# Alexandrov-Fenchel inequalities for $k$-convex star-shaped capillary
hypersurfaces in the half-space

Guofang Wang, Liangjun Weng and Chao Xia  Mathematisches Institut, Albert-
Ludwigs-Universität Freiburg, Freiburg im Breisgau, 79104, Germany
<EMAIL_ADDRESS>School of Mathematical Sciences, Anhui
University, Hefei, 230601, P. R. China<EMAIL_ADDRESS>School of
Mathematical Sciences, Xiamen University, Xiamen, 361005, P. R. China
<EMAIL_ADDRESS>
###### Abstract.

This paper is a continuation of our previous work [24] concerning Alexandrov-
Fenchel inequalities for capillary hypersurfaces in the half-space. We obtain,
in this paper, a new class of the Alexandrov-Fenchel inequalities for
hypersurface with capillary boundary in the half-space under mild assumptions,
the hypersurfaces are $k$-convex and star-shaped, which is optimal compared to
the counterpart result of the closed case by Guan-Li [10]. In particular, we
give an affirmative answer to the question in [24, Conjecture 1.5] provided
$\theta\in(0,\frac{\pi}{2}]$.

###### Key words and phrases:

Capillary hypersurfaces, Alexandrov-Fenchel inequality, inverse curvature
flow.

###### 2020 Mathematics Subject Classification:

Primary: 53E40 Secondary: 53C21, 35K96, 53C24

##  1\. Introduction

In this paper, we continue to consider Alexandrov-Fenchel inequalities for
capillary hypersurfaces in the half-space, started in [24]. Let $\Sigma$ be a
compact hypersurface in the Euclidean closed half-space
$\bar{\mathbb{R}}^{n+1}_{+}:=\\{x\in\mathbb{R}^{n+1}\,|\,x_{n+1}\geq 0\\},$
with boundary $\partial\Sigma$ lying in $\partial\mathbb{R}^{n+1}_{+}$, where
$e_{n+1}$ is the $(n+1)$-th coordinate in $\mathbb{R}^{n+1}$. Such a
hypersurface is called a capillary hypersurface, if $\Sigma$ intersects with
$\partial{{\mathbb{R}}}_{+}^{n+1}$ at a constant contact angle
$\theta\in(0,\pi)$ along $\partial\Sigma$. Denote $\widehat{\Sigma}$ the
bounded domain in $\bar{\mathbb{R}}^{n+1}_{+}$ enclosed by
$\Sigma\subset\bar{\mathbb{R}}^{n+1}_{+}$ and $\partial\mathbb{R}^{n+1}_{+}$.
The boundary of $\widehat{\Sigma}$ consists of two parts: one is $\Sigma$ and
the other, which will be denoted by $\widehat{\partial\Sigma}$, lies on
$\partial\mathbb{R}^{n+1}_{+}$. Both have a common boundary, namely
$\partial\Sigma$. In [24], we first introduced a new class of
quermassintegrals for capillary hypersurfaces
$\Sigma\subset\bar{\mathbb{R}}^{n+1}_{+}$, namely

$\displaystyle\mathcal{V}_{0,\theta}(\widehat{\Sigma}):=|\widehat{\Sigma}|,\qquad\mathcal{V}_{1,\theta}(\widehat{\Sigma}):=\frac{1}{n+1}(|\Sigma|-\cos\theta|\widehat{\partial\Sigma}|),$

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

$\displaystyle\mathcal{V}_{k+1,\theta}(\widehat{\Sigma}):=\frac{1}{n+1}\left(\int_{\Sigma}H_{k}dA-\frac{\cos\theta\sin^{k}\theta}{n}\int_{\partial\Sigma}H_{k-1}^{\partial\Sigma}ds\right),$
(1.1)

where $H_{k}$ is the normalized $k$-th mean curvature of
$\Sigma\subset\bar{\mathbb{R}}^{n+1}_{+}$ and $H_{k-1}^{\partial\Sigma}$ is
the normalized $(k-1)$-th mean curvature of
$\partial\Sigma\subset\widehat{\partial\Sigma}\subset\partial\mathbb{R}^{n+1}_{+}$.
We use the convention that $H_{0}^{\partial\Sigma}\equiv 1$. As was shown in
[24, Theorem 2.6], the first variation of
$\mathcal{V}_{k,\theta}(\widehat{\Sigma})$ gives rise to $H_{k}$. Then we
obtained a capillary Alexandrov-Fenchel inequality between
$\mathcal{V}_{n,\theta}(\widehat{\Sigma})$ and
$\mathcal{V}_{k,\theta}(\widehat{\Sigma})$ for compact convex capillary
hypersurfaces in $\bar{\mathbb{R}}^{n+1}_{+}$, see [24, Theorem 1.2]. Finally
we proposed the following conjecture in [24, Conjecture 1.5].

Conjecture. Let $n\geq 2$ and $\Sigma\subset\bar{\mathbb{R}}^{n+1}_{+}$ be a
convex capillary hypersurface with a contact angle $\theta\in(0,{{\pi}})$.
Then there holds

$\displaystyle\frac{\mathcal{V}_{k,\theta}(\widehat{\Sigma})}{\textbf{b}_{\theta}}\geq\left(\frac{\mathcal{V}_{l,\theta}(\widehat{\Sigma})}{\textbf{b}_{\theta}}\right)^{\frac{n+1-k}{n+1-l}},\quad\forall
0\leq l<k\leq n,$ (1.2)

with equality holding if and only if $\Sigma$ is a capillary spherical cap.

Here $(n+1)\mathbf{b}_{\theta}$ is the area of $\mathcal{C}_{\theta,1}$, the
capillary spherical cap with radius $1$. For $k=1$ and $l=0$, (1.2) is just
the relative isoperimetric inequality which holds true for any hypersurfaces
in $\bar{\mathbb{R}}^{n+1}_{+}$. The family of capillary spherical caps lying
entirely in $\bar{\mathbb{R}}^{n+1}_{+}$ and intersecting
$\partial{\mathbb{R}}^{n+1}_{+}$ with a constant contact angle
$\theta\in(0,\pi)$ is given by

$\displaystyle\mathcal{C}_{\theta,r}:=\Big{\\{}x\in\bar{\mathbb{R}}^{n+1}_{+}\big{|}|x-r\cos\theta
e|=r\Big{\\}},\leavevmode\nobreak\ r\leavevmode\nobreak\ \in[0,\infty),$ (1.3)

which has radius $r$ and centered at $r\cos\theta e$. Here we denote by
$e=-e_{n+1}$, which is the unit outward normal of
$\partial{\mathbb{R}}^{n+1}_{+}\subset\bar{\mathbb{R}}^{n+1}_{+}$.

In [24], we proved (1.2) for $k=n$ and any $0\leq l<n$. The approach we used
is to study the following locally constrained inverse type curvature flow

$\displaystyle\left\\{\begin{array}[]{llll}(\partial_{t}x)^{\perp}&=&\left\\{\left(1+\cos\theta\langle\nu,e\rangle\right)\frac{H_{k-1}}{H_{k}}-\langle
x,\nu\rangle\right\\}\nu,&\hbox{ in }M\times[0,T^{\prime}),\\\
\langle\nu,e\rangle&=&-\cos\theta&\hbox{ on }\partial
M\times[0,T^{\prime}),\end{array}\right.$ (1.6)

where $\nu$ is the unit outward normal of
$\Sigma\subset\bar{\mathbb{R}}^{n+1}_{+}$, $M$ denotes a compact orientable
smooth manifold of dimension $n$ with boundary $\partial M$ and
$T^{\prime}>0$.

The introduction of this flow is motivated by the original idea of Guan-Li
[10, 11] (see also [2, 4, 12]) and the Minkowski formulas for capillary
hypersurface in [24]. For more description of Guan-Li’s idea for compact
hypersurface with free boundary or capillary boundary case, we refer to [18,
20, 24, 25]. Using these Minkowski formulas one can easily show that flow
(1.6) preserves $\mathcal{V}_{k,\theta}(\widehat{\Sigma})$ and decreases
$\mathcal{V}_{k+1,\theta}(\widehat{\Sigma})$. However in [24] we were only
able to show the preservation of the convexity in the case $k=n$, which is
important for the global convergence of the flow. Nevertheless we believed
that it should be true for a general $k$.

In this paper, we confirm the above conjecture for any $k$ and $l$, provided
that $\theta\in(0,\frac{\pi}{2}]$. Moreover, we can weaken the convexity
condition in (1.2) to the $k$-convexity, together with the star-shapedness,
which are at the moment as the optimal and natural assumptions in the case of
closed hypersurface by Guan-Li [10, Theorem 2].

######  Definition 1.1.

A hypersurface $\Sigma$ is called $k$-convex for some integer $1\leq k\leq n$
if its principal curvature $\kappa(x)\in\bar{\Gamma}^{k}_{+}$ for all
$x\in\Sigma$, where $\Gamma^{k}_{+}$ is Garding’s cone

$\displaystyle\Gamma_{+}^{k}:=\\{\lambda\in\mathbb{R}^{n}:H_{j}(\lambda)>0,\quad\forall
1\leq j\leq k\\}.$

We say that $\Sigma$ is a strictly $k$-convex hypersurface if
$\kappa(x)\in\Gamma^{k}_{+}$ for all $x\in\Sigma$.

Our main result in this paper is the following.

######  Theorem 1.2.

Let $n\geq 2$ and $\Sigma\subset\bar{\mathbb{R}}^{n+1}_{+}$ be a $k$-convex
and star-shaped capillary hypersurface with a contact angle
$\theta\in(0,\frac{{\pi}}{2}]$. Then there holds

$\displaystyle\frac{\mathcal{V}_{l+1,\theta}(\widehat{\Sigma})}{\textbf{b}_{\theta}}\geq\left(\frac{\mathcal{V}_{l,\theta}(\widehat{\Sigma})}{\textbf{b}_{\theta}}\right)^{\frac{n-l}{n+1-l}},\quad\forall
0\leq l\leq k,$ (1.7)

with equality if and only if $\Sigma$ is a capillary spherical cap.

It is clear that the conditions given in Theorem 1.2 is weaker than the
convexity condition. Hence we confirm the above conjecture, provided that
$\theta\in(0,\frac{\pi}{2}]$. Moreover for $k=1$, we obtain the following
Minkowski inequality for star-shaped and mean convex capillary hypursurfaces
in $\bar{\mathbb{R}}^{n+1}_{+}$ for the whole range ${\theta\in(0,\pi)}$.

######  Theorem 1.3.

Let $n\geq 2$ and $\Sigma\subset\bar{\mathbb{R}}^{n+1}_{+}$ be a star-shaped
and mean convex capillary hypersurface with a contact angle
${\theta\in(0,\pi)}$, then

$\displaystyle\int_{\Sigma}HdA-\sin\theta\cos\theta|\partial\Sigma|\geq
n(n+1)^{\frac{1}{n}}\mathbf{b}_{\theta}^{\frac{1}{n}}(|\Sigma|-\cos\theta|\widehat{\partial\Sigma}|)^{\frac{n-1}{n}},$
(1.8)

with equality if and only if $\Sigma$ is a capillary spherical cap.

