# Volume preserving Gauss curvature flow of convex hypersurfaces in the
hyperbolic space

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.

We consider the volume preserving flow of smooth, closed and convex
hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1}(n\geq 2)$ with the
speed given by arbitrary positive power $\alpha$ of the Gauss curvature. We
prove that if the initial hypersurface is convex, then the smooth solution of
the flow remains convex and exists for all positive time $t\in[0,\infty)$.
Moreover, we apply a result of Kohlmann which characterises the geodesic ball
using the hyperbolic curvature measures and an argument of Alexandrov
reflection to prove that the flow converges to a geodesic sphere exponentially
in the smooth topology. This can be viewed as the first result for non-local
type volume preserving curvature flows for hypersurfaces in the hyperbolic
space with only convexity required on the initial data.

###### Key words and phrases:

Volume preserving Gauss Curvature flow, Hyperbolic space, Convex, Curvature
measures, Alexandrov reflection.

###### 2010 Mathematics Subject Classification:

53C44; 53C42

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Klein model and Projection
  4. 4 A priori estimate
  5. 5 Long time existence
  6. 6 Monotonicity and Hausdorff convergence
  7. 7 Proofs of Theorem 1.2 and Corollary 1.3

##  1\. Introduction

Let $X_{0}:M^{n}\to\mathbb{H}^{n+1}(n\geq 2)$ be a smooth embedding such that
$M_{0}=X_{0}(M)$ is a closed and convex hypersurface in the hyperbolic space
$\mathbb{H}^{n+1}$. We consider the smooth family of embeddings
$X:M^{n}\times[0,T)\to\mathbb{H}^{n+1}$ satisfying

$\left\\{\begin{aligned} \frac{\partial}{\partial
t}X(x,t)=&~{}(\phi(t)-K^{\alpha})\nu(x,t),\\\
X(\cdot,0)=&~{}X_{0}(\cdot),\end{aligned}\right.$ (1.1)

where $\alpha>0$, $\nu$ is the unit outer normal of $M_{t}=X(M,t)$, $K$ is the
Gauss curvature of $M_{t}$ and

$\phi(t)=\frac{1}{|M_{t}|}\int_{M_{t}}K^{\alpha}\,\mathrm{d}\mu_{t}$ (1.2)

such that the domain $\Omega_{t}$ enclosed by $M_{t}$ has a fixed volume
$|\Omega_{t}|=|\Omega_{0}|$ along the flow (1.1).

######  Definition 1.1.

Let $M^{n}$ be a smooth closed hypersurface in the hyperbolic space
$\mathbb{H}^{n+1}$. Denote the principal curvatures of $M$ by
$\kappa=(\kappa_{1},\cdots,\kappa_{n})$.

(i). $M$ is said to be horospherically convex (or simply called $h$-convex),
if its principal curvatures satisfy $\kappa_{i}\geq 1$ for all $i=1,\cdots,n$
everywhere on $M$. Equivalently, $M$ is h-convex if for any point $p\in M$
there exists a horosphere enclosing $M$ and touching $M$ at $p$. We also say
that a bounded domain $\Omega\subset\mathbb{H}^{n+1}$ is $h$-convex if its
boundary $\partial\Omega$ is $h$-convex.

(ii). $M$ is said to be positively curved (or called having positive sectional
curvatures), if its principal curvatures satisfy

$\kappa_{i}\kappa_{j}>1,\qquad\forall~{}i\neq j$

everywhere on $M$.

(iii). $M$ is said to be convex if its principal curvatures satisfy
$\kappa_{i}>0$ for all $i=1,\cdots,n$ everywhere on $M$. We also say that a
bounded domain $\Omega\subset\mathbb{H}^{n+1}$ is convex if its boundary
$\partial\Omega$ is convex.

As the main result of this paper, we prove the following convergence result
for the flow (1.1) with convex initial hypersurfaces:

######  Theorem 1.2.

Let $X_{0}:M^{n}\to\mathbb{H}^{n+1}(n\geq 2)$ be a smooth embedding such that
$M_{0}=X_{0}(M)$ is a closed and convex hypersurface in $\mathbb{H}^{n+1}$.
Then for any $\alpha>0$, the volume preserving flow (1.1) has a unique smooth
convex solution $M_{t}$ for all time $t\in[0,\infty)$, and the solution
$M_{t}$ converges smoothly and exponentially as $t\to\infty$ to a geodesic
sphere of radius $\rho_{\infty}$ which encloses the same volume as $M_{0}$.

The volume preserving mean curvature flow

$\frac{\partial}{\partial t}X(x,t)=(\phi(t)-H)\nu(x,t)$ (1.3)

was introduced by Huisken [18] in 1987 for convex hypersurfaces in the
Euclidean space $\mathbb{R}^{n+1}$, and it has been proved that for any smooth
convex initial hypersurface, the solution of the flow (1.3) converges smoothly
to a round sphere. The analogue of the flow (1.3) in the hyperbolic space
$\mathbb{H}^{n+1}$ was studied by Cabezas-Rivas and Miquel [8] in 2007
assuming a stronger condition that the initial hypersurface is $h$-convex.
There are further generalizations of the flow (1.3) in $\mathbb{H}^{n+1}$ with
the mean curvature $H$ replaced by more general curvature functions including
powers of the $k$th mean curvature $\sigma_{k}(\kappa),k=1,\cdots,n$ (see [2,
5, 17, 22, 30] for instance). In all cases, the $h$-convexity is assumed on
the initial hypersurfaces. This is mainly because that the $h$-convexity is
convenient for the analysis of the curvature evolution equations and so that
the tensor maximum principle or the constant rank theorem can be applied to
derive that the $h$-convexity is preserved along the flow. Moreover, the
$h$-convexity is strong enough geometrically such that the outer radius of the
enclosed domain is uniformly controlled by its inner radius (see [6]), and
then the a prior $C^{0}$ estimate of the flow can be proved easily. In a
recent work [1], the first author with Andrews and Chen proved the smooth
convergence of volume preserving $k$th mean curvature flows in
$\mathbb{H}^{n+1}$ for initial hypersurfaces with positive sectional
curvatures. This condition is weaker than $h$-convexity but still stronger
than the convexity $\kappa_{i}>0.$

