\varepsilon^{2}g_{ij})\left(n\langle
x,e\rangle+2n\cos\theta\langle\nu,e\rangle-3\langle X_{e},\nu\rangle H\right)$
$\displaystyle+(S_{ij}+\varepsilon g_{ij})(n\langle\nu,e\rangle+\langle
X_{e},\nu\rangle|A|^{2}+H\langle x,e\rangle)-g_{ij}H\langle\nu,e\rangle$
$\displaystyle-2\varepsilon(S_{ij}+\varepsilon g_{ij})(\langle
x+\cos\theta\nu,e\rangle-\langle X_{e},\nu\rangle H).$ (4.4)

We first check that the boundary condition (4.2) is satisfied whenever
$S_{ij}\geq 0$ and $S_{ij}\xi^{j}=0$ at some point
$p_{0}\in\partial\Sigma_{t_{0}}$ for some vector $\xi\in
T_{p_{0}}\Sigma_{t_{0}}$. Let $\\{e_{\alpha}\\}_{\alpha=1}^{n-1}$ be a local
orthonormal frame of $\partial\Sigma_{t}\subset\Sigma_{t}$ so that
$\\{\mu\\}\cup\\{e_{\alpha}\\}_{\alpha}$ forms an orthonormal frame of
$\Sigma_{t}$. We have either $\xi=\mu$ or $\xi=e_{\alpha}$ for some $\alpha$
on $\partial\Sigma_{t_{0}}$. By Proposition 2.1, we have:

  1. (i)

If $\xi=e_{\alpha}$, then $h_{\mu\mu}\geq h_{\alpha\alpha}=\varepsilon>0$ and

$\displaystyle\left(\nabla_{\mu}S_{ij}\right)\xi^{i}\xi^{j}=$
$\displaystyle\nabla_{\mu}h_{\alpha\alpha}=\tilde{h}_{\alpha\alpha}\left(h_{\mu\mu}-h_{\alpha\alpha}\right)$
$\displaystyle=$ $\displaystyle\left(\frac{1}{\sin\theta}+\cot\theta
h_{\alpha\alpha}\right)(h_{\mu\mu}-\varepsilon)\geq 0.$

  2. (ii)

If $\xi=\mu$, then $h_{\alpha\alpha}\geq h_{\mu\mu}=\varepsilon>0$ for all
$\alpha=1,\cdots,n-1$. By $\nabla_{\mu}H=0$, we get

$\displaystyle 0=$
$\displaystyle\nabla_{\mu}h_{\mu\mu}+\sum_{\alpha}\nabla_{\mu}h_{\alpha\alpha}=\nabla_{\mu}h_{\mu\mu}+\sum_{\alpha}\left(h_{\mu\mu}-h_{\alpha\alpha}\right)\tilde{h}_{\alpha\alpha}$
$\displaystyle=$
$\displaystyle\nabla_{\mu}h_{\mu\mu}+\sum_{\alpha}\left(\varepsilon-
h_{\alpha\alpha}\right)\left(\frac{1}{\sin\theta}+\cot\theta
h_{\alpha\alpha}\right),$

and hence

$\left(\nabla_{\mu}S_{ij}\right)\xi^{i}\xi^{j}=\nabla_{\mu}h_{\mu\mu}=\sum_{\alpha}(h_{\alpha\alpha}-\varepsilon)\left(\frac{1}{\sin\theta}+\cot\theta
h_{\alpha\alpha}\right)\geq 0.$

Therefore, in both cases, the boundary condition (4.2) is satisfied.

We next check the condition (4.1) whenever $S_{ij}\geq 0$ and
$S_{ij}\xi^{i}=0$ at a point $(p_{0},t_{0})$. We choose an orthonormal frame
$\\{e_{1},\cdots,e_{n}\\}$ around $p_{0}$ such that $h_{ij}$ is diagonal at
$(p_{0},t_{0})$ with principal curvatures
$\kappa_{1}\leq\kappa_{2}\leq\cdots\leq\kappa_{n}$ in the increasing order and
$e_{i}$ corresponds to the principal direction of $\kappa_{i}$, and the null
vector $\xi=e_{1}$. The lower order terms in (4) involving $S_{ij}$ and
$(S^{2})_{ij}$ can be ignored due to the null vector condition. To apply
Theorem 4.1, we need to check that

$\displaystyle Q_{1}:=$ $\displaystyle 2\nabla_{1}H\Big{(}\langle
e_{1},e\rangle\langle x,\nu\rangle-\langle e_{1},x\rangle\langle
e,\nu\rangle+\varepsilon\langle X_{e},e_{1}\rangle\Big{)}$
$\displaystyle-(H-n\varepsilon)\langle
e,\nu\rangle+\varepsilon\Big{(}(|A|^{2}-\varepsilon H)\langle
X_{e},\nu\rangle+(H-n\varepsilon)\langle x,e\rangle\Big{)}$
$\displaystyle+2\langle
X_{e},\nu\rangle\sup_{\Gamma}g^{k\ell}(2\Gamma_{k}^{p}\nabla_{\ell}S_{1p}-\Gamma_{k}^{p}\Gamma_{\ell}^{q}S_{pq})\geq
0.$ (4.5)

There are two cases:

  1. (1)

$\varepsilon=\kappa_{1}=\cdots=\kappa_{n}$ at $(p_{0},t_{0})$;

  2. (2)

there exists an $1\leq m<n$ such that
$\varepsilon=\kappa_{1}=\cdots=\kappa_{m}<\kappa_{m+1}\leq\cdots\leq\kappa_{n}$
at $(p_{0},t_{0})$.

We consider the two cases separately.

For case (1), we have $H=n\varepsilon$ and $|A|^{2}-\varepsilon H=0$. So the
second line of (4) vanishes. Moreover, by Lemma 4.2,
$\nabla_{k}S_{\ell\ell}=\nabla_{k}h_{\ell\ell}=0$ for all $1\leq k,\ell\leq
n$. This implies that $\nabla_{1}H=0$. Then $Q_{1}\geq 0$ by simply choosing
$\Gamma_{k}^{p}=0$ for all $k,p=1,\cdots,n$.