Our approach for Theorems 1.2 and 1.3 is through the locally constrained
inverse type curvature flow (1.6). We study the remaining cases $1\leq k<n$ of
flow (1.6), since $k=n$ was considered in [24]. The new key observation is
that we can weaken the convexity condition to the optimal one as in Gerhardt
[8] and Urbas [23], i.e. the $k$-convexity and star-shapedness.

######  Theorem 1.4.

Assume the initial hypersurface $\Sigma_{0}$ is a star-shaped and strictly
$k$-convex capillary hypersurface in $\bar{\mathbb{R}}^{n+1}_{+}$ with a
contact angle $\theta\in(0,{{\pi}})$. If either

  1. (i)

$k=1$, or

  2. (ii)

$k\geq 2$, $\theta\in(0,\frac{\pi}{2}]$,

then flow (1.6) exists for all time with uniform $C^{\infty}$-estimates.
Moreover, $x(\cdot,t)$ smoothly converges to a uniquely determined capillary
spherical cap $\mathcal{C}_{\theta,r}$ around $e$, as $t\to\infty$.

It is crucial to show that the flow preserves the star-shapedness. In order to
prove it, we prove a uniform lower bound for a test function $\bar{u}$ defined
in (3.9), which is a variant of $\langle x,\nu\rangle$. Another key point is
the uniform lower and upper bounds for $H_{k}/H_{k-1}$. For this we use
$P=\bar{u}H_{k}/H_{k-1}$ in (3.17). With these two points we can show the
uniform estimates of the mean curvature (and hence all principal curvatures)
when $k\geq 2$, provided $\theta\in(0,\frac{\pi}{2}]$, by using the methods
given in [20] and [24]. The constraint $\theta\in(0,\frac{\pi}{2}]$ is only
used to show $\nabla_{\mu}H\leq 0$ on the boundary, which is crucial to obtain
the bound of $H$. For the case $k=1$, instead to use the above methods, we
alternatively use the methods given in [17], which utilized a special
divergence structure of the mean curvature, to obtain the estimates directly
for the second derivatives of $x$ (more precisely of the radial function for
the star-shaped hypersurfaces, see Section 4.2). Therefore, for such a case,
one does not need the condition $\theta\in(0,\frac{\pi}{2}]$. It would be
interesting to see if this idea works for the general $k$.

Now by virtue of the monotonicity of corresponding quermassintegrals and the
convergence result in Theorem 1.4, we get easily Theorems 1.2 and Theorem 1.3.

In view of our work and the work for closed hypersurfaces, we propose the
following

Conjecture 1.  Let $n\geq 2$ and $\Sigma\subset\bar{\mathbb{R}}^{n+1}_{+}$ be
a $k$-convex capillary hypersurface with a contact angle
$\theta\in(0,{{\pi}})$, there holds

$\displaystyle\frac{\mathcal{V}_{l+1,\theta}(\widehat{\Sigma})}{\textbf{b}_{\theta}}\geq\left(\frac{\mathcal{V}_{l,\theta}(\widehat{\Sigma})}{\textbf{b}_{\theta}}\right)^{\frac{n-l}{n+1-l}},\quad\forall
0\leq l\leq k,$ (1.9)

with equality if and only if $\Sigma$ is a capillary spherical cap.

When $k=1$ and $l=0$, (1.9) is the capillary version (1.8) of the classical
Minkowski inequality. In this case, Conjecture 1 is equivalent to ask if (1.8)
is true for mean convex capillary hypersurfaces. The Minkowski inequality for
closed mean convex hypersurfaces is still an open problem, though there are
many efforts into it. See [10] and [1, 9] for instance.

The Alexandrov-Fenchel inequalities for hypersurfaces without boundary play an
important role in differential geometry and have a lot of applications (cf
[3]). For a recent application in the Penrose type inequalities, we refer to
[7]. We believe that our inequality (1.7) will also have many applications.
Here we provide a direct application of Theorem 1.2.

######  Corollary 1.5.

Let $n\geq 2$ and $\Sigma\subset\bar{\mathbb{R}}^{n+1}_{+}$ be a convex,
capillary hypersurface with a contact angle $\theta\in(0,{\frac{\pi}{2}]}$.
Assume that $\mathcal{C}$ is a capillary spherical cap given in (1.3)
satisfying
$\mathcal{V}_{n,\theta}(\widehat{\mathcal{C}})=\mathcal{V}_{n,\theta}(\widehat{\Sigma}).$
For $t>0$, define

$\displaystyle\Sigma^{t}:=\\{x+t(\nu+\cos\theta e)\,|\,x\in\Sigma\\}.$

Then we have

$\displaystyle|\Sigma^{t}|-\cos\theta|\widehat{\Sigma^{t}}|=(n+1)\mathcal{V}_{1,\theta}(\widehat{\Sigma^{t}})\leq(n+1)\mathcal{V}_{1,\theta}({\widehat{\mathcal{C}}^{t}})=|{\mathcal{C}}^{t}|-\cos\theta|\widehat{{\mathcal{C}}^{t}}|.$
(1.10)

The corresponding result of Corollary 1.5 for closed hypersurfaces is
sometimes called Sz.-Nagy’s inequality [21], and it has applications in
obtaining reverse Faber-Krahn inequality for eigenvalue problems [19] and the
optimal insulation problem [5]. We believe that Corollary 1.5 can also be
applied to such problems for domains whose boundary is a capillary
hypersurface.

The paper is organized as follows: In Section 2, some relevant evolution
equations are collected. In Section 3, we prove the preservation of the star-
shapedness and obtain the uniform a priori estimates for flow (1.6). Section 4
is devoted to show the long-time existence and the global convergence of flow
(1.6), i.e. we prove Theorem 1.4. As a consequence, we finish the proof of
Theorem 1.2 and Theorem 1.3. Corollary 1.5 is proved in Section 5.

##  2\. Preliminaries

We follow the notation in [24] closely. For
$\kappa=(\kappa_{1},\kappa_{2}\cdots,\kappa_{n})\in\mathbb{R}^{n}$, let
$\sigma_{k}(\kappa),k=1,\cdots,n,$ be the $k$-th elementary symmetric
polynomial functions and $H_{k}(\kappa)$ be its normalization
$H_{k}(\kappa):=\frac{1}{\binom{n}{k}}\sigma_{k}(\kappa)$. Let $\Sigma_{t}$ be
a family of smooth, embedding hypersurfaces with capillary boundary in
$\bar{\mathbb{R}}^{n+1}_{+}$, given by the embeddings
$x(\cdot,t):M\to\bar{\mathbb{R}}^{n+1}_{+}$, which evolves by the general flow

$\displaystyle\partial_{t}x=f\nu+T,$ (2.1)

with $T\in T\Sigma_{t}$ and a general velocity function $f$. For a fixed $f$,
we choose $T|_{\partial\Sigma_{t}}=f\cot\theta\mu$, which implies that the
restriction of $x(\cdot,t)$ on $\partial M$ is contained in $\mathbb{R}^{n}$.
We refer to [24, Section 2] for more discussions.

Along flow (2.1), we have the following evolution equations for the induced
metric $g_{ij}$, the unit outward normal $\nu$, the second fundamental form
$(h_{ij})$, the Weingarten curvature tensor $(h^{i}_{j})$, the mean curvature
$H$, $F:=F(h^{j}_{i})$ and $k$-th mean curvature
$\sigma_{k}:=\sigma_{k}(\kappa)$ of the hypersurfaces
$\Sigma_{t}:=x(M,t)\subset\bar{\mathbb{R}}^{n+1}_{+}$. (cf. [25, Proposition
2.11] for a proof)

######  Proposition 2.1 ([25]).

Along flow (2.1), it holds that

  1. (1)

$\partial_{t}g_{ij}=2fh_{ij}+\nabla_{i}T_{j}+\nabla_{j}T_{i}$.

  2. (2)

$\partial_{t}\nu=-\nabla f+h(e_{i},T)e_{i}$.

  3. (3)

$\partial_{t}h_{ij}=-\nabla^{2}_{ij}f+fh_{ik}h_{j}^{k}+\nabla_{T}h_{ij}+h_{j}^{k}\nabla_{i}T_{k}+h_{i}^{k}\nabla_{j}T_{k}.$

  4. (4)

$\partial_{t}h^{i}_{j}=-\nabla^{i}\nabla_{j}f-fh_{j}^{k}h^{i}_{k}+\nabla_{T}h^{i}_{j}.$

  5. (5)

$\partial_{t}H=-\Delta f-|h|^{2}f+\langle\nabla H,T\rangle$.

  6. (6)

$\partial_{t}\sigma_{k}=-\frac{\partial\sigma_{k}}{\partial
h_{i}^{j}}\nabla^{i}\nabla_{j}f-f(\sigma_{1}\sigma_{k}-(k+1)\sigma_{k+1})+\langle\nabla\sigma_{k},T\rangle$.

  7. (7)

$\partial_{t}F=-F^{j}_{i}\nabla^{i}\nabla_{j}f-fF^{j}_{i}h_{j}^{k}h^{i}_{k}+\langle\nabla
F,T\rangle$, where $F^{i}_{j}:=\frac{\partial F}{\partial h^{j}_{i}}$.

##  3\. $k$-convexity and star-shapedness

In order to prove (1.7), we follow the procedure as in [24] by using the flow
approach, see also [10, 20, 25] etc. Let $M$ denote a compact orientable
smooth manifold of dimension $n$ with boundary $\partial M$ and
$x(\cdot,t):M\to\bar{\mathbb{R}}^{n+1}_{+}$ be a family of embedding
hypersurfaces with capillary boundary $x_{|\partial M}:\partial
M\to\partial\mathbb{R}_{+}^{n+1}$ and $\nu$ its unit normal vector field. We
consider the following flow

$\displaystyle\left\\{\begin{array}[]{llll}(\partial_{t}x)^{\perp}&=&\left\\{\left(1+\cos\theta\langle\nu,e\rangle\right)\frac{H_{k-1}}{H_{k}}-\langle
x,\nu\rangle\right\\}\nu:=f\nu,&\hbox{ in }M\times[0,T^{\prime}),\\\
\langle\nu,e\rangle&=&-\cos\theta&\hbox{ on }\partial
M\times[0,T^{\prime}),\\\ x(\cdot,0)&=&x_{0}(\cdot)&\text{ in
}M,\end{array}\right.$ (3.4)

where $x_{0}:M\to\bar{\mathbb{R}}^{n+1}_{+}$ is an initial capillary
hypersurface, which is star-shaped and strictly $k$-convex.

Let $\Sigma_{t}=x(M,t)$ and $\Sigma_{0}=x_{0}(M)$. If there is no confusion,
we may omit the subscript $t$ and do not distinguish $M$ and $\Sigma$.

The key feature of flow (3.4) is that it preserves
$\mathcal{V}_{k,\theta}(\widehat{\Sigma})$ while monotonically decreasing
$\mathcal{V}_{k+1,\theta}(\widehat{\Sigma})$ (cf. [24, Proposition 3.1]).
Under the convexity assumption, we have proved the long-time existence and
convergence of flow (3.4) when $k=n$ in [24]. Hence we only consider the case
$k<n$.