An open question in the field is whether the convexity ($\kappa_{i}>0$,
$i=1,\cdots,n$) is sufficient to guarantee the smooth convergence of the
volume preserving curvature flow in the hyperbolic space. Our result in
Theorem 1.2 provides the first affirmative answer to this question. The curve
case ($n=1$) of Theorem 1.2 was also treated recently by the first and second
authors in [31], where the idea is that in this case there is only one
curvature and we can calculate the curvature evolution explicitly to derive
the convexity preserving immediately. However, the higher dimensional case
($n\geq 2$) as stated in Theorem 1.2 requires on some new ideas. The key step
is again to show the convexity preserving, but the tensor maximum principle no
longer works when applying to the evolution of the second fundamental form
directly. Instead, we shall use the projection method via the Klein model of
the hyperbolic space that was described earlier by the first author and
Andrews in [2]. Based on this, we can treat the flow (1.1) as an equivalent
flow in the Euclidean space and this allows us to derive a time-dependent
positive lower bound on the principal curvatures, and a time-dependent upper
bound on the Gauss curvature $K$. A continuation argument then implies that
the flow exists for all positive time $t>0$.

To study the asymptotical behavior of the flow as time $t\to\infty$, we shall
use some machinery from the theory of convex bodies in $\mathbb{H}^{n+1}$. In
particular, we use the Blaschke selection theorem to show that for a
subsequence of times $t_{i}\to\infty$, the enclosed domain $\Omega_{t_{i}}$ of
$M_{t_{i}}$ converges in Hausdorff sense to a limit convex domain
$\hat{\Omega}$. We then apply the monotonicity of a certain quermassintegral
to show that $\hat{\Omega}$ satisfies

$\Phi_{0}(\hat{\Omega},\beta)=c\Phi_{n}(\hat{\Omega},\beta)$ (1.4)

for any Borel set $\beta$ in $\mathbb{H}^{n+1}$, where $\Phi_{0}$ and
$\Phi_{n}$ are the hyperbolic curvature measures of $\hat{\Omega}$, and $c$ is
a constant. A theorem of Kohlmann [19] which characterises the geodesic ball
in the hyperbolic space $\mathbb{H}^{n+1}(n\geq 2)$ using the equation (1.4)
for curvature measures can be used to conclude that $\hat{\Omega}$ is a
geodesic ball. Moreover, if we denote the center of the inner ball of
$\Omega_{t}$ as $p_{t}$, then using the Alexandrov reflection argument and the
subsequential Hausdorff convergence, we can deduce that for all time
$t\to\infty$, the points $p_{t}$ converge to a fixed point $p$, which is the
center of the ball $\hat{\Omega}$. With the help of this, we can prove the
uniform positive bounds for the principal curvatures and then obtain the
smooth convergence of the flow to a geodesic sphere.

The convergence result in Theorem 1.2 has an application in the Alexandrov-
Fenchel inequalities for quermassintegrals in the hyperbolic space (see §2 for
the definitions). Along the flow (1.1), we find that the $(n-1)$th
quermassintegral $\mathcal{A}_{n-1}(\Omega_{t})$ is monotone decreasing in
time $t$. As the volume $|\Omega_{t}|$ is preserved, the smooth convergence
proved in Theorem 1.2 yields the following Alexandrov-Fenchel inequality of
quermassintegrals for convex domains in the hyperbolic space:

######  Corollary 1.3.

Suppose that $\Omega\subset\mathbb{H}^{n+1}$ is a bounded weakly convex domain
with smooth boundary $\partial\Omega$. Then the following inequality holds:

$\mathcal{A}_{n-1}(\Omega)\geq\psi_{n}\left(|\Omega|\right),$ (1.5)

where $\psi_{n}:[0,\infty)\to\mathbb{R}$ is a strictly increasing function
such that the equality is achieved for geodesic balls. Moreover, the equality
holds in (1.5) if and only if $\Omega$ is a geodesic ball.

Here we say that a hypersurface is weakly convex if all principal curvatures
$\kappa_{i}\geq 0$. We remark that the inequality (1.5) for convex domains in
the hyperbolic space was proved also by Brendle, Guan and Li in a preprint [7]
using a different method. The argument we present here provides a new proof of
(1.5).

The paper is organized as follows: In $\S$2, we collect some preliminaries
including the geometry of hypersufaces in the hyperbolic space, the evolution
equations along the flow (1.1), the quermassintegrals and curvature measures
in the hyperbolic space. In §3, we project the flow to the Euclidean space via
the Klein model. This projection has the advantage that the second fundamental
forms of the corresponding solutions have a simple relation. In order to show
that convexity is preserving along the flow (1.1), it suffices to show that
the corresponding flow (3.10) in the Euclidean space preserves the convexity.
In $\S$4.1, we give the a priori $C^{0}$ and $C^{1}$ estimates of the flow
(1.1) and the projected flow (3.10), which follows by a similar argument as in
previous work [2, 1]. In $\S$4.2, we prove the positive lower bound for the
principal curvatures along the flow (1.1), by proving the corresponding
estimate along the flow (3.10) in the Euclidean space. In §4.3, we adapt
Tso’s[29] technique to prove an upper bound on the Gauss curvature on any
finite time interval. Since we only assumed convexity, the terms involving
global term $\phi(t)$ need to be carefully treated. This implies two-sided
curvature bounds of the solution on any finite time interval, and then we
obtain the long time existence of the flow (1.1) in $\S$5. In §6, we show the
subsequential Hausdorff convergence of $M_{t}$ and the convergence of the
center of the inner ball of $\Omega_{t}$ to a fixed point. Finally, in §7, we
complete the proofs of Theorem 1.2 and Corollary 1.3.