For case (2), we have $S_{11}=\cdots S_{mm}=\kappa_{1}-\varepsilon=0$ at
$(x_{0},t_{0})$ and $S_{\ell\ell}>0$ for $\ell=m+1,\cdots,n$. Then

$\displaystyle|A|^{2}-\varepsilon H>0,\quad H-n\varepsilon>0.$

Again by Lemma 4.2, the minimality of $S_{11},\cdots,S_{mm}$ implies that
$\nabla_{k}S_{\ell\ell}=0$ for all $k=1,\cdots,n$ and $\ell=1,\cdots,m$. Then
the last term in (4) can be explicitly computed as follows

$\displaystyle\sup_{\Gamma}g^{k\ell}(2\Gamma_{k}^{p}\nabla_{\ell}S_{1p}-\Gamma_{k}^{p}\Gamma_{\ell}^{q}S_{pq})$
$\displaystyle=$
$\displaystyle\sup_{\Gamma}\sum_{k=1}^{n}\left(\sum_{\ell=1}^{m}2\Gamma^{\ell}_{k}\nabla_{k}S_{1\ell}+\sum_{\ell=m+1}^{n}(2\Gamma^{\ell}_{k}\nabla_{k}S_{1\ell}-(\Gamma_{k}^{\ell})^{2}S_{\ell\ell})\right)$
$\displaystyle=$
$\displaystyle\sup_{\Gamma}\sum_{k=1}^{n}\left(\sum_{\ell=1}^{m}2\Gamma^{\ell}_{k}\nabla_{k}h_{1\ell}+\sum_{\ell=m+1}^{n}\left(\frac{|\nabla_{k}S_{1\ell}|^{2}}{S_{\ell\ell}}-\left(\Gamma_{k}^{\ell}-\frac{\nabla_{k}S_{1\ell}}{S_{\ell\ell}}\right)^{2}S_{\ell\ell}\right)\right)$
$\displaystyle\geq$
$\displaystyle\sum_{k=1}^{n}\sum_{\ell=m+1}^{n}\frac{|\nabla_{k}S_{1\ell}|^{2}}{S_{\ell\ell}}$
$\displaystyle\geq$
$\displaystyle\sum_{\ell=m+1}^{n}\frac{|\nabla_{1}h_{\ell\ell}|^{2}}{\kappa_{\ell}-\varepsilon},$
(4.6)

where in the first inequality we have chosen
$\Gamma_{k}^{\ell}=\frac{\nabla_{k}S_{1\ell}}{S_{\ell\ell}}$ for
$\ell=m+1,\cdots,n$ and $\Gamma_{k}^{\ell}=0$ for $\ell=1,\cdots,m$, and in
the last inequality we used the Codazzi equation
$\nabla_{k}h_{1\ell}=\nabla_{1}h_{k\ell}$ and discarded the terms for
$k\neq\ell$.

For the first term of (4), we calculate that

$\displaystyle\Big{\langle}e_{1},e\langle x,\nu\rangle-x\langle
e,\nu\rangle+\varepsilon X_{e}\Big{\rangle}^{2}$ $\displaystyle\leq$
$\displaystyle|e\langle x,\nu\rangle-x\langle e,\nu\rangle+\varepsilon
X_{e}|^{2}-\langle\nu,e\langle x,\nu\rangle-x\langle e,\nu\rangle+\varepsilon
X_{e}\rangle^{2}$ $\displaystyle=$ $\displaystyle\langle
x,\nu\rangle^{2}+|x|^{2}\langle e,\nu\rangle^{2}-2\langle x,\nu\rangle\langle
e,\nu\rangle\langle x,e\rangle$ $\displaystyle+2\varepsilon\langle
X_{e},e\langle x,\nu\rangle-x\langle
e,\nu\rangle\rangle+\varepsilon^{2}(|X_{e}|^{2}-\langle X_{e},\nu\rangle^{2})$
$\displaystyle=$ $\displaystyle-2\langle X_{e},\nu\rangle\langle
e,\nu\rangle+\langle x,\nu\rangle^{2}-\langle e,\nu\rangle^{2}$
$\displaystyle+2\varepsilon\langle X_{e},e\langle x,\nu\rangle-x\langle
e,\nu\rangle\rangle+\varepsilon^{2}(|X_{e}|^{2}-\langle
X_{e},\nu\rangle^{2}).$ (4.7)

By (2.9) and Proposition 2.5, we have

$\displaystyle\langle x,\nu\rangle^{2}-\langle e,\nu\rangle^{2}=\langle
x+e,\nu\rangle\langle x-e,\nu\rangle\leq-\delta_{1}\delta_{4},$ (4.8)

where $\delta_{1},\delta_{4}$ are constants in Proposition 2.5. Since

$\displaystyle\langle X_{e},\nu\rangle\leq|X_{e}|\leq 2,\quad|e\langle
x,\nu\rangle-x\langle e,\nu\rangle|\leq 2,$ (4.9)

it follows from (4), (4.8) and (4.9) that

$\displaystyle\langle e_{1},e\langle x,\nu\rangle-x\langle
e,\nu\rangle+\varepsilon X_{e}\rangle^{2}\leq-2\langle X_{e},\nu\rangle\langle
e,\nu\rangle-\delta_{1}\delta_{4}+4\varepsilon^{2}+8\varepsilon.$

Note that $\delta_{1},\delta_{4}$ depend only on $R_{1},R_{2}$ and $\theta$.
By choosing $\varepsilon>0$ small enough such that
$4\varepsilon^{2}+8\varepsilon\leq\delta_{1}\delta_{4}$, we have

$\langle e_{1},e\langle x,\nu\rangle-x\langle e,\nu\rangle+\varepsilon
X_{e}\rangle^{2}\leq-2\langle X_{e},\nu\rangle\langle e,\nu\rangle.$ (4.10)

Then using (4) and (4.10), we obtain the following estimate on $Q_{1}$:

$\displaystyle Q_{1}\geq$
$\displaystyle-2|\sum_{\ell=m+1}^{n}\nabla_{1}h_{\ell\ell}|\sqrt{-2\langle
X_{e},\nu\rangle\langle
e,\nu\rangle}-\sum_{\ell=m+1}^{n}(\kappa_{\ell}-\varepsilon)\langle
e,\nu\rangle$ $\displaystyle\quad+2\langle
X_{e},\nu\rangle\sum_{\ell=m+1}^{n}\frac{|\nabla_{1}h_{\ell\ell}|^{2}}{\kappa_{\ell}-\varepsilon}$
$\displaystyle\geq$
$\displaystyle\sum_{\ell=m+1}^{n}\left(\sqrt{\kappa_{\ell}-\varepsilon}\sqrt{-\langle
e,\nu\rangle}-\sqrt{2\langle
X_{e},\nu\rangle}\frac{|\nabla_{1}h_{\ell\ell}|}{\sqrt{\kappa_{\ell}-\varepsilon}}\right)^{2}$
$\displaystyle\geq$ $\displaystyle 0.$ (4.11)

In summary, in both two cases, we have ${Q}_{1}\geq 0$ and thus the condition
(4.1) in Theorem 4.1 is satisfied. Therefore, we conclude that $S_{ij}\geq 0$
is preserved for $t>0$. This implies that $h_{ij}\geq\varepsilon g_{ij}>0$ for
$t\in[0,T)$. ∎

##  5\. Proofs of Theorem 1.1 and Theorem 1.2

In this section, we prove the long time existence and smooth convergence of
the flow (1.5) for strictly convex hypersurfaces in the unit ball with
$\theta$-capillary boundary. As an application, we prove a family of new
inequalities for quermassintegrals $W_{k,\theta}(\widehat{\Sigma})$.

###  5.1. Long-time existence

Let $\Sigma\subset\bar{\mathbb{B}}^{n+1}(n\geq 2)$ be a smooth properly
embedded, strictly convex hypersurface with $\theta$-capillary boundary
($\theta\in(0,\frac{\pi}{2}]$), given by an embedding:
$x:\bar{\mathbb{B}}^{n}\to\Sigma\subset\bar{\mathbb{B}}^{n+1}$. By Theorem
4.3, the solution $\Sigma_{t}$ of the flow (3.4) starting from $\Sigma$
remains to be strictly convex for $t>0$. Proposition 2.5 implies that $\langle
X_{e},\nu\rangle\geq\delta_{2}>0$ and hence $\Sigma_{t}$ is star-shaped for
$t\geq 0$. Then we can reduce the flow (3.4) to a scalar parabolic equation
with oblique boundary condition as in [WW2020, Wang-Xia2019]

We briefly review the transformation of the flow (1.5) to a scalar parabolic
equation on the hemisphere $\bar{\mathbb{S}}_{+}^{n}$, and refer the readers
to [WW2020, Wang-Xia2019] for more details. Assume that $e$ is the $(n+1)$-th
coordinate vector $e_{n+1}$. Consider the following Möbius transformation:

$\displaystyle\varphi:~{}\bar{\mathbb{B}}^{n+1}\quad$
$\displaystyle\longrightarrow\quad\bar{\mathbb{R}}_{+}^{n+1},$
$\displaystyle(x^{\prime},x_{n+1})$
$\displaystyle\longmapsto\frac{2(x^{\prime},0)+(1-|x^{\prime}|^{2}-x_{n+1}^{2})e}{|x^{\prime}|^{2}+(x_{n+1}-1)^{2}}:=(y^{\prime},y_{n+1})=\tilde{y},$
(5.1)

where $x=(x^{\prime},x_{n+1})$ with
$x^{\prime}=(x_{1},\cdots,x_{n})\in\mathbb{R}^{n}$ and $x_{n+1}=\langle
x,e\rangle$. Then $\varphi$ maps $\mathbb{S}^{n}=\partial\mathbb{B}^{n+1}$ to
$\partial\mathbb{R}_{+}^{n+1}:=\\{(y^{\prime},y_{n+1})\in\mathbb{R}^{n+1}:y_{n+1}=0\\}$
and

$\displaystyle\varphi^{\ast}(\delta_{\mathbb{R}^{n+1}_{+}})=\frac{4}{(|x^{\prime}|^{2}+(x_{n+1}-1)^{2})^{2}}\delta_{\mathbb{B}^{n+1}}:=e^{-2w}\delta_{\mathbb{B}^{n+1}},$

which implies that $\varphi$ is a conformal transformation from
$(\bar{\mathbb{B}}^{n+1},\delta_{\mathbb{B}^{n+1}})$ to
$(\bar{\mathbb{R}}_{+}^{n+1},\delta_{\mathbb{R}_{+}^{n+1}})$. Then a properly
embedded hypersurface $\Sigma_{t}=x(\bar{\mathbb{B}}^{n},t)$ in
$(\bar{\mathbb{B}}^{n+1},\delta_{\mathbb{B}^{n+1}})$ can be identified with
$\widetilde{\Sigma}_{t}:=\tilde{y}(\bar{\mathbb{B}}^{n},t)$ in
$(\bar{\mathbb{R}}_{+}^{n+1},(\varphi^{-1})^{\ast}\delta_{\mathbb{R}_{+}^{n+1}})$,
where $\tilde{y}=\varphi\circ x$.

Note that $X_{e}$ is a conformal vector field such that
$\varphi_{\ast}(X_{e})=-\tilde{y}$. For a hypersurface
$\Sigma\subset\bar{\mathbb{B}}^{n+1}$ with capillary boundary
$\partial\Sigma\subset\mathbb{S}^{n}$, one has

$\displaystyle e^{-2w}\langle
X_{e},\nu\rangle=\langle\varphi_{\ast}(X_{e}),\varphi_{\ast}(\nu)\rangle=|\varphi_{\ast}(\nu)|\langle\tilde{y},\tilde{\nu}\rangle,$

where $|\varphi_{\ast}(\nu)|=\frac{1}{2}(|y^{\prime}|^{2}+(y_{n+1}+1)^{2})$
and $\tilde{\nu}:=-\frac{\varphi_{\ast}(\nu)}{|\varphi_{\ast}(\nu)|}$. Hence,
the hypersurface $\varphi(\Sigma)$ is star-shaped in $\mathbb{R}_{+}^{n+1}$
with respect to the origin, i.e., $\langle\tilde{y},\tilde{\nu}\rangle>0$ on
$\varphi(\Sigma)$, if and only if $\langle X_{e},\nu\rangle>0$ on $\Sigma$. In
particular, since $\langle X_{e},\nu\rangle>0$ on $\Sigma_{t}$ by Proposition
2.5, the hypersurface $\widetilde{\Sigma}_{t}$ in
$(\bar{\mathbb{R}}_{+}^{n+1},(\varphi^{-1})^{\ast}\delta_{\mathbb{R}_{+}^{n+1}})$
can be written as a radial graph over $\bar{\mathbb{S}}^{n}_{+}$.