In this section, we show that the star-shapedness and the $k$-convexity are
preserved along flow (3.4). In order to prove this, the key is a choice of a
suitable test function.

For convenience, we define the parabolic operator for (3.4) as

$\displaystyle\mathcal{L}:=\partial_{t}-\frac{1+\cos\theta\langle\nu,e\rangle}{F^{2}}F^{ij}\nabla_{ij}^{2}-\langle
T+x-\frac{\cos\theta}{F}e,\nabla\rangle,$

where

$\displaystyle F:=\frac{H_{k}}{H_{k-1}},\leavevmode\nobreak\
\leavevmode\nobreak\ 1\leq k\leq n.$

###  3.1. Gradient estimate

Let $\Sigma_{0}$ be an initial capillary hypersurface which is star-shaped and
$k$-convex. The star-shapedness of $\Sigma_{0}$ implies that there exist
$0<r_{1}<r_{2}<\infty$ such that

$\Sigma_{0}\subset\widehat{\mathcal{C}_{\theta,r_{2}}}\setminus\widehat{\mathcal{C}_{\theta,r_{1}}},$

where $\widehat{\mathcal{C}_{\theta,r}}$ denotes the bounded domain in
$\bar{\mathbb{R}}^{n+1}_{+}$ enclosed by $\mathcal{C}_{\theta,r}$ and
$\partial\bar{\mathbb{R}}^{n+1}_{+}$. Following the argument in [24,
Proposition 4.2], we have the following height estimate, which follows from
the maximum principle (or the avoidable principle) since
$\mathcal{C}_{\theta,r}$ is a stationary solution of the flow.

######  Proposition 3.1.

For any $t\in[0,T^{\prime})$, along flow (3.4), $1\leq k\leq n$, it holds

$\displaystyle\Sigma_{t}\subset\widehat{\mathcal{C}_{\theta,r_{2}}}\setminus\
\widehat{\mathcal{C}_{\theta,r_{1}}},$ (3.5)

where $\mathcal{C}_{\theta,r}$ defined by (1.3) and $r_{1},r_{2},c_{0}$ only
depend on $\Sigma_{0}$.

Next we show the the star-shapedness is preserved along flow (3.4).

######  Proposition 3.2.

Let $\Sigma_{0}$ be a star-shaped and strictly $k$-convex hypersurface with
capillary boundary in $\bar{\mathbb{R}}^{n+1}_{+}$ and $\theta\in(0,\pi)$,
then there exists $c_{0}>0$ depending only on $\Sigma_{0}$, such that the
solution $\Sigma_{t}$ of flow (3.4) satisfies

$\displaystyle\langle x,\nu\rangle(p,t)\geq c_{0},$ (3.6)

for all $(p,t)\in M\times[0,T^{\prime})$.

###### Proof.

A direct computation gives

$\displaystyle\langle x,\nu\rangle_{;ij}=h_{ij}+h_{ij;k}\langle
x,e_{k}\rangle-(h^{2})_{ij}\langle x,\nu\rangle.$

Using Proposition 2.1, we have

$\displaystyle\partial_{t}\langle x,\nu\rangle$ $\displaystyle=$
$\displaystyle\langle f\nu+T,\nu\rangle+\langle x,-\nabla
f+h(e_{i},T)e_{i}\rangle$ $\displaystyle=$
$\displaystyle\frac{1+\cos\theta\langle\nu,e\rangle}{F}-\langle
x,\nu\rangle-\langle x,e_{k}\rangle\Big{[}\frac{\cos\theta\langle
h_{ki}e_{i},e\rangle}{F}$
$\displaystyle-(1+\cos\theta\langle\nu,e\rangle)\frac{F_{;k}}{F^{2}}-\langle
x,h_{ki}e_{i}\rangle\Big{]}+h(x^{T},T),$

which yields

$\displaystyle\mathcal{L}\langle
x,\nu\rangle=\Big{[}\big{(}1+\cos\theta\langle\nu,e\rangle\big{)}\frac{F^{ij}(h^{2})_{ij}}{F^{2}}-1\Big{]}\langle
x,\nu\rangle.$ (3.7)

It is easy to see

$\displaystyle\langle\nu,e\rangle_{;kl}=h_{kl;s}\langle
e_{s},e\rangle-(h^{2})_{kl}\langle\nu,e\rangle.$

Combining it with

$\displaystyle\langle e,\nabla f\rangle$ $\displaystyle=$
$\displaystyle\langle
e,e_{i}\rangle\left(\frac{1+\cos\theta\langle\nu,e\rangle}{F}\right)_{;i}-\langle
e,e_{i}\rangle\langle x,\nu\rangle_{;i}$ $\displaystyle=$
$\displaystyle\frac{\cos\theta}{F}h(e^{T},e^{T})-F^{-2}(1+\cos\theta\langle\nu,e\rangle)\langle\nabla
F,e\rangle-h(x^{T},e^{T}),$

we have

$\displaystyle\partial_{t}\langle\nu,e\rangle$ $\displaystyle=$
$\displaystyle-\langle\nabla f,e\rangle+h(e_{i},T)\langle e_{i},e\rangle$
$\displaystyle=$
$\displaystyle-\frac{\cos\theta}{F}h(e^{T},e^{T})+F^{-2}(1+\cos\theta\langle\nu,e\rangle)\langle\nabla
F,e\rangle+h(x^{T},e^{T})+h(T,e^{T}).$

It follows

$\displaystyle\mathcal{L}\langle\nu,e\rangle=(1+\cos\theta\langle\nu,e\rangle)F^{-2}F^{ij}(h^{2})_{ij}\langle\nu,e\rangle.$
(3.8)

Now we introduce the function mentioned in the introduction

$\displaystyle\bar{u}:=\frac{\langle
x,\nu\rangle}{1+\cos\theta\langle\nu,e\rangle}.$ (3.9)

One can check that it satisfies

$\displaystyle\mathcal{L}\bar{u}$ $\displaystyle=$
$\displaystyle\frac{1}{1+\cos\theta\langle\nu,e\rangle}\mathcal{L}u-\frac{\langle
x,\nu\rangle}{(1+\cos\theta\langle\nu,e\rangle)^{2}}\mathcal{L}(1+\cos\theta\langle\nu,e\rangle)$
$\displaystyle+\frac{2(1+\cos\theta\langle\nu,e\rangle)}{F^{2}}F^{ij}\Bigg{[}\frac{\langle
x,\nu\rangle_{;i}(1+\cos\theta\langle\nu,e\rangle)_{;j}}{(1+\cos\theta\langle\nu,e\rangle)^{2}}$
$\displaystyle-\langle
x,\nu\rangle\frac{(\cos\theta\langle\nu,e\rangle)_{;i}(\cos\theta\langle\nu,e\rangle)_{;j}}{(1+\cos\theta\langle\nu,e\rangle)^{3}}\Bigg{]}.$

Combining with equations (3.7) and (3.8), we get

$\displaystyle\mathcal{L}\bar{u}=\frac{\langle
x,\nu\rangle}{1+\cos\theta\langle\nu,e\rangle}\left(\frac{F^{ij}(h^{2})_{ij}}{F^{2}}-1\right)+2\frac{F^{ij}}{F^{2}}{\bar{u}}_{;i}(1+\cos\theta\langle\nu,e\rangle)_{;j}.$
(3.10)

On $\partial M$, we have

$\displaystyle\nabla_{\mu}\langle x,\nu\rangle=\langle
x,h(\mu,\mu)\mu\rangle=\cos\theta h(\mu,\mu)\langle
x,\overline{\nu}\rangle=\cot\theta h(\mu,\mu)\langle x,\nu\rangle$

and

$\displaystyle\nabla_{\mu}\big{(}1+\cos\theta\langle\nu,e\rangle\big{)}=-\sin\theta
h(\mu,\mu)\langle\nu,e\rangle.$

Altogether yields

$\displaystyle\nabla_{\mu}\bar{u}=0,\ \text{ on }\ \partial M.$ (3.11)

Since

$\displaystyle F^{ij}(h^{2})_{ij}\geq F^{2}$ (3.12)

for $F:=\frac{H_{k}}{H_{k-1}}$ (cf. [2] or [4, Lemma 3.2]), (3.10) gives

$\displaystyle\mathcal{L}\bar{u}\geq 0,\quad\text{ mod }\nabla\bar{u},$

which, together with (3.11), implies

$\displaystyle\bar{u}\geq\min_{M}\bar{u}(\cdot,0)$

by the maximum principle. Since $1+\cos\theta\langle\nu,e\rangle\geq
1-\cos\theta>0$, then

$\displaystyle\langle
x,\nu\rangle=(1+\cos\theta\langle\nu,e\rangle)\cdot\bar{u}\geq c_{0}>0,$

for some positive constant $c_{0}$, which depends only on $\Sigma_{0}$. ∎

###  3.2. Curvature estimate

In this subsection, we show the uniform positive lower bound and the upper
bound of $F=\frac{H_{k}}{H_{k-1}}$, $1\leq k\leq n-1$. As a consequence, the
$k$-convexity is preserved along flow (3.4).

The evolution equation of $F$ was computed in [24, Proposition 4.3].

######  Proposition 3.3 ([24]).

Along flow (3.4), it holds

$\displaystyle\mathcal{L}F$ $\displaystyle=$
$\displaystyle\frac{2\cos\theta}{F^{2}}F^{ij}F_{;j}h_{ik}\langle
e_{k},e\rangle-\frac{2(1+\cos\theta\langle\nu,e\rangle)}{F^{3}}F^{ij}F_{;i}F_{;j}$
$\displaystyle+F\left(1-\frac{F^{ij}(h^{2})_{ij}}{F^{2}}\right),$

and

$\displaystyle\nabla_{\mu}F=0,\quad\text{ on }\partial M.$ (3.14)

######  Proposition 3.4.

If $\Sigma_{t}$ solves flow (3.4), then

$\displaystyle F(p,t)\leq\max_{M}F(\cdot,0).$ (3.15)

###### Proof.

(3.12) and (3.3) impliy

$\displaystyle\mathcal{L}F=F\left(1-\frac{F^{ij}(h^{2})_{ij}}{F^{2}}\right)\leq
0,\quad\text{ mod }\nabla F.$

Since $\nabla_{\mu}F=0$ on $\partial M$, the conclusion directly follows from
the maximum principle.

∎

######  Proposition 3.5.

If $\Sigma_{t}$ solves flow (3.4) with an initial hypersurface $\Sigma_{0}$
being a star-shaped and strictly $k$-convex capillary hypersurface in
$\bar{\mathbb{R}}^{n+1}_{+}$, then

$\displaystyle F(p,t)\geq C,\quad\quad\forall(p,t)\in M\times[0,T^{\prime}),$
(3.16)

where the positive constant $C$ depends only on the initial datum.