######  Acknowledgments.

The research was surpported by National Key R and D Program of China
2021YFA1001800 and 2020YFA0713100, National Natural Science Foundation of
China NSFC11721101 and Research grant KY0010000052 from University of Science
and Technology of China. B. Yang would like to thank Professor Jiayu Li for
his constant support.

##  2\. Preliminaries

In this section, we collect some preliminary results concerning the geometry
of hypersurfaces in the hyperbolic space, the evolution equations for
geometric quantities along the flow (1.1), the quermassintegrals and curvature
measures in the hyperbolic space.

###  2.1. Hyperbolic space

The hyperbolic space $\mathbb{H}^{n+1}$ can be viewed as a warped product
manifold $(\mathbb{R}_{+}\times\mathbb{S}^{n},g_{\mathbb{H}^{n+1}})$ with

$g_{\mathbb{H}^{n+1}}=d\rho^{2}+\sinh^{2}\rho g_{\mathbb{S}^{n}},$

where $g_{\mathbb{S}^{n}}$ is the round metric on unit sphere
$\mathbb{S}^{n}$. Let $D$ be the Levi-Civita connection on $\mathbb{H}^{n+1}$.
The vector field $V=\sinh\rho\partial_{\rho}$ is a conformal Killing field
satisfying $DV=\cosh\rho g_{\mathbb{H}^{n+1}}.$

Let $\Omega$ be a convex domain in $\mathbb{H}^{n+1}$ with a smooth boundary
$M=\partial\Omega$. Then $M$ is a smooth convex hypersurface in
$\mathbb{H}^{n+1}$. We denote by $g_{ij},h_{ij}$ and $\nu$ the induced metric,
the second fundamental form and the unit outward normal vector of $M$
respectively. As $M$ is convex, there exists a point $p_{0}\in\Omega$ such
that $M$ is star-shaped with respect to $p_{0}$ and can be written as a radial
graph $M=\\{(\rho(\theta),\theta),~{}\theta\in\mathbb{S}^{n}\\}$ with respect
to $p_{0}$ for a smooth function $\rho\in C^{\infty}(\mathbb{S}^{n})$.
Equivalently, the support function of $M$ with respect to $p_{0}\in\Omega$
defined by

$u=\langle V,\nu\rangle=\langle\sinh\rho\partial_{\rho},\nu\rangle$

is positive everywhere on $M$. It is well known that (see e.g.[16])

$\displaystyle g_{ij}$
$\displaystyle=\rho_{i}\rho_{j}+\sinh^{2}\rho\sigma_{ij},$ $\displaystyle
h_{ij}$
$\displaystyle=\frac{1}{\sqrt{\sinh^{2}\rho+|\bar{\nabla}\rho|^{2}}}(-(\sinh\rho)\rho_{ij}+2(\cosh\rho)\rho_{i}\rho_{j}+\sinh^{2}\rho\cosh\rho\sigma_{ij}),$
$\displaystyle\nu=$
$\displaystyle\frac{1}{\sqrt{1+|\bar{\nabla}\rho|^{2}/\sinh^{2}\rho}}\left(1,-\frac{\rho_{1}}{\sinh^{2}\rho},\cdots,-\frac{\rho_{n}}{\sinh^{2}\rho}\right),$

where $\bar{\nabla}$ is the covariant derivative on $\mathbb{S}^{n}$ with
respect to the round metric $g_{\mathbb{S}^{n}}=(\sigma_{ij})$ and
$\rho_{i}=\bar{\nabla}_{i}\rho,\rho_{ij}=\bar{\nabla}_{i}\bar{\nabla}_{j}\rho$.
It follows that the Gauss curvature $K$ of $M$ can be expressed as a function
of $\rho$ and its first and second derivatives:

$K=\frac{\det h_{ij}}{\det
g_{ij}}=\frac{\det(-\sinh\rho\rho_{ij}+2\cosh\rho\rho_{i}\rho_{j}+\sinh^{2}\rho\cosh\rho\sigma_{ij})}{(\sinh^{2}\rho+|\bar{\nabla}\rho|^{2})^{\frac{n+2}{2}}(\sinh\rho)^{2n-2}}.$
(2.1)

###  2.2. Evolution equations

Let $M_{t}$ be a smooth solution to the curvature flow (1.1) in the hyperbolic
space $\mathbb{H}^{n+1}$. We have the following evolution equations (see [2])
for the induced metric $g_{ij}$, the area element $d{\mu_{t}}$ and the speed
function $K^{\alpha}$:

$\displaystyle\frac{\partial}{\partial t}g_{ij}$
$\displaystyle=2\left(\phi(t)-K^{\alpha}\right)h_{ij},$ (2.2)
$\displaystyle\frac{\partial}{\partial t}d\mu_{t}$
$\displaystyle=H\left(\phi(t)-K^{\alpha}\right)d\mu_{t},$ (2.3)
$\displaystyle\frac{\partial}{\partial t}K^{\alpha}$ $\displaystyle=\alpha
K^{\alpha-1}\dot{K}^{ij}\left(\nabla_{i}\nabla_{j}{K^{\alpha}}+(K^{\alpha}-\phi(t))(h_{i}^{k}h_{kj}-\delta_{ij})\right),$
(2.4)

where $\dot{K}^{ij}$ denote the derivatives of $K$ with respect to the
components of the second fundamental form, and $\nabla$ denotes the Levi-
Civita connection on $M_{t}$ with respect to the induced metric $g_{ij}$. On
the time interval when $M_{t}$ is star-shaped with respect to some point
$p_{0}$, the support function
$u(x,t)=\langle{\sinh\rho_{p_{0}}(x)\partial_{\rho_{p_{0}}},\nu}\rangle$ of
$M_{t}$ with respect to $p_{0}$ evolves by (see [2, Lemma 4.3]):

$\frac{\partial}{\partial t}{u}=\alpha
K^{\alpha-1}\dot{K}^{ij}\nabla_{i}\nabla_{j}{u}+\cosh{\rho_{p_{0}}}(x)\left(\phi(t)-(n\alpha+1)K^{\alpha}\right)+\alpha
K^{\alpha}Hu.$ (2.5)