In $\mathbb{R}_{+}^{n+1}$, we use the polar coordinate
$\tilde{y}=(\rho,z)\in[0,\infty)\times\bar{\mathbb{S}}_{+}^{n}$, where $\rho$
is the distance from $\tilde{y}$ to the origin and we write
$z=(\beta,\xi)\in[0,\frac{\pi}{2}]\times\mathbb{S}^{n-1}$ for the spherical
polar coordinate of $z\in\mathbb{S}^{n}$. Then

$\displaystyle\left\\{\begin{aligned}
\rho^{2}=&|y^{\prime}|^{2}+y_{n+1}^{2},\\\
y_{n+1}=&\rho\cos\beta,\quad|y^{\prime}|=\rho\sin\beta.\end{aligned}\right.$
(5.2)

Let $u:=\log\rho$ and $v:=\sqrt{1+|\nabla^{\mathbb{S}}u|^{2}}$, where
$\nabla^{\mathbb{S}}$ is the Levi-Civita connection on $\mathbb{S}^{n}_{+}$
with respect to the round metric $\sigma$. We have

$\displaystyle\langle
X_{e},\nu\rangle=e^{w}\frac{\rho}{v}=\frac{2\rho}{\sqrt{1+|\nabla^{\mathbb{S}}u|^{2}}(\rho^{2}+2\rho\cos\beta+1)}.$
(5.3)

Furthermore, up to a time-dependent tangential diffeomorphism, one can rewrite
the flow (1.5) equivalently as the following scalar parabolic equation on
$\bar{\mathbb{S}}_{+}^{n}$:

$\left\\{\begin{aligned} \frac{\partial}{\partial
t}u&=F(\nabla^{2}_{\mathbb{S}}u,\nabla^{\mathbb{S}}u,\rho,\beta)\qquad&\text{in}\quad\mathbb{S}_{+}^{n}\times[0,T),\\\
\nabla^{\mathbb{S}}_{\partial_{\beta}}u&=-\cos\theta\sqrt{1+|\nabla^{\mathbb{S}}u|^{2}}\qquad&\text{on}\quad\partial\mathbb{S}_{+}^{n}\times[0,T),\\\
u(\cdot,0)&=u_{0}(\cdot)\qquad&\text{on}\quad\mathbb{S}_{+}^{n},\end{aligned}\right.$
(5.4)

where $u_{0}=\log\rho_{0}$, $\rho_{0}$ is radial function of the initial
hypersurface $\varphi(\Sigma_{0})$, $\partial_{\beta}$ is the unit outward
normal of $\partial\mathbb{S}_{+}^{n}$ on $\bar{\mathbb{S}}_{+}^{n}$ and

$\displaystyle F(\nabla^{2}_{\mathbb{S}}u,\nabla^{\mathbb{S}}u,\rho,\beta):=$
$\displaystyle\operatorname{div}_{\mathbb{S}^{n}_{+}}\left(\frac{\nabla^{\mathbb{S}}u}{\rho
e^{w}v}\right)-\frac{n+1}{v}\sigma\left(\nabla^{\mathbb{S}}u,\nabla^{\mathbb{S}}(\frac{1}{\rho
e^{w}})\right)$
$\displaystyle-\frac{n\cos\theta}{2}\cdot\frac{\rho^{2}-1}{\rho}\sin\beta\sigma(\nabla^{\mathbb{S}}u,\partial_{\beta})$
$\displaystyle+\frac{n\cos\theta}{2}\cdot\frac{\rho^{2}\cos\beta+2\rho+\cos\beta}{\rho}.$

As $\theta\in(0,\frac{\pi}{2}]$, we have $\cos\theta\in[0,1)$. Then we have a
uniformly oblique boundary condition in (5.4).

The barrier estimate in Lemma 3.1 and the convexity of $\Sigma_{t}$ imply the
uniform $C^{1}$ estimate of $\Sigma_{t}$. In fact, under the transformation
(5.1), using (5.2) we have

$\displaystyle\rho^{2}=$
$\displaystyle\frac{|x^{\prime}|^{4}+(1-x_{n+1}^{2})^{2}+2(1+x_{n+1}^{2})|x^{\prime}|^{2}}{|x^{\prime}|^{4}+(1-x_{n+1})^{4}+2(1-x_{n+1})^{2}|x^{\prime}|^{2}}$
$\displaystyle=$ $\displaystyle
1+\frac{4x_{n+1}}{|x^{\prime}|^{2}+(1-x_{n+1})^{2}}.$

By Proposition 2.5, we have $x_{n+1}=\langle
x,e\rangle\geq\cos\theta+\delta_{0}$ and $\langle
x-e,\nu\rangle\geq\delta_{1}>0$. Then

$\displaystyle\rho^{2}\geq$ $\displaystyle 1+2(\cos\theta+\delta_{0})\geq
1+2\delta_{0},$ $\displaystyle\rho^{2}\leq$ $\displaystyle
1+\frac{4}{|x-e|^{2}}\leq 1+\frac{4}{|\langle x-e,\nu\rangle|^{2}}\leq
1+\frac{4}{\delta_{1}^{2}}.$

Since $u=\log\rho$, we obtain the uniform $C^{0}$ estimate of $u$, i.e.,

$0<c_{1}\leq u\leq c_{2}<\infty$

for some constants $c_{1},c_{2}$ depending on $\Sigma_{0}$. For the $C^{1}$
estimate, using $\langle X_{e},\nu\rangle\geq\delta_{2}>0$ on $\Sigma_{t}$ and
(5.3) we obtain that