###### Proof.

Define

$\displaystyle P:=F\bar{u}.$ (3.17)

Using (3.10) and (3.3), we obtain

$\displaystyle\mathcal{L}P$ $\displaystyle=$
$\displaystyle\bar{u}\mathcal{L}F+F\mathcal{L}\bar{u}-2(1+\cos\theta\langle\nu,e\rangle)\frac{F^{ij}F_{;i}{\bar{u}}_{;j}}{F^{2}}$
$\displaystyle=$ $\displaystyle\bar{u}\left[2\cos\theta
F^{-2}F^{ij}F_{;j}h_{ik}\langle
e_{k},e\rangle-2(1+\cos\theta\langle\nu,e\rangle)F^{-3}F^{ij}F_{;i}F_{;j}\right]$
$\displaystyle+\bar{u}F\left(1-\frac{F^{ij}(h^{2})_{ij}}{F^{2}}\right)+F\Bigg{[}\frac{\langle
x,\nu\rangle}{1+\cos\theta\langle\nu,e\rangle}\left(\frac{F^{ij}(h^{2})_{ij}}{F^{2}}-1\right)$
$\displaystyle+2\frac{F^{ij}}{F^{2}}{\bar{u}}_{;i}(1+\cos\theta\langle\nu,e\rangle)_{;j}\Bigg{]}-2(1+\cos\theta\langle\nu,e\rangle)F^{-2}F^{ij}F_{;i}{\bar{u}}_{;j}$
$\displaystyle=$ $\displaystyle
2\frac{F^{ij}}{F^{2}}\left[(1+\cos\theta\langle\nu,e\rangle)_{;i}-(1+\cos\theta\langle\nu,e\rangle)\frac{F_{;i}}{F}\right]P_{;j}.$

On $\partial M$, (3.11) and (3.14) imply

$\displaystyle\nabla_{\mu}P=0.$ (3.18)

By applying the maximum principle, we get

$\displaystyle P\geq\min_{M}F\bar{u}(\cdot,0),$

which, together with the boundedness of $\bar{u}$ proved in Proposition 3.1,
implies the desired estimate of $F$. ∎

##  4\. Long-time existence and convergence

The short-time existence of flow (3.4) is now well-known. Let $T^{*}$ be the
maximal time of the existence of a smooth solution to (3.4) in the class of
star-shaped and strictly $k$-convex hypersurfaces. In order to establish the
long-time existence of flow (3.4), we need to obtain the uniform height
estimate, the gradient estimate and the second order derivative estimate (or
curvature estimates) for the solution of flow (3.4). Then the long-time
existence and the uniform $C^{\infty}$-estimates follow from the standard
nonlinear parabolic PDE theory with a strictly oblique boundary condition (cf.
[6, 14, 22]). We will discuss the proof for the case $2\leq k<n$ separately
from $k=1$ in this section, since for the latter case we do not need the
constraint $\theta\leq\frac{\pi}{2}$, by using a different method.

Assume that a capillary hypersurfaces
$\Sigma\subset\bar{\mathbb{R}}^{n+1}_{+}$ is star-shaped with respect to the
origin. One can reparametrize it as a graph over
$\overline{\mathbb{S}}^{n}_{+}$. Namely, there exists a positive function
$\rho$ defined on $\overline{\SS}^{n}_{+}$ such that

$\displaystyle\Sigma=\left\\{\rho(X)X|X\in\overline{\SS}^{n}_{+}\right\\},$

where $X:=(X_{1},\ldots,X_{n})$ is a local coordinate of
$\overline{\SS}^{n}_{+}$.

We denote $\nabla^{0}$ the Levi-Civita connection on $\SS^{n}_{+}$ with
respect to the standard round metric $\sigma:=g_{{}_{\SS^{n}_{+}}}$,
$\partial_{i}:=\partial_{X_{i}}$,
$\sigma_{ij}:=\sigma(\partial_{i},\partial_{j})$,
$\rho_{i}:=\nabla^{0}_{i}\rho$, and
$\rho_{ij}:=\nabla^{0}_{i}\nabla^{0}_{j}\rho$. The induced metric $g$ on
$\Sigma$ is given by

$\displaystyle
g_{ij}=\rho^{2}\sigma_{ij}+\rho_{i}\rho_{j}=e^{2\varphi}\left(\sigma_{ij}+\varphi_{i}\varphi_{j}\right),$

where $\varphi(X):=\log\rho(X)$. Its inverse $g^{-1}$ is given by

$\displaystyle
g^{ij}=\frac{1}{\rho^{2}}\left(\sigma^{ij}-\frac{\rho^{i}\rho^{j}}{\rho^{2}+|\nabla^{0}\rho|^{2}}\right)=e^{-2\varphi}\left(\sigma^{ij}-\frac{\varphi^{i}\varphi^{j}}{v^{2}}\right),$

where $\rho^{i}:=\sigma^{ij}\rho_{j}$, $\varphi^{i}:=\sigma^{ij}\varphi_{j}$
and

$v:=\sqrt{1+|\nabla^{0}\varphi|^{2}}.$

The unit outward normal vector field on $\Sigma$ is given by

$\displaystyle\nu=\frac{1}{v}\left(\partial_{\rho}-\rho^{-2}\nabla^{0}\rho\right)=\frac{1}{v}\left(\partial_{\rho}-\rho^{-1}\nabla^{0}\varphi\right).$

The second fundamental form $h:=(h_{ij})$ on $\Sigma$ is

$\displaystyle
h_{ij}=\frac{e^{\varphi}}{v}\left(\sigma_{ij}+\varphi_{i}\varphi_{j}-\varphi_{ij}\right),$

and its Weingarten matrix $(h^{i}_{j})$ is

$\displaystyle
h_{j}^{i}=g^{ik}h_{kj}=\frac{1}{e^{\varphi}v}\left[\delta^{i}_{j}-(\sigma^{ik}-\frac{\varphi^{i}\varphi^{k}}{v^{2}})\varphi_{kj}\right].$

We also have for the mean curvature,

$\displaystyle
H=\frac{1}{e^{\varphi}v}\left[n-\left(\sigma^{ij}-\frac{\varphi^{i}\varphi^{j}}{v^{2}}\right)\varphi_{ij}\right].$

Next we derive the capillary boundary condition in (3.4) in terms of the
radial function $\varphi$. We use the polar coordinate in the half-space, that
is, for $x:=(x^{\prime},x_{n+1})\in\mathbb{R}^{n}\times[0,+\infty)$ and
$X:=(\beta,\xi)\in[0,\frac{\pi}{2}]\times\SS^{n-1}$, we have

$\displaystyle x_{n+1}=r\cos\beta,\quad|x^{\prime}|=r\sin\beta.$

Then

$\displaystyle
e_{n+1}=\partial_{x_{n+1}}=\cos\beta\partial_{r}-\frac{\sin\beta}{r}\partial_{\beta}.$

In these coordinates the standard Euclidean metric is given by

$|dx|^{2}=dr^{2}+r^{2}\left(d\beta^{2}+\sin^{2}\beta
g_{{}_{\SS^{n-1}}}\right).$

It follows that

$\displaystyle\langle\nu,e_{n+1}\rangle=\frac{1}{v}\left(\cos\beta+\sin\beta\nabla_{\partial_{\beta}}^{0}\varphi\right).$

Along $\partial\SS^{n}_{+}$, i.e., $\beta=\frac{\pi}{2}$, it holds

$\displaystyle e=-e_{n+1}=\frac{1}{r}\partial_{\beta},$

then

$\displaystyle-\cos\theta=\langle\nu,e\rangle=\langle\frac{1}{v}\left(\partial_{r}-r^{-1}\nabla^{0}\varphi\right),\frac{1}{r}\partial_{\beta}\rangle=-\frac{\nabla^{0}_{\partial_{\beta}}\varphi}{v},$

that is,

$\displaystyle\nabla^{0}_{\partial_{\beta}}\varphi=\cos\theta\sqrt{1+|\nabla^{0}\varphi|^{2}}.$
(4.1)

In the following, we derive the full curvature estimates (which in turn the
$C^{2}$ estimates of $\varphi$) of the flow (3.4) for the case $2\leq k<n$
separately from $k=1$.

###  4.1. Case $2\leq k\leq n$ and $\theta\in(0,\frac{\pi}{2}]$

If $k\geq 2$, by Definition 1.1, we know $H>0,\sigma_{2}>0$. Since the square
norm of the second fundamental form satisfies

$\displaystyle|h|^{2}=H^{2}-2\sigma_{2}\leq H^{2},$

then the uniform upper bound of $H$ gives all the curvature bounds of flow
(3.4).

By direct computation, for any $2\leq k\leq n$ we have

$\displaystyle\mathcal{L}H$ $\displaystyle=$
$\displaystyle(1+\cos\theta\langle\nu,e\rangle)F^{-2}\big{[}F^{kl,st}h_{kl;i}h_{st;i}+F^{kl}(h^{2})_{kl}H-F|h|^{2}\big{]}$
$\displaystyle-2(1+\cos\theta\langle\nu,e\rangle)F^{-3}|\nabla
F|^{2}+2\cos\theta F^{-2}F_{;i}\langle\nu,e\rangle_{;i}$
$\displaystyle-\cos\theta F^{-1}H-F^{-1}|h|^{2}+H$ $\displaystyle\leq$
$\displaystyle(1+\cos\theta\langle\nu,e\rangle)F^{-2}F^{kl}(h^{2})_{kl}H-(2+\cos\theta\langle\nu,e\rangle)F^{-1}|h|^{2}$
$\displaystyle-2(1+\cos\theta\langle\nu,e\rangle)F^{-3}|\nabla
F|^{2}+2\cos\theta F^{-2}F_{;i}\langle\nu,e\rangle_{;i}$
$\displaystyle-\cos\theta F^{-1}H-\cos\theta F^{-1}H+H$ $\displaystyle=:$
$\displaystyle(1+\cos\theta\langle\nu,e\rangle)F^{-2}F^{kl}(h^{2})_{kl}H+\text{K}_{2}+H,$
(4.2)

where the middle inequality follows from the concavity of $F$. And if
$\theta\in(0,\frac{\pi}{2}]$, we have

$\displaystyle\nabla_{\mu}H\leq 0,\leavevmode\nobreak\ \text{ on
}\leavevmode\nobreak\ \partial M.$ (4.3)

To obtain (4.3) we need to use the constraint $\theta\in(0,\frac{\pi}{2}]$,
see [24, Eq. (4.8)] for a proof. This is the only place that we need
$\theta\in(0,\frac{\pi}{2}]$. Furthermore, the term $\text{K}_{2}$ in (4.2)
satisfies

$\displaystyle\text{K}_{2}$ $\displaystyle:=$
$\displaystyle-(2+\cos\theta\langle\nu,e\rangle)F^{-1}|h|^{2}-2(1+\cos\theta\langle\nu,e\rangle)F^{-3}|\nabla
F|^{2}$ (4.4) $\displaystyle+2\cos\theta F^{-2}F_{;i}\langle\nu,e\rangle_{;i}$
$\displaystyle\leq$ $\displaystyle-c_{0}|h|^{2},$

for some universal constant $c_{0}>0$. For the proof of (4.4) we refer to [24,
Proof of Proposition 4.8].

The proof of the upper bound of mean curvature was already provided in [24,
Proposition 4.8] for the case $k=n$, hence we only consider the case $2\leq
k<n$.