###  2.3. Quermassintegrals

Let $\mathcal{K}(\mathbb{H}^{n+1})$ be the set of compact convex sets in
$\mathbb{H}^{n+1}$ with nonempty interior. For any
$\Omega\in\mathcal{K}(\mathbb{H}^{n+1})$ , the quermassintegrals of $\Omega$
are defined as follows (see [27, Definition 2.1] 111Note that the definition
for $\mathcal{A}_{k}$ given here is the same as that for $W_{k+1}$ given in
[27] up to a constant. In fact, we have
$\mathcal{A}_{k}=(n+1)\binom{n}{k}W_{k+1}$.):

$\mathcal{A}_{k}(\Omega)=(n-k)\binom{n}{k}\frac{\omega_{k}\cdots\omega_{0}}{\omega_{n-1}\cdots\omega_{n-k-1}}\int_{\mathcal{L}_{k+1}}{\chi(L_{k+1}\cap\Omega)dL_{k+1}}$
(2.6)

for $k=0,1,\dots,n-1$, where $\omega_{k}=|\mathbb{S}^{k}|$ denotes the area of
$k$-dimensional unit sphere, $\mathcal{L}_{k+1}$ is the space of
$(k+1)$-dimensional totally geodesic subspaces $L_{k+1}$ in $\mathbb{H}^{n+1}$
and $\binom{n}{k}=\frac{n!}{k!(n-k)!}$. The function $\chi$ is defined to be 1
if $L_{k+1}\cap\Omega\neq\emptyset$ and to be 0 otherwise. In particular, we
have

$\mathcal{A}_{-1}(\Omega)=|\Omega|,\qquad\mathcal{A}_{0}(\Omega)=|\partial\Omega|,\qquad\mathcal{A}_{n}(\Omega)=\frac{\omega_{n}}{n+1}.$

If the boundary $M=\partial\Omega$ is smooth (or at least of class $C^{2}$),
we can define the principal curvatures $\kappa=(\kappa_{1},\dots,\kappa_{n})$
as the eigenvalues of the Weingarten matrix $\mathcal{W}$ of $M$. For each
$k\in\\{1,\dots,n\\}$, the $k$th mean curvature $\sigma_{k}$ of $M$ is then
defined as the $k$th elementary symmetric functions of the principal
curvatures of $M$:

$\sigma_{k}=\sum_{1\leq i_{1}<\cdots<i_{k}\leq
n}{\kappa_{i_{1}}\cdots\kappa_{i_{k}}}.$

These include the mean curvature $H=\sigma_{1}$ and Gauss curvature
$K=\sigma_{n}$ as special cases. In the smooth case, the quermassintegrals and
the curvature integrals of a smooth convex domain $\Omega$ in
$\mathbb{H}^{n+1}$ are related as follows:

$\displaystyle\mathcal{A}_{1}(\Omega)=$
$\displaystyle\int_{\partial\Omega}{\sigma_{1}}d\mu-n\mathcal{A}_{-1}(\Omega),$
(2.7) $\displaystyle\mathcal{A}_{k}(\Omega)=$
$\displaystyle\int_{\partial\Omega}{\sigma_{k}}d\mu-\frac{n-k+1}{k-1}\mathcal{A}_{k-2}(\Omega),\quad
k=2,\cdots,n.$ (2.8)

The quermassintegrals for smooth domains satisfy a nice variational property
(see [4]):

$\frac{d}{dt}\mathcal{A}_{k}(\Omega_{t})=(k+1)\int_{M_{t}}\eta\sigma_{k+1}d\mu_{t},\quad
k=0,\cdots,n-1$ (2.9)

along any normal variation with velocity $\eta$.

The quermassintegrals defined by (2.6) are monotone with respect to inclusion
of convex sets. That is, if $E,F\in\mathcal{K}(\mathbb{H}^{n+1})$ satisfy
$E\subset F$, we have

$\mathcal{A}_{k}(E)\leq\mathcal{A}_{k}(F)$ (2.10)

for all $k=0,1,\cdots,n$. Moreover, they are continuous with respect to the
Hausdoff distance. Recall that the Hausdorff distance between two convex sets
$\Omega,L\in\mathcal{K}(\mathbb{H}^{n+1})$ is defined as

$\mathrm{dist}_{\mathcal{H}}(\Omega,L):=\mathrm{inf}\\{\lambda>0:\Omega\subset
B_{\lambda}(L)\,\text{and}\,L\subset B_{\lambda}(\Omega)\\},$

where
$B_{\lambda}(L):=\\{q\in\mathbb{H}^{n+1}|~{}\mathrm{d}_{\mathbb{H}^{n+1}}(q,L)<\lambda\\}$.

######  Lemma 2.1.

Suppose that
$\\{\Omega_{i}\\}_{i=1}^{\infty},\Omega\in\mathcal{K}(\mathbb{H}^{n+1})$ and
$\Omega_{i}$ converges to $\Omega$ in the Hausdorff sense. Then we have
$\lim_{i\to\infty}{\mathcal{A}_{k}(\Omega_{i})}=\mathcal{A}_{k}(\Omega)$ for
all $k=-1,0,\cdots,n-1$.