$\displaystyle|\nabla^{\mathbb{S}}u|\leq$
$\displaystyle\sqrt{1+|\nabla^{\mathbb{S}}u|^{2}}$ $\displaystyle\leq$
$\displaystyle\frac{2\rho}{\delta_{2}\sqrt{\rho^{2}+2\rho\cos\beta+1}}$
$\displaystyle\leq$
$\displaystyle\frac{2}{\delta_{2}}\sqrt{1+\frac{4}{\delta_{1}^{2}}}=:c_{3},$

where $c_{3}$ is a positive constant depending on $\Sigma_{0}$. It follows
that the scalar flow equation (5.4) is a quasilinear parabolic equation with
uniform oblique boundary condition. The higher order estimates of $u$ follows
from the classical parabolic theory for quasi-linear parabolic equations
[Ural1991, Theorem 7.4] (see also [Lieb96, §13]). As a consequence, the
solution $u$ of (5.4) exists for all time $t\in[0,\infty)$. Equivalently, we
have

######  Proposition 5.1.

Let $\Sigma\subset\bar{\mathbb{B}}^{n+1}(n\geq 2)$ be a properly embedded,
strictly convex hypersurface with $\theta$-capillary boundary
$\partial\Sigma\subset\mathbb{S}^{n}$ ($\theta\in(0,\frac{\pi}{2}]$), given by
an embedding: $x:\bar{\mathbb{B}}^{n}\to\Sigma\subset\bar{\mathbb{B}}^{n+1}$.
Then the solution $\Sigma_{t}$ of the flow (1.5) starting from $\Sigma$ has
uniform $C^{k}$ regularity estimates for all $k\geq 0$ and exists for all time
$t\in[0,\infty)$.

###  5.2. Convergence to a spherical cap

To prove the smooth convergence to the spherical cap, we explore the
monotonicity of the flow (3.4). By the case $k=1$ of (2.2), we have

$\displaystyle\frac{d}{dt}{W_{1,\theta}(\widehat{\Sigma_{t}})}=\frac{n^{2}}{n+1}\int_{\Sigma_{t}}(H_{2}-H_{1}^{2})\langle
X_{e},\nu\rangle dA_{t}\leq 0.$ (5.5)

The $C^{0}$ estimate of $\Sigma_{t}$ implies that
${W_{1,\theta}(\widehat{\Sigma_{t}})}$ is uniformly bounded from below. Then
the long-time existence and uniform $C^{\infty}$-estimates imply that

$\int_{\Sigma_{t}}{(H_{2}-H_{1}^{2})\langle X_{e},\nu\rangle dA_{t}}\to
0,\,\quad\text{as $t\to\infty$}.$

Since $\langle X_{e},\nu\rangle\geq\delta_{2}>0$ along the flow, any limit of
$\Sigma_{t}$ is totally umbilical and hence must be a spherical cap.

The limit spherical cap is uniquely determined and is independent of the
subsequence of times. In fact, for any sequence of times $t_{j}\to\infty$ such
that $\Sigma_{t_{j}}$ converges smoothly to a $\theta$-capillary spherical cap
$C_{\theta,r_{\infty}}(e_{\infty})$, the radius $r_{\infty}$ is uniquely
determined by the fact
$W_{0,\theta}(\widehat{C_{\theta,r_{\infty}}}(e_{\infty}))=W_{0,\theta}(\widehat{\Sigma})$.
Furthermore, we must have $e_{\infty}=e$, by using a similar argument as in
[Scheuer-Wang-Xia2018, Weng-Xia2022] for the locally constrained inverse
curvature flow (1.6). The idea is simply that any spherical cap
$C_{\theta,r_{\infty}}(e_{\infty})$ with the center $e_{\infty}\neq e$ is not
stationary and must evolve towards $e$ along the flow (3.4). We refer the
readers to [Scheuer-Wang-Xia2018, Weng-Xia2022] for the similar justification
of this fact along the flow (1.6).

Therefore, we conclude that along the flow (3.4), the solution $\Sigma_{t}$
converges smoothly to the unique spherical cap $C_{\theta,r_{\infty}}(e)$.
This completes the proof of Theorem 1.1.

###  5.3. Proof of Theorem 1.2

Firstly, we assume that $\Sigma$ is strictly convex, then by Proposition 2.5,
there exists a constant vector $e\in\mathbb{S}^{n}$ such that $\langle
X_{e},\nu\rangle>0$ holds everywhere on $\Sigma$. Start the flow (3.4) from
$\Sigma$. By Theorem 1.1, the solution $\Sigma_{t}$ is strictly convex and
converges smoothly to a spherical cap $C_{\theta,r_{\infty}}(e)$ as
$t\rightarrow\infty$. By the monotonicity in Proposition 2.3, we have

$\displaystyle W_{0,\theta}(\widehat{\Sigma})=$ $\displaystyle
W_{0,\theta}(\widehat{C_{\theta,r_{\infty}}}(e))=f_{0}(r_{\infty})$
$\displaystyle W_{k,\theta}(\widehat{\Sigma})\geq$ $\displaystyle
W_{k,\theta}(\widehat{C_{\theta,r_{\infty}}}(e))=f_{k}(r_{\infty}).$

Since $f_{k}$ is strictly increasing, we can rewrite the above two equations
as

$\displaystyle W_{k,\theta}(\widehat{\Sigma})\geq f_{k}\circ
f_{0}^{-1}\left(W_{0,\theta}(\widehat{\Sigma})\right).$ (5.6)

If equality holds in (5.6), then all $\Sigma_{t}$ are spherical caps with
$\theta$-capillary boundary for $t>0$ and in particular the initial
hypersurface $\Sigma$ is also a spherical cap with $\theta$-capillary
boundary.

When $\Sigma$ is convex, we may assume that it is not a flat ball, otherwise
the statement is trivially true. By Theorem A.3, we can approximate $\Sigma$
by a family of strictly convex hypersurfaces $\Sigma_{\varepsilon}$ as
$\varepsilon\to 0$. Then the inequality (1.8) follows by approximation. The
equality characterization can be proved by using an argument of [GL09] and
noting the fact that $\Sigma$ has an interior strictly convex point.