For the case $2\leq k<n$, we note that the only difference in the evolution
equation $\mathcal{L}H$ compared to $k=n$ lies in the handling of the
following term

$\displaystyle(1+\cos\theta\langle\nu,e\rangle)F^{-2}F^{kl}(h^{2})_{kl}H,$
(4.5)

while the boundary condition remains the same. Now we compare these two cases:

  * •

For the case of $k=n$, we have

$\displaystyle F^{ij}(h^{2})_{ij}=F^{2},$

which follows that the term (4.5) in the evolution equation $\mathcal{L}H$
(i.e. (4.2)) is just a linear term with respect to $H$, and hence it is
trivially controlled by the square norm of the second fundamental form
$|h|^{2}$ coming from the term K2 in (4.4).

  * •

For the case of $2\leq k<n$, by diagonalizing the symmetric $(1,1)$-tensor
$(h_{i}^{j})$ under a proper coordinates chart such that its eigenvalues
satisfy $\kappa_{1}\geq\cdots\geq\kappa_{n}$ at the interior maximum point of
$H$, we have

$\kappa_{1}\leq H\leq n\kappa_{1},$

where in the first inequality we have used $k\geq 2$ and
$\kappa\in\Gamma_{k}$. Indeed $\kappa\in\Gamma_{2}$ implies
$\sum\limits_{j\geq 2}\kappa_{j}>0$. Furthermore, we know that (cf. [4, Lemma
3.4 or Eq. (3.15)])

$\displaystyle F^{-2}F^{ij}(h^{2})_{ij}\leq
1+C(n,k)\frac{(\kappa_{1}-\kappa_{n})^{2}}{\kappa_{1}^{2}},$

which implies

$\displaystyle F^{-2}F^{ij}(h^{2})_{ij}\leq
C+C\frac{\kappa_{n}^{2}}{\kappa_{1}^{2}}\leq C+Cn^{2}\frac{|h|^{2}}{H^{2}}\leq
C(1+n^{2}),$

for $k\geq 2$. Hence like the case $k=n$, we can also get the bound of $H$, by
applying (4.4).

In conclusion, for any $2\leq k\leq n$, following the argument in [24] and the
above discussion, we establish the full curvature estimate of flow (3.4).

######  Corollary 4.1.

If $\Sigma_{0}$ is star-shaped and and strictly $k$-convex capillary
hypersurface with a contact angle $\theta\in(0,\frac{{\pi}}{2}]$ and $k\geq
2$, then the solution $\Sigma_{t},t\in[0,T^{*})$ of flow (3.4), has a uniform
curvature bound. That is, there exists $C>0$ depending only on $\Sigma_{0}$,
such that the principal curvatures of $\Sigma_{t}$,

$|\kappa_{i}|(p,t)\leq C,\leavevmode\nobreak\ \leavevmode\nobreak\ \forall
1\leq i\leq n$

for all $(p,t)\in M\times[0,T^{*})$.

###  4.2. Case $k=1$, $\theta\in(0,\pi)$

For $k=1$ in (3.4),

$f:=\frac{n\left(1+\cos\theta\langle\nu,e\rangle\right)}{H}-\langle
x,\nu\rangle.$

Hence (3.4) is a modified inverse mean curvature flow. To prove its existence
for all time, we follow the approach in [17, Section 4] and [14, Chapter V],
which proves the Schauder estimates without having direct access to a second
derivative estimate of $\varphi$.

Consider flow (3.4) for $k=1$ for radial graphs over $\bar{\SS}^{n}_{+}$. It
is known that the scalar function $\varphi(z,t):=\log\rho(X(z,t),t)$ satisfies

$\displaystyle\begin{array}[]{rcll}\partial_{t}\varphi&=&G(\varphi_{ij},\varphi_{k}),&\text{
in }\SS^{n}_{+}\times[0,T^{*}),\\\
\nabla_{\partial_{\beta}}^{0}\varphi&=&\cos\theta\sqrt{1+|\nabla^{0}\varphi|^{2}},&\text{
on }\partial\SS^{n}_{+}\times[0,T^{*}),\\\
\varphi(\cdot,0)&=&\varphi_{0}(\cdot),&\text{ on }\SS^{n}_{+},\end{array}$
(4.9)

where $\partial_{\beta}$ is the outward normal of
$\partial\SS^{n}_{+}\subset\bar{\SS}^{n}_{+}$, $\varphi_{0}$ is the
corresponding radial function of $x_{0}(M)$ over $\overline{\SS}^{n}_{+}$ and

$\displaystyle
G(\varphi_{ij},\varphi_{k}):=\frac{nv^{2}}{n-\left(\sigma^{ij}-\frac{\varphi^{i}\varphi^{j}}{v^{2}}\right)\varphi_{ij}}\left[1-\frac{\cos\theta}{v}\left(\cos\beta+\sin\beta\nabla^{0}_{\partial_{\beta}}\varphi\right)\right]-1.$

Note that $G$ does not explicitly depend on the solution $\varphi$ itself, but
only on its first and second derivatives, which is an important fact. Set

$\displaystyle G^{ij}:=\frac{\partial
G}{\partial\varphi_{ij}},G^{k}:=\frac{\partial G}{\partial\varphi_{k}}.$

Recall that $T^{*}$ is the maximal time such that there exists some
$\varphi\in C^{2,1}(\bar{\SS}^{n}_{+}\times[0,T^{*}))\cap
C^{\infty}(\bar{\SS}^{n}_{+}\times(0,T^{*}))$ which solves (4.9). In the
sequel, we will prove a priori estimates for the solutions of $\varphi$ on
$[0,T^{\prime}]$ for any $T^{\prime}<T^{*}$.

####  4.2.1. Gradient estimates

First of all, we give the estimate of
$\dot{\varphi}:=\frac{\partial\varphi}{\partial t}$.

######  Proposition 4.2.

If $\varphi$ solves (4.9) and $\theta\in(0,\pi)$, then

$\displaystyle\min_{\bar{\SS}^{n}_{+}}\frac{nv_{0}}{e^{\varphi_{0}}H_{0}}\left[1-\frac{\cos\theta}{v}\left(\cos\beta+\sin\beta\nabla^{0}_{\partial_{\beta}}\varphi_{0}\right)\right]\leq\dot{\varphi}+1$
$\displaystyle\qquad\leq\max_{\bar{\SS}^{n}_{+}}\frac{nv_{0}}{e^{\varphi_{0}}H_{0}}\left[1-\frac{\cos\theta}{v}\left(\cos\beta+\sin\beta\nabla^{0}_{\partial_{\beta}}\varphi_{0}\right)\right],\quad\text{in}\quad\SS^{n}_{+}\times[0,T^{\prime}],$

where $H_{0}:=H(\cdot,0)$ and $v_{0}:=v(\cdot,0)$.

###### Proof.

Assume $\dot{\varphi}$ admits the maximum value at some point
$(p_{0},t_{0})\in\bar{\SS}^{n}_{+}\times[0,T^{\prime}]$, that is

$\displaystyle\dot{\varphi}(p_{0},t_{0})=\max_{\bar{\SS}^{n}_{+}\times[0,T^{\prime}]}\dot{\varphi}.$

We claim that $t_{0}=0$. If not, we have $t_{0}>0$. A direct computation gives

$\displaystyle\frac{\partial\dot{\varphi}}{\partial
t}=\sum_{i,j=1}^{n}G^{ij}\dot{\varphi}_{ij}+\sum_{k=1}^{n}G^{k}\dot{\varphi}_{k}\quad\text{in}\quad\SS^{n}_{+}\times[0,T^{\prime}].$

The maximum principle yields that $p_{0}\in\partial\SS^{n}_{+}$. Choosing an
orthonormal frame around $p_{0}$ such that $\\{e_{\alpha}\\}_{\alpha=1}^{n-1}$
are the tangential vector fields on $\partial\SS^{n}_{+}$ and
$e_{n}:=\partial_{\beta}$ the unit normal along
$\partial\SS^{n}_{+}\subset\bar{\SS}^{n}_{+}$. Then we have
$(\dot{\varphi})_{\alpha}(p_{0},t_{0})=0$. Combining it with the boundary
equation in (4.9) yields that

$\displaystyle(\dot{\varphi})_{n}(p_{0},t_{0})$ $\displaystyle=$
$\displaystyle\frac{d}{dt}\big{(}\cos\theta\sqrt{1+|\nabla^{0}\varphi|^{2}}\big{)}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\frac{\varphi_{i}(\varphi_{i})_{t}}{\sqrt{1+|\nabla^{0}\varphi|^{2}}}\cos\theta=\frac{\varphi_{n}{(\dot{\varphi})}_{n}}{\sqrt{1+|\nabla^{0}\varphi|^{2}}}\cos\theta=\cos^{2}\theta{(\dot{\varphi})}_{n},$

Since $|\cos\theta|<1$, we obtain $({\dot{\varphi}})_{n}(p_{0},t_{0})=0$,
which is a contradiction to the Hopf boundary lemma. Therefore $t_{0}=0$, as a
result,

$\displaystyle\dot{\varphi}\leq\max_{\bar{\SS}^{n}_{+}}\dot{\varphi_{0}}.$

Similarly, we have

$\displaystyle\dot{\varphi}\geq\min_{\bar{\SS}^{n}_{+}}\dot{\varphi_{0}}.$

Both, together with

$\displaystyle
G=\frac{nv}{e^{\varphi}H}\left[1-\frac{\cos\theta}{v}\left(\cos\beta+\sin\beta\nabla^{0}_{\partial_{\beta}}\varphi\right)\right]-1,$

give the desired estimate. Hence we complete the proof. ∎

Proposition 3.2 says that the star-shapedness is preserved along flow (3.4).
In particular, combining with Proposition 3.1, Proposition 4.2 yields the
uniform gradient estimate of $\varphi$ in (4.9). That is

######  Proposition 4.3.

If $\varphi$ solves (4.9), then

$\displaystyle\|\varphi\|_{C^{1}(\bar{\SS}^{n}_{+}\times[0,T^{\prime}])}\leq
C,$

where $C>0$ depends only on the initial datum.

####  4.2.2. Hölder estimates and long-time existence

In this subsection, we follow the argument in [17] to obtain the higher
regularity of flow (3.4) for $k=1$, i.e. (4.9), by checking carefully the case
$\theta\in(0,\pi)$. First, we estimate the Hölder norm of the spatial
derivative of $\varphi$.

######  Proposition 4.4.

If $\varphi$ solves (4.9), then there exists some $\gamma>0$ such that

$\displaystyle[\nabla^{0}\varphi]_{x,\gamma}+[\nabla^{0}\varphi]_{t,\frac{\gamma}{2}}\leq
C,$

where $[\psi]_{z,\gamma}$ denotes the $\gamma$-Hölder semi-norm of $\psi$ in
$\bar{\SS}^{n}_{+}\times[0,T^{\prime}]$ with respect to the $z$-variable and
$C:=C(\|\varphi_{0}\|_{C^{2+\alpha}(\bar{\SS}^{n}_{+})},n,\gamma)$ .