To see this, we use an expression of the measure $dL_{k+1}$. Every totally
geodesic subspace $L_{k+1}$ in $\mathbb{H}^{n+1}$ is determined by its
orthogonal subspace $L_{n-k}[o]$, which passes through the origin
$o\in\mathbb{H}^{n+1}$, and by the intersection point $m=L_{k+1}\cap
L_{n-k}[o]$. In this way, $\mathcal{L}_{k+1}$ can be identified as a bundle
over the Grassmannian manifolds $G_{n+1,n-k}$ which consists of all subspaces
$L_{n-k}[o]$, and then $dL_{k+1}$ is written as (see [25])

$dL_{k+1}=\cosh^{k+1}(\rho)d\mu_{n-k}dL_{n-k}[o],$ (2.11)

where $d\mu_{n-k}$ is the volume element on $L_{n-k}[o]$, $dL_{n-k}[o]$ is the
volume element on $G_{n+1,n-k}$ and $\rho=d_{\mathbb{H}^{n+1}}(x,o)$ denotes
the distance of a point $x$ in $L_{n-k}[o]$ to the origin. Then we can rewrite
the integral in (2.6) equivalently as

$\int_{\mathcal{L}_{k+1}}{\chi(L_{k+1}\cap\Omega)dL_{k+1}}=\int_{G_{n+1,n-k}}\left(\int_{\Pi_{L_{n-k}[o]}(\Omega)}\cosh^{k+1}(\rho)d\mu_{n-k}\right)dL_{n-k}[o],$
(2.12)

where $\Pi_{L_{n-k}[o]}:\mathbb{H}^{n+1}\to\mathbb{H}^{n+1}$ is the orthogonal
projection over $L_{n-k}[o]\in G_{n+1,n-k}$ along geodesics. Then the
conclusion in Lemma 2.1 follows from (2.12) immediately.

###  2.4. Curvature measures in $\mathbb{H}^{n+1}$

For any $\Omega\in\mathcal{K}(\mathbb{H}^{n+1})$ and $\varepsilon>0$, define
the parallel set

$\Omega_{\varepsilon}=\\{x\in\mathbb{H}^{n+1}|~{}d_{\mathbb{H}^{n+1}}(x,\Omega)\leq\varepsilon\\}.$

The maps $f_{\Omega}:\mathbb{H}^{n+1}\setminus\Omega\to\partial\Omega$ and
$F_{\Omega}:\mathbb{H}^{n+1}\setminus\Omega\to
T_{\partial\Omega}\mathbb{H}^{n+1}$:

$d_{\mathbb{H}^{n+1}}(f_{\Omega}(x),x)=d_{\mathbb{H}^{n+1}}(x,\Omega)\quad\text{and}\quad
x=\mathrm{exp}_{f_{\Omega}(x)}^{\mathbb{H}^{n+1}}(d(\Omega,x)F_{\Omega}(x))$

are well-defined. For any Borel set $\beta\subset\mathbb{H}^{n+1}$, define the
local parallel set

$M_{\varepsilon}(\Omega,\beta)=f_{\Omega}^{-1}(\beta\cap\partial\Omega)\cap(\Omega_{\varepsilon}\setminus\Omega).$

$\Omega$$\Omega_{\varepsilon}$$M_{\varepsilon}(\Omega,\beta)$$\beta\cap\partial\Omega$
Figure 1. Local parallel set $M_{\varepsilon}(\Omega,\beta)$

Following [20], given a convex set $\Omega$ and $\varepsilon>0$, let us define
a Radon measure $\mu_{\varepsilon}$ on the Borel $\sigma$-algebra
$\mathcal{B}(\mathbb{H}^{n+1})$ of the hyperbolic space by

$\mu_{\varepsilon}(\Omega,\beta)=\mathrm{Vol}_{\mathbb{H}^{n+1}}(M_{\varepsilon}(\Omega,\beta)).$

Set

$l_{k+1}(t)=\int_{0}^{t}{\sinh^{k}(s)\cosh^{n-k}(s)ds},\quad k=0,1,\cdots,n$

and $l_{0}\equiv 1$. Then there holds the following Steiner-type formula:

$\mu_{\varepsilon}(\Omega,\beta)=\sum_{k=0}^{n}l_{n+1-k}(\varepsilon)\Phi_{k}(\Omega,\beta),\quad\forall\beta\in\mathcal{B}(\mathbb{H}^{n+1}).$

The coefficients $\Phi_{0}(\Omega,\cdot),\cdots,\Phi_{n}(\Omega,\cdot)$ are
called the curvature measures of the convex body $\Omega$, which are Borel
measures on $\mathbb{H}^{n+1}$.

When the boundary $\partial\Omega$ is smooth, $\Phi_{k}(\Omega,\cdot)$ has a
nice expression:

$\Phi_{k}(\Omega,\beta)=\int_{\partial\Omega\cap\beta}\sigma_{n-k}d\mu,$

where $\sigma_{n-k}$ is the $(n-k)$th mean curvature of $\partial\Omega$. We
refer [20] to the readers for a general integral representation for
$\Phi_{k}(\Omega,\cdot)$.

As in the classical setting of convex bodies in the Euclidean space, the
curvature measures introduced by Kohlmann are solid enough to be weakly
continuous with respect to the topology induced on
$\mathcal{K}(\mathbb{H}^{n+1})$.

######  Theorem 2.2 ([14]).

Let $\\{\Omega_{j}\\}_{j=1}^{\infty}\subset\mathcal{K}(\mathbb{H}^{n+1})$ be a
sequence of convex sets such that $\Omega_{j}\to\Omega$ as $j\to\infty$ in the
Hausdorff topology. Then for every $k=0,\cdots,n$ we have

$\Phi_{k}(\Omega_{j},\cdot)\to\Phi_{k}(\Omega,\cdot)$

as $j\to\infty$, weakly in the sense of measure.

The following characterization of geodesic balls via the hyperbolic curvature
measures was proved by Kohlmann [19] and can be viewed as a generalization of
the classical Alexandrov Theorem in differential geometry.

######  Theorem 2.3 ([19]).

Let $n\geq 2$ and $k\in\\{0,\dots,n-2\\}$. Assume that $\Omega$ is a compact,
connected, locally convex subset of $\mathbb{H}^{n+1}$ with nonempty interior
satisfying

$\Phi_{k}(\Omega,\beta)=c\Phi_{n}(\Omega,\beta)$ (2.13)

for any Borel set $\beta\in\mathcal{B}(\mathbb{H}^{n+1})$, where $c>0$ is a
constant, then $\Omega$ is a geodesic ball.