###  5.4. Further question

To establish the Alexandrov-Fenchel type inequalities comparing
quermassintegrals $W_{k,\theta}$ and $W_{\ell,\theta}$ for $k>\ell$, it’s also
natural to study the following fully nonlinear locally constrained curvature
flow

$\left\\{\begin{aligned} \left(\partial_{t}x\right)^{\bot}&=\Big{(}\langle
x+\cos\theta\nu,e\rangle-\frac{H_{k}}{H_{k-1}}\langle
X_{e},\nu\rangle\Big{)}\nu&\text{in}\quad\bar{\mathbb{B}}^{n}\times[0,T),\\\
\langle\bar{N}\circ
x,\nu\rangle&=-\cos\theta&\text{on}\quad\partial\bar{\mathbb{B}}^{n}\times[0,T),\\\
x(\cdot,0)&=x_{0}(\cdot)&\text{on}\quad\bar{\mathbb{B}}^{n},\end{aligned}\right.$
(5.7)

for $k=2,\cdots,n$. Along the flow (5.7), $W_{k-1,\theta}$ is preserved while
$W_{k,\theta}$ is decreasing. Using the maximum principle, the two-sided
positive bounds on the curvature function $H_{k}/H_{k-1}$ can be proved.
Furthermore, we can prove that the strict convexity is preserved along the
flow (5.7) by using the similar argument as in §4. The curvature upper bound
is not yet available. However, we still expect that this is true and then this
would give the smooth convergence of the flow to a spherical cap.

##  Appendix A Approximation result

In this appendix, we prove that a convex hypersurface $\Sigma$ with
$\theta$-capillary boundary in the unit ball can be approximated by a sequence
of strictly convex hypersurfaces with $\theta$-capillary boundary in the unit
ball in the $C^{2,\alpha}$ sense. The free boundary case (i.e.
$\theta=\frac{\pi}{2}$) has been treated by Lambert and Scheuer in [Lambert-
Scheuer2017]. We adapt their idea and include a proof for the capillary case.

Firstly, we show the following lemma which could be considered as a natural
generalization of [Lambert-Scheuer2017, Lemma 3.1].

######  Lemma A.1.

Let $\Sigma\subset\bar{\mathbb{B}}^{n+1}$ be a convex hypersurface with
$\theta$-capillary boundary for $\theta\in(0,\frac{\pi}{2}]$. Then there
exists a point $e\in\operatorname{int}(\widehat{\partial\Sigma})$, such that
$\widehat{\partial\Sigma}\subset\widehat{\partial C_{\theta,\infty}}(e)$.
Moreover, either $\partial\Sigma=\partial C_{\theta,\infty}(e)$, or
$\partial\Sigma\subset\operatorname{int}(\widehat{\partial
C_{\theta,\infty}}(e))$ and in this case there exists an interior point in
$\Sigma$ such that $h_{ij}>0$ at this point.

###### Proof.

The case $\theta=\frac{\pi}{2}$ is treated in [Lambert-Scheuer2017, Lemma
3.1]. In the following, we assume $\theta\in(0,\frac{\pi}{2})$.

Step 1. We first show that there exists
$e\in\operatorname{int}(\widehat{\partial\Sigma})$ such that
$\widehat{\partial\Sigma}\subset\widehat{\partial C_{\theta,\infty}}(e)$.

By Proposition 2.1 and $\Sigma$ is convex, one has
$\widehat{h}_{\alpha\beta}\geq\cot\theta\delta_{\alpha\beta}>0$ on
$\partial\Sigma$. By [NAndo, Theorem 1.2], we know that
$\widehat{\partial\Sigma}\subset\bigcap_{q\in\partial\Sigma}\widehat{\mathcal{S}_{q,\theta}}$,
where $\mathcal{S}_{q,\theta}$ is the circumscribed sphere of $\partial\Sigma$
in $\mathbb{S}^{n}$ with radius $\theta$ such that it touches $\partial\Sigma$
tangentially at $q$. On the other hand, by the convexity of $\partial\Sigma$,
we have
$\widehat{\partial\Sigma}=\bigcap_{q\in\partial\Sigma}\widehat{\mathcal{S}_{q,\frac{\pi}{2}}}$
which contains $\bigcap_{q\in\partial\Sigma}\widehat{\mathcal{S}_{q,\theta}}$.
Therefore, we have
$\widehat{\partial\Sigma}=\bigcap_{q\in\partial\Sigma}\widehat{\mathcal{S}_{q,\theta}}$.

Let $\bar{B}_{r}(e)\subset\mathbb{S}^{n}$ be the closed geodesic ball with the
smallest radius $r$ and center $e$ such that
$\widehat{\partial\Sigma}\subset\bar{B}_{r}(e)$. It is obvious that
$r\leq\theta$. We show that
$e\in\operatorname{int}(\widehat{\partial\Sigma})$. Let
$C=S_{r}(e)\cap\partial\Sigma$, where $S_{r}(e)=\partial B_{r}(e)$. The
minimality of $r$ implies that $C$ must support $\bar{B}_{r}(e)$: for each
hyperplane of $\mathbb{S}^{n}$ which passes through $e$, $S_{r}(e)$ is cutted
into two hemispheres $S_{r}(e)=S_{1}\cup S_{2}$ and must satisfes
$\bar{S_{i}}\cap C\neq\emptyset,\ i=1,2.$ This leads to the following two
cases.

Case 1: If two points $q_{1},q_{2}$ of $C$ are antipodal on $S_{r}(e)$, then
$e$ is the midpoint of the minimal geodesic $\gamma$ joining $q_{1}$ and
$q_{2}$. Then $e\in\gamma\subset\widehat{\partial\Sigma}$ by convexity. If
$e\in\partial\Sigma$, then $\gamma\subset\widehat{\mathcal{S}_{e,\theta}}$
implies that $\theta=\frac{\pi}{2}$ and $\gamma\subset\partial\Sigma$ by the
convexity of $\partial\Sigma$, contradicting with $\theta<\frac{\pi}{2}$.
Therefore, we have $e\in\operatorname{int}(\widehat{\partial\Sigma})$.