###### Proof.

The a priori estimates of $|\nabla^{0}\varphi|$ and $|\partial_{t}\varphi|$
imply the bound for $[\varphi]_{x,\gamma}$ and
$[\varphi]_{t,\frac{\gamma}{2}}$, due to Proposition 4.2 and 4.3. Together
with [14, Chapter II, Lemma 3.1] one can give the bound for
$[\nabla^{0}\varphi]_{t,\frac{\gamma}{2}}$ provided that we have a bound for
$[\nabla^{0}\varphi]_{x,\gamma}$. Hence it suffices to bound
$[\nabla^{0}\varphi]_{x,\gamma}$ in the following. In fact, we fix $t$ and
rewrite (4.9) as a quasilinear elliptic PDE with a strictly oblique boundary
value condition

$\displaystyle\text{div}_{\sigma}\left(\frac{\nabla^{0}\varphi}{\sqrt{1+|\nabla^{0}\varphi|^{2}}}\right)=\frac{n}{v}-\frac{n}{1+\dot{\varphi}}\left[v-{\cos\theta}\left(\cos\beta+\sin\beta\nabla^{0}_{\partial_{\beta}}\varphi\right)\right],$
(4.10)

and on $\partial\SS^{n}_{+}$,

$\nabla_{\partial_{\beta}}^{0}\varphi=\cos\theta\sqrt{1+|\nabla^{0}\varphi|^{2}}.$

The right-hand side of (4.10) is a bounded function in $x$, due to Proposition
4.2 and 4.3. Thus for any $\Omega^{\prime}\subset\subset\SS^{n}_{+}$, using
[15, Chapter 3, Theorem 14.1], one can prove an interior estimate of form

$\displaystyle[\nabla^{0}\varphi]_{\gamma,\Omega^{\prime}}\leq
C(\text{dist}_{\sigma}(\Omega^{\prime},\partial\SS^{n}_{+}),|\nabla^{0}\varphi|,|\dot{\varphi}|),$

for some $\gamma>0$. And the boundary Hölder estimate of $\nabla^{0}\varphi$
can be obtained by adapting the similar procedure as in [15, Chapter 10,
Section 2]. Hence we prove the estimate for $[\nabla^{0}\varphi]_{x,\gamma}$,
which yields the conclusion.

∎

Next we estimate the Hölder norm of the time derivative of $\varphi$.

######  Proposition 4.5.

If $\varphi$ solves (4.9), then there exists some $\gamma>0$ such that

$\displaystyle[\partial_{t}\varphi]_{x,\gamma}+[\partial_{t}\varphi]_{t,\frac{\gamma}{2}}\leq
C,$

where $[\psi]_{z,\gamma}$ denotes the $\gamma$-Hölder semi-norm of $\psi$ in
$\bar{\SS}^{n}_{+}\times[0,T^{\prime}]$ with respect to the $z$-variable and
$C:=C(\|\varphi_{0}\|_{C^{2+\alpha}(\bar{\SS}^{n}_{+})},n,\gamma)$ .

###### Proof.

Recall that for $k=1$, we have

$\displaystyle\mathcal{L}:=\partial_{t}-\frac{n(1+\cos\theta\langle\nu,e\rangle)}{H^{2}}\Delta-\langle
T+x-\frac{n\cos\theta}{H}e,\nabla\rangle.$

The evolution equations (3.3), (3.7) and (3.8) have the form

$\displaystyle\mathcal{L}H$ $\displaystyle=$ $\displaystyle
2nH^{-2}\langle\nabla
H,\nabla(1+\cos\theta\langle\nu,e\rangle)\rangle-2n(1+\cos\theta\langle\nu,e\rangle)H^{-3}|\nabla
H|^{2}$ (4.11) $\displaystyle+H\left(1-\frac{n|h|^{2}}{H^{2}}\right),$
$\displaystyle\mathcal{L}\langle
x,\nu\rangle=\Big{[}nH^{-2}\big{(}1+\cos\theta\langle\nu,e\rangle\big{)}|h|^{2}-1\Big{]}\langle
x,\nu\rangle,$ (4.12)

and

$\displaystyle\mathcal{L}(1+\cos\theta\langle\nu,e\rangle)=n\frac{|h|^{2}}{H^{2}}(1+\cos\theta\langle\nu,e\rangle)\cos\theta\langle\nu,e\rangle.$
(4.13)

Note that

$\dot{\varphi}=\frac{n(1+\cos\theta\langle\nu,e\rangle)}{uH}-1.$

Combining it with (4.11), (4.12) and (4.13), through some tedious
calculations, we know that the evolution equation of

$\Phi:=\dot{\varphi},$

has a nice divergence structure with respect to the induced metric $g$, which
is uniformly controlled (due to the uniform estimates for $\varphi$ and
$\nabla^{0}\varphi$, see Proposition 4.3). That is,

$\displaystyle\partial_{t}\Phi$ $\displaystyle=$
$\displaystyle\text{div}_{g}(\frac{n(1+\cos\theta\langle\nu,e\rangle)\nabla\Phi}{H^{2}})-\frac{2n(1+\cos\theta\langle\nu,e\rangle)}{H^{2}(\Phi+1)}|\nabla\Phi|^{2}$
$\displaystyle+\langle
T+x-\frac{n\cos\theta}{H}e,\nabla\Phi\rangle+\frac{n}{H^{2}}\langle\nabla(1+\cos\theta\langle\nu,e\rangle),\nabla\Phi\rangle.$

Moreover by using (3.11) and (3.14) for $k=1$, we know that $\Phi$ satisfies
the Neumann boundary value condition, as

$\displaystyle\nabla_{\mu}\Phi=\nabla_{\mu}\left[\frac{n(1+\cos\theta\langle\nu,e\rangle)}{uH}-1\right]=0.$

In order to adopt the approach as in [14, Chapter V], we use the weak
formulation of the quasi-linear parabolic equation (4.2.2), that is

$\displaystyle\int_{t_{0}}^{t_{1}}\int_{M}\eta\partial_{t}\Phi d\mu_{t}dt$
$\displaystyle=$
$\displaystyle\int_{t_{0}}^{t_{1}}\int_{M}\Bigg{[}-\frac{n(1+\cos\theta\langle\nu,e\rangle)}{H^{2}}\langle\nabla\eta,\nabla\Phi\rangle$
(4.15)
$\displaystyle-\frac{2n(1+\cos\theta\langle\nu,e\rangle)}{H^{2}(\Phi+1)}|\nabla\Phi|^{2}\eta+\frac{n}{H^{2}}\langle\nabla(1+\cos\theta\langle\nu,e\rangle),\nabla\Phi\rangle\eta$
$\displaystyle+\eta\langle
T+x-\frac{n\cos\theta}{H}e,\nabla\Phi\rangle\Bigg{]}d\mu_{t}dt,$

for the test function $\eta$ and $0<t_{0}<t_{1}<T^{\prime}$.

Choose a specific test function $\eta$ defined by

$\displaystyle\eta:=\xi^{2}(1+\cos\theta\langle\nu,e\rangle)(\Phi+1),$ (4.16)

where $\xi$ is an arbitrary smooth function with values in $[0,1]$. Then the
left hand side of (4.15) is equal to

$\displaystyle\int_{t_{0}}^{t_{1}}\int_{M}\eta\partial_{t}\Phi d\mu_{t}dt$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{M}\xi^{2}(\Phi+1)^{2}(1+\cos\theta\langle\nu,e\rangle)d\mu_{t}\Big{|}_{t_{0}}^{t_{1}}$
$\displaystyle-\int_{t_{0}}^{t_{1}}\int_{M}(\Phi+1)^{2}\xi\frac{\partial\xi}{\partial
t}(1+\cos\theta\langle\nu,e\rangle)d\mu_{t}dt.$

The sum of the first three terms on the right hand side in (4.15) equals

$\displaystyle\int_{t_{0}}^{t_{1}}\int_{M}\left[-\frac{2n(1+\cos\theta\langle\nu,e\rangle)^{2}}{H^{2}}(\Phi+1)\xi\langle\nabla\xi,\nabla\Phi\rangle-\frac{3n(1+\cos\theta\langle\nu,e\rangle)^{2}}{H^{2}}\xi^{2}|\nabla\Phi|^{2}\right]d\mu_{t}dt.$

Plugging these back into (4.15) yields that

$\displaystyle\frac{1}{2}\int_{M}\xi^{2}(\Phi+1)^{2}(1+\cos\theta\langle\nu,e\rangle)d\mu_{t}\Big{|}_{t_{0}}^{t_{1}}+\int_{t_{0}}^{t_{1}}\int_{M}\frac{3n(1+\cos\theta\langle\nu,e\rangle)^{2}}{H^{2}}\xi^{2}|\nabla\Phi|^{2}d\mu_{t}dt$
$\displaystyle=$
$\displaystyle\int_{t_{0}}^{t_{1}}\int_{M}\left[(\Phi+1)\frac{\partial\xi}{\partial
t}+\xi\langle
T+x-\frac{n\cos\theta}{H}e,\nabla\Phi\rangle-\frac{2n(1+\cos\theta\langle\nu,e\rangle)}{H^{2}}\langle\nabla\xi,\nabla\Phi\rangle\right]\cdot$
$\displaystyle\qquad\xi(\Phi+1)(1+\cos\theta\langle\nu,e\rangle)d\mu_{t}dt.$

Notice that Proposition 3.4 implies that

$\displaystyle\frac{(1+\cos\theta\langle\nu,e\rangle)^{2}\xi^{2}|\nabla\Phi|^{2}}{H^{2}}\geq\frac{(1+\cos\theta\langle\nu,e\rangle)^{2}\xi^{2}|\nabla\Phi|^{2}}{\max
H^{2}}.$

For any $\varepsilon>0$, again using Proposition 4.3, Proposition 3.5,
Proposition 3.4, and Young’s inequality, we have

$\displaystyle\int_{t_{0}}^{t_{1}}\int_{M}\xi^{2}\langle
T+x-\frac{n\cos\theta}{H}e,\nabla\Phi\rangle(\Phi+1)(1+\cos\theta\langle\nu,e\rangle)d\mu_{t}dt$
$\displaystyle\leq$
$\displaystyle\int_{t_{0}}^{t_{1}}\int_{M}\left[\varepsilon\xi^{2}|\nabla\Phi|^{2}(1+\cos\theta\langle\nu,e\rangle)^{2}+\frac{1}{4\varepsilon}\xi^{2}\Big{|}T+x-\frac{n\cos\theta}{H}e\Big{|}(\Phi+1)^{2}\right]d\mu_{t}dt$
$\displaystyle=$
$\displaystyle\int_{t_{0}}^{t_{1}}\int_{M}\left[\varepsilon\xi^{2}|\nabla\Phi|^{2}(1+\cos\theta\langle\nu,e\rangle)^{2}+C_{\varepsilon}\xi^{2}(\Phi+1)^{2}\right]d\mu_{t}dt$

and

$\displaystyle-\int_{t_{0}}^{t_{1}}\int_{M}\frac{2n(1+\cos\theta\langle\nu,e\rangle)}{H^{2}}\langle\nabla\xi,\nabla\Phi\rangle\xi(\Phi+1)(1+\cos\theta\langle\nu,e\rangle)d\mu_{t}dt$
$\displaystyle\leq$
$\displaystyle\int_{t_{0}}^{t_{1}}\int_{M}\left[\varepsilon\xi^{2}|\nabla\Phi|^{2}(1+\cos\theta\langle\nu,e\rangle)^{2}+\frac{n^{2}}{\varepsilon
H^{4}}|\nabla\xi|^{2}(\Phi+1)^{2}(1+\cos\theta\langle\nu,e\rangle)^{2}\right]d\mu_{t}dt$
$\displaystyle=$
$\displaystyle\int_{t_{0}}^{t_{1}}\int_{M}\left[\varepsilon\xi^{2}|\nabla\Phi|^{2}(1+\cos\theta\langle\nu,e\rangle)^{2}+C_{\varepsilon}|\nabla\xi|^{2}(\Phi+1)^{2}\right]d\mu_{t}dt,$

where $C_{\varepsilon}>0$ is a uniform constant.