Note that there is no smoothness assumption on the domain $\Omega$ in Theorem
2.3. When the boundary $\partial\Omega$ is smooth, the equation (2.13) means
that the $(n-k)$th mean curvature $\sigma_{n-k}$ is a constant on
$\partial\Omega$, and the conclusion of Theorem 2.3 in this case reduces to
the classical result of Montiel and Ros [24].

##  3\. Klein model and Projection

To investigate the flow (1.1) for convex hypersurfaces, it is convenient to
project it to the Euclidean space $\mathbb{R}^{n+1}$ and apply the Gauss map
parametrization as in the work [2, §5] by the first author and Andrews. We
briefly review the argument here and refer the readers to [2] for details.

Let us denote by $\mathbb{R}^{1,n+1}$ the Minkowski spacetime, that is the
vector space $\mathbb{R}^{n+2}$ endowed with the Minkowski spacetime metric
$\langle\cdot,\cdot\rangle$ given by

$\langle X,X\rangle~{}=~{}-X_{0}^{2}+\sum_{i=1}^{n+1}X_{i}^{2}$

for any vector $X=(X_{0},X_{1},\cdots,X_{n+1})\in\mathbb{R}^{n+2}$. The
hyperbolic space $\mathbb{H}^{n+1}$ is then

$\mathbb{H}^{n+1}=\\{X\in\mathbb{R}^{1,n+1}~{}|~{}\langle X,X\rangle=-1,\
X_{0}>0\\}.$

The Klein model parametrizes the hyperbolic space using the unit disc, which
induces a projection from an embedding $X:M^{n}\rightarrow\mathbb{H}^{n+1}$ to
an embedding $Y:M^{n}\rightarrow B_{1}(0)\subset\mathbb{R}^{n+1}$ by

$X~{}=~{}\frac{(1,Y)}{\sqrt{1-|Y|^{2}}}.$ (3.1)

Let $\nu\in T\mathbb{H}^{n+1},g^{X}_{ij},h^{X}_{ij},K^{X}$ and
$N\in\mathbb{R}^{n+1},g^{Y}_{ij},h^{Y}_{ij},K^{Y}$ denote the unit normal
vectors, induced metrics, the second fundamental forms and Gauss curvatures of
$X(M^{n})\subset\mathbb{H}^{n+1}$ and $Y(M^{n})\subset\mathbb{R}^{n+1}$
respectively. We have the following relations:

$\displaystyle h^{X}_{ij}$
$\displaystyle=\frac{h^{Y}_{ij}}{\sqrt{\left(1-|Y|^{2}\right)\left(1-\langle
N,Y\rangle^{2}\right)}},$ (3.2) $\displaystyle g^{X}_{ij}$
$\displaystyle=\frac{1}{1-|Y|^{2}}\left(g^{Y}_{ij}+\frac{\langle
Y,\partial_{i}Y\rangle\langle Y,\partial_{j}Y\rangle}{1-|Y|^{2}}\right).$
(3.3)

It follows from (3.2) that $X(M^{n})$ is convex in $\mathbb{H}^{n+1}$ is
equivalent to that $Y(M^{n})$ is convex in $\mathbb{R}^{n+1}$. If we take the
determinant on both sides of (3.2) and (3.3), the Gauss curvatures satisfy

$K^{X}=\left(\frac{1-|Y|^{2}}{1-\langle
N,Y\rangle^{2}}\right)^{\frac{n+2}{2}}K^{Y}.$ (3.4)

Suppose that $X:M^{n}\times[0,T)\to\mathbb{H}^{n+1}$ is a smooth convex
solution to the flow (1.1). Then the corresponding solution
$Y:M^{n}\times[0,T)\to B_{1}(0)\subset\mathbb{R}^{n+1}$ satisfies the
evolution equation:

$\frac{\partial}{\partial t}Y=\sqrt{(1-|Y|^{2})(1-\langle
N,Y\rangle^{2})}\Big{(}\phi(t)-(K^{X})^{\alpha}\Big{)}N.$ (3.5)

Since each $Y_{t}(M^{n})=Y(M^{n},t)\subset B_{1}(0)\subset\mathbb{R}^{n+1}$ is
convex, we can parametrize $Y_{t}(M^{n})$ using the Gauss map and the support
function $s(z):=\langle Y_{t}(N^{-1}(z)),z\rangle$, where
$N^{-1}:\mathbb{S}^{n}\to M^{n}$ is the inverse map of the Gauss map. Then
$Y_{t}(M^{n})$ is given by the embedding $Y:\mathbb{S}^{n}\to\mathbb{R}^{n+1}$
with (see [32, §2])

$Y(z)=~{}s(z)z+\bar{\nabla}s$ (3.6)

where $\bar{\nabla}$ is the gradient with respect to the round metric
$g_{\mathbb{S}^{n}}$ on $\mathbb{S}^{n}$. The derivative of this map is given
by

$\partial_{i}Y=~{}\mathfrak{r}_{ik}\sigma^{kl}\partial_{l}z$

in local coordinates, where $\mathfrak{r}_{ij}$ is given as follows

$\mathfrak{r}_{ij}=~{}\bar{\nabla}_{i}\bar{\nabla}_{j}s+s\sigma_{ij}.$ (3.7)

The eigenvalues $\mathfrak{r}_{i}$ of $\mathfrak{r}_{ij}$ with respect to the
round metric $g_{\mathbb{S}^{n}}$ are the inverse of the principal curvatures
$\kappa_{i}^{Y}$, i.e., $\mathfrak{r}_{i}=1/{\kappa_{i}^{Y}}$, and are called
the principal radii of curvature. The determinant of (3.7) gives the Gauss
curvature of $Y_{t}(M^{n})$:

$K^{Y}=\frac{1}{\mathrm{det}(\mathfrak{r}_{ij})}.$ (3.8)

By the identity (3.4), we obtain

$K^{X}=\left(\frac{1-(s^{2}+|\bar{\nabla}s|^{2})}{1-s^{2}}\right)^{\frac{n+2}{2}}K^{Y}.$
(3.9)

The evolution equation (3.5) is equivalent to the following scalar parabolic
equation:

$\frac{\partial}{\partial
t}s=~{}\sqrt{(1-s^{2}-|\bar{\nabla}s|^{2})(1-s^{2})}\Big{(}\phi(t)-(K^{X})^{\alpha}\Big{)}$

on the round sphere $\mathbb{S}^{n}$ for the support function $s(z,t)$ of
$Y(M^{n},t)$.