Case 2: If $C$ contains no pair of antipodal points of $S_{r}(e)$, since $C$
support $B_{r}(e)$, then $C$ contains at least three points
$q_{1},q_{2},q_{3}$ which would not lie simultaneously in any hemisphere of
$S_{r}(e)$. Then the convex hull of $q_{1},q_{2},q_{3}$, which is a spherical
triangle, would not be contained in any half ball of $B_{r}(e)$. Then we must
have
$e\in\operatorname{int}(\operatorname{conv}(q_{1},q_{2},q_{3}))\subset\operatorname{int}(\widehat{\partial\Sigma})$.

Step 2. We next show that either $\partial\Sigma=\partial
C_{\theta,\infty}(e)$ or
$\partial\Sigma\subset\operatorname{int}(\widehat{\partial
C_{\theta,\infty}}(e))$.

For any $q_{1},q_{2}\in\partial\Sigma$, we take $p_{1},p_{2}\in\mathbb{S}^{n}$
such that $\widehat{\mathcal{S}_{q_{i},\theta}}=\bar{B}_{\theta}(p_{i})$ for
$i=1,2$. If $p_{1}\neq p_{2}$, then

$\widehat{\partial\Sigma}\subset\bar{B}_{\theta}(p_{1})\cap\bar{B}_{\theta}(p_{2}).$

Let $o$ be the midpoint of the geodesic between $p_{1}$ and $p_{2}$. For any
$z\in\partial(\bar{B}_{\theta}(p_{1})\cap\bar{B}_{\theta}(p_{2}))$, we draw
the geodesics from the point $z$ to the points $o$, $p_{1}$ and $p_{2}$,
respectively. Denote by $\angle zop_{i}$ the angle of the spherical triangle
$\Delta zop_{i}$ at the vertex $o$ for $i=1,2$, respectively. Without loss of
generality, we assume that $z\in\partial B_{\theta}(p_{1})$, then $\angle
zop_{1}\geq\pi/2$, see Figure 3.

$p_{1}$$p_{2}$$\theta$$o$$z$$\bar{B}_{\theta}(p_{2})$$\bar{B}_{\theta}(p_{1})$
Figure 3. The intersection of $\bar{B}_{\theta}(p_{1})$ and
$\bar{B}_{\theta}(p_{2})$

Let $d^{\mathbb{S}}$ be the distance function on $\mathbb{S}^{n}$. By the Law
of Cosines on the sphere [Whittlesey], we have

$\displaystyle\cos d^{\mathbb{S}}(z,o)\cos d^{\mathbb{S}}(o,p_{1})+\cos\angle
zop_{1}\sin d^{\mathbb{S}}(z,o)\sin d^{\mathbb{S}}(o,p_{1})=\cos
d^{\mathbb{S}}(z,p_{1}).$

Then we get

$\displaystyle\cos d^{\mathbb{S}}(z,o)>$ $\displaystyle\cos
d^{\mathbb{S}}(z,o)\cos d^{\mathbb{S}}(o,p_{1})$ $\displaystyle\geq$
$\displaystyle\cos d^{\mathbb{S}}(z,p_{1})=\cos\theta,$

and hence $d^{\mathbb{S}}(z,o)<\theta$. Thus, the minimality of
$\bar{B}_{r}(e)$ implies that either $r=\theta$ or $r<\theta$.

In the latter case, we have $\widehat{\partial\Sigma}\subset\bar{B}_{r}(e)$
with $r<\theta$. Then we have $\langle x,e\rangle\geq\cos r$ and
$0\geq\langle\bar{\nu},e\rangle\geq-\sin r$ on $\partial\bar{\mathbb{B}}^{n}$.
Using the relation (2.1), we obtain

$\displaystyle D_{\mu}\langle x,e\rangle=\langle\mu,e\rangle=$
$\displaystyle\sin\theta\langle\bar{N}\circ
x,e\rangle+\cos\theta\langle\bar{\nu},e\rangle$ $\displaystyle\geq$
$\displaystyle\sin\theta\cos r-\cos\theta\sin r$ $\displaystyle=$
$\displaystyle\sin(\theta-r)>0,\quad\text{on $\partial\bar{\mathbb{B}}^{n}$}.$

Then the strict minimum of $\langle x,e\rangle$ is attained at some interior
point in $\mathbb{B}^{n}$. By attaching a large supporting sphere to $\Sigma$
from below, we obtain an interior strictly convex point in $\mathbb{B}^{n}$. ∎

Let $\Sigma\subset\bar{\mathbb{B}}^{n+1}$ be a convex hypersurface with
capillary boundary supported on $\mathbb{S}^{n}$ at a contact angle
$\theta\in(0,\frac{\pi}{2}]$, which is given by the embedding
$x_{0}:\bar{\mathbb{B}}^{n}\to\Sigma\subset\bar{\mathbb{B}}^{n+1}$. We
consider the mean curvature flow:

$\left\\{\begin{aligned}
\partial_{t}x&=-H\nu+V\qquad\text{in}\quad\bar{\mathbb{B}}^{n}\times[0,T),\\\
\langle\bar{N}\circ
x,\nu\rangle&=-\cos\theta\qquad\text{on}\quad\partial\bar{\mathbb{B}}^{n}\times[0,T),\\\
x(\cdot,0)&=x_{0}(\cdot)\qquad\text{on}\quad\bar{\mathbb{B}}^{n},\end{aligned}\right.$
(A.1)

and denote $\Sigma_{t}=x(\bar{\mathbb{B}}^{n},t)$, where $V$ is the tangential
component of $\partial_{t}x$ and satisfies
$V\big{|}_{\partial\Sigma_{t}}=-H\cot\theta\mu$.

The short time existence of the flow (A.1) can be obtained by a similar
argument as in [Stahl1996-2] by Stahl.

######  Theorem A.2.

For any $\alpha\in(0,1)$, the flow (A.1) admits a unique solution:

$x(\cdot,t)\in C^{\infty}(\bar{\mathbb{B}}^{n}\times(0,\delta])\cap
C^{2+\alpha,1+\frac{\alpha}{2}}(\bar{\mathbb{B}}^{n}\times[0,\delta]),$

where $\delta>0$ is a small constant.