By choosing $\varepsilon>0$ small enough, say $\varepsilon:=\frac{n}{\max
H^{2}}$, and noticing $2\geq 1+\cos\theta\langle\nu,e\rangle\geq
1-\cos\theta>0$, we conclude that

$\displaystyle\frac{1}{2}(1-\cos\theta)\int_{M}\xi^{2}(\Phi+1)^{2}d\mu_{t}\Big{|}_{t_{0}}^{t_{1}}+\frac{n(1-\cos\theta)^{2}}{\max
H^{2}}\int_{t_{0}}^{t_{1}}\int_{M}\xi^{2}|\nabla\Phi|^{2}d\mu_{t}dt$
$\displaystyle\leq$ $\displaystyle
C\int_{t_{0}}^{t_{1}}\int_{M}(\Phi+1)^{2}\left[|\nabla\xi|^{2}+\xi\left|\frac{\partial\xi}{\partial
t}\right|(1+\cos\theta\langle\nu,e\rangle)+\xi^{2}\right]d\mu_{t}dt.$

This inequality embraces the same type of the one in [14, Chapter V, §1,
Eq.(1.13)], therefore, similar to [14, Chapter V, §1] (interior estimate) and
[14, Chapter V, §7, Page 478] (boundary estimate) or [16, Chapter XIII,
section 6], the bound for $[\Phi]_{x,\gamma}=[\dot{\varphi}]_{x,\gamma}$ and
$[\Phi]_{t,\frac{\gamma}{2}}=[\dot{\varphi}]_{t,\frac{\gamma}{2}}$ can be
obtained. Moreover, all local interior and boundary estimates are independent
of the time $T^{\prime}$. And the global estimates follow from the local
results and a covering argument, which also does not depend on $T^{\prime}$. ∎

The Hölder estimate of gradient $\varphi$ implies the Hölder estimate for the
mean curvature.

######  Proposition 4.6.

If $\varphi$ solves (4.9), then there exists some $\gamma>0$ such that

$\displaystyle[H]_{x,\gamma}+[H]_{t,\frac{\gamma}{2}}\leq C,$ (4.17)

where $[\psi]_{z,\gamma}$ denotes the $\gamma$-Hölder semi-norm of $\psi$ in
$\bar{\SS}^{n}_{+}\times[0,T^{\prime}]$ with respect to the $z$-variable and
$C:=C(\|\varphi_{0}\|_{C^{2+\alpha}(\bar{\SS}^{n}_{+})},n,\gamma)$.

###### Proof.

From (4.9), we know

$\displaystyle
e^{\varphi}H=\frac{n}{1+\dot{\varphi}}\left[\sqrt{1+|\nabla^{0}\varphi|^{2}}-{\cos\theta}\left(\cos\beta+\sin\beta\nabla^{0}_{\partial_{\beta}}\varphi\right)\right].$

Hence the conclusion (4.17) follows from the Hölder estimates for
$|\nabla^{0}\varphi|,\dot{\varphi}$ and $\varphi$ in Proposition 4.4 and
Proposition 4.5, while the Hölder estimate for $\varphi$ follows trivially
from Proposition 4.2 and Proposition 4.3. ∎

With the preparation above, we obtain the full second order derivative
estimates and high order estimates for $\varphi$.

######  Proposition 4.7.

If $\varphi$ solves (4.9), then for every $t_{0}\in(0,T^{\prime})$, there
exists some $\gamma>0$ such that

$\displaystyle\|\varphi\|_{C^{2+\gamma,1+\frac{\gamma}{2}}(\bar{\SS}^{n}_{+}\times[0,T^{\prime}])}\leq
C(\|\varphi_{0}\|_{C^{2+\alpha}(\bar{\SS}^{n}_{+})},n,\gamma),$ (4.18)

and

$\displaystyle\|\varphi\|_{C^{2l+\gamma,l+\frac{\gamma}{2}}(\bar{\SS}^{n}_{+}\times[t_{0},T^{\prime}])}\leq
C(\|\varphi(\cdot,t_{0})\|_{C^{2l+\alpha}(\bar{\SS}^{n}_{+})},n,\gamma),$
(4.19)

for any $2\leq l\in\mathbb{N}$.

###### Proof.

To get the Schauder estimates (4.18), we rewrite the evolution equation for
$\varphi$ with respect to the induced metric $g$, which is uniformly
controlled (due to the uniform estimate for $\varphi$ and $\nabla^{0}\varphi$,
see Proposition 4.3). Noticing that $v:=\sqrt{1+|\nabla^{0}\varphi|^{2}}$ and

$\displaystyle
e^{\varphi}vH=n-\left(\sigma^{ij}-\frac{\varphi^{i}\varphi^{j}}{v^{2}}\right)\varphi_{ij}=n-e^{2\varphi}\Delta_{g}\varphi,$

we have

$\displaystyle\partial_{t}\varphi$ $\displaystyle=$
$\displaystyle\frac{nv}{e^{\varphi}H}\left[1-\frac{\cos\theta}{v}\left(\cos\beta+\sin\beta\nabla^{0}_{\partial_{\beta}}\varphi\right)\right]-1$
$\displaystyle=$
$\displaystyle-\frac{n}{e^{2\varphi}H^{2}}(n-e^{2\varphi}\Delta_{g}\varphi)\left[1-\frac{\cos\theta}{v}\left(\cos\beta+\sin\beta\nabla^{0}_{\partial_{\beta}}\varphi\right)\right]$
$\displaystyle+\frac{2nv}{e^{\varphi}H}\left[1-\frac{\cos\theta}{v}\left(\cos\beta+\sin\beta\nabla^{0}_{\partial_{\beta}}\varphi\right)\right]-1.$

It follows

$\displaystyle\partial_{t}\varphi$ $\displaystyle=$
$\displaystyle\frac{n}{H^{2}}\left[1-\frac{\cos\theta}{v}\left(\cos\beta+\sin\beta\nabla^{0}_{\partial_{\beta}}\varphi\right)\right]\Delta_{g}\varphi$
$\displaystyle+\left(\frac{2nv}{e^{\varphi}H}-\frac{n^{2}}{e^{2\varphi}H^{2}}\right)\left[1-\frac{\cos\theta}{v}\left(\cos\beta+\sin\beta\nabla^{0}_{\partial_{\beta}}\varphi\right)\right]-1,$

which is a linear, uniformly parabolic equation with Hölder coefficients, due
to Proposition 4.6 and Proposition 4.4. Moreover, the Neumann boundary value
condition in (4.9) satisfies the strictly oblique property (due to
$|\cos\theta|<1$) and its coefficient has uniform Hölder bounds (due to again
Proposition 4.4). Hence (4.18) follows from the linear parabolic PDE theory
with a strictly oblique boundary value problem (cf. [16, Chapter V, Theorem
4.9 and Theorem 4.23], see also [14, Chapter IV, Theorem 5.3]).

Next we show (4.19). By differentiating both sides of (4.2.2) with respect to
$t$ and $\partial_{j}$, $1\leq j\leq n$, respectively, one can easily get
evolution equations of $\dot{\varphi}$ and $\nabla^{0}_{j}\varphi$
respectively, which, using the estimate (4.18), can be treated as uniformly
parabolic PDEs on the time interval $[t_{0},T^{\prime}]$. At the initial time
$t_{0}$, all compatibility conditions are satisfied and the initial function
$\varphi(\cdot,t_{0})$ is smooth, which implies the
$C^{3+\gamma,(3+\gamma)/2}$-estimate for $\nabla^{0}_{j}\varphi$ and the
$C^{2+\gamma,(2+\gamma)/2}$-estimate for $\dot{\varphi}$. So, we have the
$C^{4+\gamma,(4+\gamma)/2}$-estimate for $\varphi$. From [16, Chapter 4,
Theorem 4.3, Exercise 4.5] and the above argument, it is not difficult to know
that the constants are independent of the time $T^{\prime}$. Higher regularity
of (4.19) can be proven by induction over $2\leq l\in\mathbb{N}$. Hence we
complete the proof. ∎

With the estimates given above, the long-time existence of inverse mean
curvature type flow (4.9) or (3.4) ($k=1$) follows directly. That is.

######  Corollary 4.8.

If $\Sigma_{0}$ is star-shaped and strictly mean convex capillary hypersurface
with a contact angle $\theta\in(0,\pi)$, then the solution of flow (3.4) for
$k=1$ exists for all time with uniform $C^{\infty}$-estimates.

We are now ready to prove Theorem 1.4 by using above uniform a priori
estimates.

###### Proof of Theorem 1.4.

Following the parabolic theory with a strictly oblique boundary value
condition (cf. [6, 16, 22]), by Corollary 4.1 for $k\geq 2$, we know the
solution of (3.4) exists for all time with uniform $C^{\infty}$-estimate
($k=1$ due to Corollary 4.8). And the convergence result of Theorem 1.4 can be
proved in the similar way as [24, Proposition 4.12], by adopting the
monotonicity property of quermassintegrals along flow (3.4). ∎

Finally, by applying the convergence result of flow (3.4) and the monotonicity
of $\mathcal{V}_{k,\theta}$, i.e. Theorem 1.4 and [24, Proposition 3.1], one
can finish the proof of Theorems 1.2 and 1.3 for the strictly $k$-convex case.
When $\Sigma$ is only $k$-convex, we use the approximation argument as in [10]
to prove Theorems 1.2 and 1.3, while the equality characterization in Theorems
1.2 and 1.3 can be established along the same way as [20, 24]. For the concise
of this paper, we leave it to the interested reader and omit it here.

##  5\. An application

We observe that the quermassintegrals for capillary hypersurfaces introduced
in [24] can be in fact obtained by considering suitable parallel hypersufaces.
This class of parallel hypersufaces was already used in [13] to prove the
Heintze-Karcher inequality for capillary hypersurfaces.

For any $t>0$, define

$\displaystyle\Sigma^{t}:=\\{x+t(\nu+\cos\theta e)\,|x\in\Sigma\\}.$ (5.1)

If $\Sigma$ is a convex capillary hypersurface in
$\bar{\mathbb{R}}^{n+1}_{+}$, so is $\Sigma^{t}$ for any $t>0$. One can show
the followings formulas, which have their own interest.