We summarize the results as the following lemma:

######  Lemma 3.1.

Assume that $X(M^{n},t)\subset\mathbb{H}^{n+1},t\in[0,T)$ is a smooth convex
solution to the flow (1.1). Then up to a tangential diffeomorphism, the
corresponding solution $Y(M^{n},t)\subset B_{1}(0)\subset\mathbb{R}^{n+1}$ and
its support function $s(z,t)$ satisfy the following equations:

$\displaystyle\frac{\partial}{\partial t}Y=$
$\displaystyle~{}\sqrt{(1-|Y|^{2})(1-\langle
N,Y\rangle^{2})}\Big{(}\phi(t)-(K^{X})^{\alpha}\Big{)}N,$ (3.10)
$\displaystyle\frac{\partial}{\partial t}s=$
$\displaystyle~{}\sqrt{(1-s^{2}-|\bar{\nabla}s|^{2})(1-s^{2})}\Big{(}\phi(t)-(K^{X})^{\alpha}\Big{)},$
(3.11)

where the Gauss curvature $K^{X}$ is related to the Gauss curvature $K^{Y}$
via the identities (3.4) and (3.9).

For simplicity of the notation, we denote $A$ and $B$ as the following
functions:

$\left\\{\begin{aligned}
A=A(s,\bar{\nabla}s)&:=\sqrt{(1-s^{2}-|\bar{\nabla}s|^{2})(1-s^{2})},\\\
B=B(s,\bar{\nabla}s)&:=(1-s^{2}-|\bar{\nabla}s|^{2})^{\frac{n+2}{2}\alpha+\frac{1}{2}}(1-s^{2})^{-\frac{n+2}{2}\alpha+\frac{1}{2}}.\end{aligned}\right.$
(3.12)

Then (3.11) can be simplified as:

$\frac{\partial}{\partial t}s=A\phi(t)-B(K^{Y})^{\alpha}.$ (3.13)

In §4.2, we will estimate the upper bound of the largest eigenvalues of
$\mathfrak{r}_{ij}$ along the equation (3.13) and this implies a positive
lower bound on the principal curvatures of the solution $M_{t}$ to the
original flow (1.1). In the following lemma, we derive the evolution equation
of $\mathfrak{r}_{ij}$ along the flow (3.13):

######  Lemma 3.2.

Let $s(z,t)$ be a smooth solution of the flow (3.13). Choose a local
orthonormal frame $e_{1},\cdots,e_{n}$ around the point $z\in\mathbb{S}^{n}$
such that $(\mathfrak{r}_{ij})$ is diagonal at $(z,t)$ and $\mathfrak{r}_{11}$
corresponds to the largest eigenvalue of $(\mathfrak{r}_{ij})$. Along the
equation (3.13), we have the following evolution equation of
$\mathfrak{r}_{11}$:

$\displaystyle\frac{\partial}{\partial
t}\mathfrak{r}_{11}-F^{k\ell}\bar{\nabla}_{k}\bar{\nabla}_{\ell}\mathfrak{r}_{11}=$
$\displaystyle\phi(t)\Big{(}A_{s_{1}s_{1}}{\mathfrak{r}_{11}}^{2}+2A_{ss_{1}}s_{1}\mathfrak{r}_{11}-2A_{s_{1}s_{1}}\mathfrak{r}_{11}s+A_{s}\mathfrak{r}_{11}+A_{s_{k}}\bar{\nabla}_{k}{\mathfrak{r}_{11}}$
$\displaystyle+A+A_{ss}{s_{1}}^{2}-2A_{ss_{1}}ss_{1}-A_{s}s+A_{s_{1}s_{1}}s^{2}-A_{s_{1}}s_{1}\Big{)}$
$\displaystyle-(K^{Y})^{\alpha}\Big{(}B_{s_{1}s_{1}}{\mathfrak{r}_{11}}^{2}+2B_{ss_{1}}s_{1}\mathfrak{r}_{11}-2B_{s_{1}s_{1}}\mathfrak{r}_{11}s+B_{s}\mathfrak{r}_{11}$
$\displaystyle+B+B_{ss}{s_{1}}^{2}-2B_{ss_{1}}ss_{1}-B_{s}s+B_{s_{1}s_{1}}s^{2}-B_{s_{1}}s_{1}$
$\displaystyle+B_{s_{k}}\bar{\nabla}_{k}{\mathfrak{r}_{11}}+\alpha
B\mathfrak{r}^{kk}\mathfrak{r}^{\ell\ell}(\bar{\nabla}_{1}{\mathfrak{r}_{k\ell}})^{2}+{\alpha}^{2}B(\sum_{k}\mathfrak{r}^{kk}\bar{\nabla}_{1}\mathfrak{r}_{kk})^{2}$
$\displaystyle-2\alpha\mathfrak{r}^{kk}\bar{\nabla}_{1}\mathfrak{r}_{kk}(B_{s_{1}}\mathfrak{r}_{11}+B_{s}s_{1}-B_{s_{1}}s)+\alpha
B\mathfrak{r}^{kk}(\mathfrak{r}_{11}-\mathfrak{r}_{kk})\Big{)},$ (3.14)

where $F^{k\ell}=\alpha B(K^{Y})^{\alpha}\mathfrak{r}^{k\ell}$. The notations
such as $A_{s},A_{s_{k}}$ are partial derivatives of $A$ with respect to its
first and second arguments.