We prove the following approximation result:

######  Theorem A.3.

Let $\Sigma\subset\bar{\mathbb{B}}^{n+1}$ be a convex hypersurface with
capillary boundary supported on $\mathbb{S}^{n}$ at a contact angle
$\theta\in(0,\frac{\pi}{2}]$. Suppose that $\Sigma_{t}$, $t\in[0,\delta)$ is a
solution to the flow (A.1) starting from $\Sigma$. Then either $\Sigma$ is a
flat ball $C_{\theta,\infty}$, or $\Sigma_{t}$ is strictly convex for all time
$t>0$ as long as the flow exists.

###### Proof.

By Lemma A.1, there exists a point
$e\in\operatorname{int}(\widehat{\partial\Sigma})$ such that
$\widehat{\partial\Sigma}\subset\widehat{\partial C_{\theta,\infty}}(e)$, and
either $\partial\Sigma=\partial C_{\theta,\infty}(e)$ or
$\partial\Sigma\subset\operatorname{int}(\widehat{\partial
C_{\theta,\infty}}(e))$.

For the first case $\partial\Sigma=\partial C_{\theta,\infty}(e)$, by item (2)
of Proposition 2.1, we have $h_{\alpha\beta}=0$ for
$\alpha,\beta=1,\cdots,n-1$ on the boundary $\partial\Sigma$. If
$h_{\mu\mu}>0$ on $\partial\Sigma$, then item (4) of Proposition 2.1 implies
that $\nabla_{\mu}h_{\alpha\beta}>0$ on $\partial\Sigma$ and then
$h_{\alpha\beta}<0$ at some interior points sufficiently close to the boundary
$\partial\Sigma$. So we must have $h_{\mu\mu}=0$ on $\partial\Sigma$. This
implies that if $\Sigma$ is not $C_{\theta,\infty}(e)$, then there exists an
interior point where the second fundamental form is positive definite. For the
second case $\partial\Sigma\subset\operatorname{int}(\widehat{\partial
C_{\theta,\infty}}(e))$, by Lemma A.1, there is also a strictly convex point
in the interior of $\Sigma$.

In the following, we assume that $\Sigma$ is not the flat ball
$C_{\theta,\infty}(e)$. Let

$\chi(x,t)=\min_{|\xi|=1}{h_{ij}{\xi}^{i}{\xi}^{j}},\quad x\in\Sigma_{t}$

be the smallest principal curvature at the point $x\in\Sigma_{t}$. Since
$h_{ij}$ is smooth, the function $\chi(x,t)$ is Lipschitz continuous in space
and by a cut-off function argument, we find a smooth function
$\phi_{0}:\bar{\mathbb{B}}^{n}\to\mathbb{R}$ such that
$0\leq\phi_{0}\leq\chi(x,0)$ and there exists an interior point $y$ such that
$\phi_{0}(y)>0$. We extend the function $\phi_{0}$ to
$\phi:\bar{\mathbb{B}}^{n}\times[0,\delta^{\prime})\to\mathbb{R}$ by solving a
linear parabolic PDE:

$\left\\{\begin{aligned} \frac{\partial}{\partial
t}\phi&=\Delta\phi+\nabla_{V}{\phi},\qquad\text{in}\quad\bar{\mathbb{B}}^{n}\times[0,\delta^{\prime}),\\\
\nabla_{\mu}\phi&=0\qquad\text{on}\quad\partial\bar{\mathbb{B}}^{n}\times[0,\delta^{\prime}),\\\
\phi(\cdot,0)&={\phi}_{0}(\cdot)\qquad\text{on}\quad\bar{\mathbb{B}}^{n},\end{aligned}\right.$
(A.2)

where $\Delta$ and $\nabla$ are Laplacian operator and Levi-Civita connection
with respect to the induced metric on $\Sigma_{t}=x(\bar{\mathbb{B}}^{n},t)$
of the flow (A.1). The solution $\phi$ of (A.2) exists at least for a short
time interval $[0,\delta^{\prime})$. By the strong maximum principle for
scalar functions (see [Stahl1996-2, Corollary 3.2]), we have $\phi(\cdot,t)>0$
in $\bar{\mathbb{B}}^{n}$ and $t\in(0,\delta^{\prime})$.

We take $\tau=\frac{1}{2}\min\\{\delta,\delta^{\prime}\\}$ and consider the
tensor:

$M_{ij}=h_{ij}-\phi g_{ij}$ (A.3)

for time $t\in[0,\tau)$. By the construction of $\phi_{0}$, we see that
$M_{ij}\geq 0$ at time $t=0$. We now apply the tensor maximum principle (i.e.
Theorem 4.1) to deduce that $M_{ij}\geq 0$ is preserved along the flow (A.1).

By a direct computation using Proposition 3.2 for $f=-H$, we have:

$\frac{\partial}{\partial t}M_{ij}=\Delta M_{ij}+\nabla_{V}M_{ij}+N_{ij},$
(A.4)

where

$N_{ij}=|A|^{2}M_{ij}+|A|^{2}\phi g_{ij}-2HM_{i}^{\ell}M_{j\ell}-2H\phi
M_{ij}+M_{j}^{k}\nabla_{i}{V_{k}}+M_{i}^{k}\nabla_{j}{V_{k}}.$ (A.5)

We easily see that whenever $M_{ij}\geq 0$ and $M_{ij}\xi^{i}=0$ at a point,
we have

$N_{ij}\xi^{i}\xi^{j}=|A|^{2}\phi\geq 0$ (A.6)

and thus the condition (4.1) is satisfied. The boundary condition (4.2) can be
checked similarly as in Theorem 4.3. Hence Theorem 4.1 implies that
$M_{ij}\geq 0$ is preserved along the flow (A.1) for time $t\in[0,\tau)$, and
it follows that $\Sigma_{t}$ is strictly convex for time interval $(0,\tau)$.

To show the strict convexity for the whole time interval $(0,\delta)$, we fix
a time $t_{0}\in(0,\tau)$. Since $\Sigma_{t}$ is strictly convex at time
$t_{0}$, then there exists a constant $\varepsilon>0$, such that
$h_{ij}\geq\varepsilon g_{ij}$ holds everywhere on $\Sigma_{t_{0}}$. A similar
procedure as above can be used to show that $\tilde{M}_{ij}=h_{ij}-\varepsilon
g_{ij}\geq 0$ is preserved along the flow (A.1) for all time $t>t_{0}$ as long
as the flow (A.1) exists, which finishes the proof. ∎

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