######  Proposition 5.1.

Steiner type formulas hold for capillary hypersurfaces

$\displaystyle\mathcal{V}_{1,\theta}(\widehat{\Sigma^{t}})$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{n}\binom{n}{k}t^{k}\mathcal{V}_{k+1,\theta}(\widehat{\Sigma}),$
(5.2) $\displaystyle\mathcal{V}_{0,\theta}(\widehat{\Sigma^{t}})$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{n+1}\binom{n+1}{k}t^{k}\mathcal{V}_{k,\theta}(\widehat{\Sigma}).$
(5.3)

###### Proof.

First of all, we have a simple observation that

$\displaystyle(n+1)\mathcal{V}_{1,\theta}(\widehat{\Sigma})=|\Sigma|-\cos\theta|\widehat{\partial\Sigma}|=\int_{\Sigma}(1+\cos\theta\langle\nu,e\rangle)dA.$
(5.4)

To show (5.2), we pick an orthonormal frame $\\{e_{i}\\}_{i=1}^{n}$ of
$T_{x}\Sigma$ such that the second fundamental form $h_{ij}(x)$ of $\Sigma$ is
diagonal at $x$, i.e. $h_{ij}(x):=\kappa_{i}(x)\delta_{ij}$. Note that
$(x+t(\nu+\cos\theta e))_{i}=(1+th_{ii})e_{i}$, then the area element of
$\Sigma^{t}$ at $x+t(\nu+\cos\theta e)$ is

$\displaystyle
dA_{\Sigma^{t}}=\prod_{i=1}^{n}(1+\kappa_{i}t)dA=\sum_{k=0}^{n}\sigma_{k}(\kappa)t^{k}dA,$

where $dA_{\Sigma^{t}}$ and $dA$ are the area element with respect to the
induced metric of $\Sigma^{t}$ and $\Sigma$ in $\bar{\mathbb{R}}^{n+1}_{+}$
respectively. Therefore, we have

$\displaystyle(n+1)\mathcal{V}_{1,\theta}(\widehat{\Sigma^{t}})$
$\displaystyle=$
$\displaystyle\int_{\Sigma^{t}}(1+\cos\theta\langle\nu,e\rangle)dA_{\Sigma^{t}}$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{n}\int_{\Sigma}(1+\cos\theta\langle\nu,e\rangle)t^{k}\sigma_{k}(\kappa)dA$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{n}\int_{\Sigma}(1+\cos\theta\langle\nu,e\rangle)t^{k}\binom{n}{k}H_{k}dA$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{n}(n+1)\binom{n}{k}t^{k}\mathcal{V}^{\prime}_{k+1,\theta}(\widehat{\Sigma}),$

where we denote

$\displaystyle(n+1)\mathcal{V}^{\prime}_{k+1,\theta}(\widehat{\Sigma}):=\int_{\Sigma}(1+\cos\theta\langle\nu,e\rangle)H_{k}dA.$
(5.5)

Then we only need to show that
$\mathcal{V}^{\prime}_{k+1,\theta}=\mathcal{V}_{k+1,\theta}$, which will be
done in the following Proposition 5.2. Hence we have (5.2).

To show (5.3), note that

$\displaystyle(n+1)\mathcal{V}_{0,\theta}(\widehat{\Sigma})=(n+1)|\widehat{\Sigma}|=\int_{\Sigma}\langle
x,\nu\rangle dA.$

Hence we have as above

$\displaystyle(n+1)\mathcal{V}_{0,\theta}(\widehat{\Sigma^{t}})$
$\displaystyle=$ $\displaystyle\int_{\Sigma^{t}}\langle x+t(\nu+\cos\theta
e),\nu\rangle dA_{\Sigma^{t}}=\int_{\Sigma^{t}}(\langle
x,\nu\rangle+t(1+\cos\theta\langle\nu,e\rangle)dA_{\Sigma^{t}}.$ (5.6)
$\displaystyle=$ $\displaystyle\sum_{k=0}^{n}\int_{\Sigma}\langle x,\nu\rangle
t^{k}\binom{n}{k}H_{k}dA+t\sum_{k=0}^{n}(n+1)\binom{n}{k}t^{k}\mathcal{V}^{\prime}_{k+1,\theta}(\widehat{\Sigma})$
$\displaystyle=$ $\displaystyle\int_{\Sigma}\langle x,\nu\rangle
dA+\sum_{k=1}^{n}t^{k}\binom{n}{k}\int_{\Sigma}(1+\cos\theta\langle\nu,e\rangle)H_{k-1}dA$
$\displaystyle+\sum_{k=1}^{n+1}(n+1)\binom{n}{k-1}t^{k}\mathcal{V}^{\prime}_{k,\theta}(\widehat{\Sigma})$
$\displaystyle=$
$\displaystyle(n+1)\sum_{k=0}^{n+1}\binom{n+1}{k}t^{k}\mathcal{V}^{\prime}_{k,\theta}(\widehat{\Sigma}),$

where we have used the Minkowski formulas [24, Proposition 2.5] in the third
equality for $1\leq k\leq n$. Hence (5.3) follows also from the following
Proposition 5.2. ∎

######  Proposition 5.2.

Let $\Sigma$ be a compact, capillary hypersurface in
$\bar{\mathbb{R}}^{n+1}_{+}$ with a constant contact angle $\theta\in(0,\pi)$,
there holds

$\displaystyle\mathcal{V}_{k+1,\theta}(\widehat{\Sigma})=\mathcal{V}^{\prime}_{k+1,\theta}(\widehat{\Sigma})=\frac{1}{n+1}\int_{\Sigma}(1+\cos\theta\langle\nu,e\rangle)H_{k}(\kappa)dA,\quad
0\leq k\leq n.$ (5.7)

###### Proof.

Case 1. $k=0$. It is (5.4).

Case 2. $1\leq k<n$. First of all, it is clear that by (1.1), the definition
of $\mathcal{V}_{k+1,\theta}$, we only need to show

$\displaystyle\int_{\Sigma}\langle\nu,e\rangle
H_{k}dA=-\frac{\sin^{k}\theta}{n}\int_{\partial\Sigma}H^{\partial\Sigma}_{k-1}ds,\leavevmode\nobreak\
\leavevmode\nobreak\ 1\leq k\leq n-1.$ (5.8)

This case follows from the proof of the Minkowski formulas in [24, Proposition
2.5]. We use all the notations, definitions and computations given there.
Define

$P_{e}:=\langle\nu,e\rangle x-\langle x,\nu\rangle e,$

which is in fact a tangential vector field on $\Sigma$. One can check

$\displaystyle\nabla_{i}(P_{e})_{j}$ $\displaystyle=$
$\displaystyle\langle\nu,e\rangle g_{ij}+h_{il}\langle e,e_{l}\rangle\langle
x,e_{j}\rangle-h_{il}\langle x,e_{l}\rangle\langle e,e_{j}\rangle,\quad\hbox{
in }\Sigma,$ (5.9) $\displaystyle\langle P_{e}^{T},\mu\rangle$
$\displaystyle=$ $\displaystyle-\langle
x,\bar{\nu}\rangle,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\hbox{ on
}\partial\Sigma.$ (5.10)

Moreover we have $h_{\alpha\beta}=\sin\theta\widehat{h}_{\alpha\beta}$ for
$2\leq\alpha,\beta\leq n$, which implies that
$\sigma_{k}^{\mu\mu}=\sin^{k}\theta\sigma_{k}(\hat{\kappa})$, where
$\widehat{h}_{\alpha\beta}$ is the second fundamental form of
$\partial\Sigma\subset\partial\bar{\mathbb{R}}_{+}^{n+1}$ and
$\sigma_{k}^{ij}:=\frac{\partial\sigma_{k+1}}{\partial h^{i}_{j}}$. We have

$\displaystyle(n-k)\int_{\Sigma}\langle\nu,e\rangle\sigma_{k}(\kappa)dA$
$\displaystyle=$
$\displaystyle\int_{\Sigma}\sigma_{k}^{ij}\nabla_{i}(P_{e})_{j}dA=\int_{\partial\Sigma}\sigma_{k}^{\mu\mu}\langle
P_{e},\mu\rangle ds$ $\displaystyle=$
$\displaystyle-\sin^{k}\theta\int_{\partial\Sigma}\sigma_{k}(\widehat{\kappa})\langle
x,\bar{\nu}\rangle ds$ $\displaystyle=$
$\displaystyle-\sin^{k}\theta\frac{\binom{n-1}{k}}{\binom{n-1}{k-1}}\int_{\partial\Sigma}\sigma_{k-1}(\widehat{\kappa})ds,\leavevmode\nobreak\
\leavevmode\nobreak\ \quad 1\leq k\leq n-1,$

where in the last equality we have used the usual Minkowski formula for
$\partial\Sigma$ in $\mathbb{R}^{n}=\partial\bar{\mathbb{R}}^{n+1}_{+}.$ It is
clear that the above formula is equivalent to (5.8). Hence we have proved
(5.7) for this case.

Case 3. $k=n$. Dividing (5.6) by $t^{n+1}$ and letting $t\to\infty$, one can
easily show that

$\mathcal{V}^{\prime}_{n+1,\theta}(\widehat{\Sigma})=\lim_{t\to\infty}\mathcal{V}_{0,\theta}(t^{-1}\Sigma+\mathcal{C}_{\theta,1})=|\mathcal{C}_{\theta,1}|=(n+1)\textbf{b}_{\theta}=\mathcal{V}_{n+1,\theta}(\widehat{\Sigma}).$

In conclusion, we finish the proof of (5.7).

∎

With the help of formula (5.2), Corollary 1.5 is a direct consequence of
Theorem 1.2.

###### Proof of Corollary 1.5.

First of all, we remark that
$\mathcal{V}_{n+1,\theta}(\widehat{\Sigma})=\mathcal{V}_{n+1,\theta}(\widehat{\mathcal{C}})=(n+1)\textbf{b}_{\theta}$
(cf. [24, Eq. (1.12)]). In fact $\mathcal{V}_{n+1,\theta}(\widehat{\Sigma})$
is a topological constant. Since
$\mathcal{V}_{n,\theta}(\widehat{\Sigma})=\mathcal{V}_{n,\theta}(\widehat{\mathcal{C}})$,
by Theorem 1.2 we have

$\displaystyle\mathcal{V}_{k+1,\theta}(\widehat{\Sigma})\leq\mathcal{V}_{k+1,\theta}(\widehat{\mathcal{C}}),\quad
0\leq k\leq n-1,$

which, together with (5.2), implies

$\displaystyle\mathcal{V}_{1,\theta}(\widehat{\Sigma^{t}})\leq\mathcal{V}_{1,\theta}(\widehat{\mathcal{C}^{t}}).$

Hence we complete the proof. ∎

######  Remark 5.3.

When $\Sigma$ is a $k$-convex and star-shaped capillary hypersurfaces, so is
$\Sigma^{t}$ for small $t>0$. Therefore in view of Theorem 1.2 the above proof
works (and hence Corollary 1.5 holds) for such a hypersurface with small $t$.

Acknowledgment: LW is supported by NSFC (Grant No. 12201003). CX is supported
by NSFC (Grant No. 11871406, 12271449).

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