###### Proof.

We first recall that $\mathfrak{r}_{ij}$ satisfies a Codazzi-type identity

$\bar{\nabla}_{k}\mathfrak{r}_{ij}=\bar{\nabla}_{i}\mathfrak{r}_{kj}.$ (3.15)

This follows from the Ricci identity for commuting covariant derivatives on
$\mathbb{S}^{n}$. Differentiating (3.15) and commuting derivatives, we have
the Simons’ identity for the second derivatives

$\bar{\nabla}_{(i}\bar{\nabla}_{j)}\mathfrak{r}_{k\ell}=\bar{\nabla}_{(k}\bar{\nabla}_{\ell)}\mathfrak{r}_{ij}+\mathfrak{r}_{k\ell}\sigma_{ij}-\mathfrak{r}_{ij}\sigma_{k\ell},$
(3.16)

where the brackets denote symmetrisation. Let
$(\mathfrak{r}^{ij})=(\mathfrak{r}_{ij})^{-1}$ be the inverse of
$(\mathfrak{r}_{ij})$. The spatial derivatives of $K^{Y}$ satisfy

$\displaystyle\bar{\nabla}_{i}K^{Y}=$
$\displaystyle-\frac{1}{(\mathrm{det}(\mathfrak{r}_{ij}))^{2}}\bar{\nabla}_{i}\mathrm{det}(\mathfrak{r}_{ij})$
$\displaystyle=$
$\displaystyle-\frac{1}{\mathrm{det}(\mathfrak{r}_{ij})}\mathfrak{r}^{k\ell}\bar{\nabla}_{i}\mathfrak{r}_{k\ell}$
$\displaystyle=$
$\displaystyle-K^{Y}\mathfrak{r}^{k\ell}\bar{\nabla}_{i}\mathfrak{r}_{k\ell},$
(3.17)

and

$\displaystyle\bar{\nabla}_{j}\bar{\nabla}_{i}K^{Y}=$
$\displaystyle-K^{Y}\mathfrak{r}^{k\ell}\bar{\nabla}_{j}\bar{\nabla}_{i}\mathfrak{r}_{k\ell}-\bar{\nabla}_{j}K^{Y}\mathfrak{r}^{k\ell}\bar{\nabla}_{i}\mathfrak{r}_{k\ell}-K^{Y}\bar{\nabla_{j}}\mathfrak{r}^{k\ell}\bar{\nabla}_{i}\mathfrak{r}_{k\ell}$
$\displaystyle=$
$\displaystyle-K^{Y}\mathfrak{r}^{k\ell}\bar{\nabla}_{j}\bar{\nabla}_{i}\mathfrak{r}_{k\ell}+K^{Y}\left(\mathfrak{r}^{k\ell}\bar{\nabla}_{i}\mathfrak{r}_{k\ell}\right)\left(\mathfrak{r}^{pq}\bar{\nabla}_{j}\mathfrak{r}_{pq}\right)$
$\displaystyle\quad+K^{Y}\mathfrak{r}^{kq}\mathfrak{r}^{p\ell}\bar{\nabla}_{j}\mathfrak{r}_{pq}\bar{\nabla}_{i}\mathfrak{r}_{k\ell}.$
(3.18)

On the other hand, the spatial derivatives of $A(s,\bar{\nabla}s)$ can be
calculated as

$\displaystyle\bar{\nabla}_{i}A=$ $\displaystyle A_{s}s_{i}+A_{s_{k}}s_{ki}$
$\displaystyle=$ $\displaystyle
A_{s_{k}}\mathfrak{r}_{ki}+A_{s}s_{i}-sA_{s_{k}}\sigma_{ki},$ (3.19)

and

$\displaystyle\bar{\nabla}_{j}\bar{\nabla}_{i}A=$ $\displaystyle
A_{s}s_{ij}+A_{ss}s_{i}s_{j}+A_{ss_{\ell}}s_{\ell
j}s_{i}+A_{s_{k}s}s_{j}s_{ki}+A_{s_{k}s_{\ell}}s_{ki}s_{\ell
j}+A_{s_{k}}s_{kij}$ $\displaystyle=$ $\displaystyle
A_{s}(\mathfrak{r}_{ij}-s\sigma_{ij})+A_{ss}s_{i}s_{j}+A_{ss_{\ell}}(\mathfrak{r}_{\ell
j}-s\sigma_{\ell j})s_{i}$
$\displaystyle+A_{s_{k}s}s_{j}(\mathfrak{r}_{ki}-s\sigma_{ki})+A_{s_{k}s_{\ell}}(\mathfrak{r}_{ki}-s\sigma_{ki})(\mathfrak{r}_{\ell
j}-s\sigma_{\ell j})$
$\displaystyle+A_{s_{k}}\left(\bar{\nabla}_{j}\mathfrak{r}_{ki}-s_{j}\sigma_{ki}\right)$
$\displaystyle=$ $\displaystyle
A_{s_{k}}\bar{\nabla}_{k}\mathfrak{r}_{ij}+A_{s}\mathfrak{r}_{ij}+A_{ss_{\ell}}s_{i}\mathfrak{r}_{\ell
j}+A_{s_{k}s}s_{j}\mathfrak{r}_{ki}+A_{s_{k}s_{\ell}}\mathfrak{r}_{ki}\mathfrak{r}_{\ell
j}$ $\displaystyle-A_{s_{k}s_{l}}\mathfrak{r}_{ki}\sigma_{\ell
j}s-A_{s_{k}s_{\ell}}\mathfrak{r}_{\ell
j}\sigma_{ki}s-A_{s}s\sigma_{ij}+A_{ss}s_{i}s_{j}-A_{ss_{\ell}}\sigma_{\ell
j}ss_{i}$ $\displaystyle-
A_{s_{k}s}\sigma_{ki}s_{j}s+A_{s_{k}s_{\ell}}\sigma_{ki}\sigma_{lj}s^{2}-A_{s_{k}}\sigma_{ki}s_{j},$
(3.20)

where $A_{s},A_{s_{k}}$ are partial derivatives of $A$ with respect to its
first and second arguments, and we used (3.7) and the Codazzi-type identity
(3.15). The calculation for $B(s,\bar{\nabla}s)$ is similar.

Taking the spatial derivatives to the equation (3.13) and using
(3.17)$-$(3.20), we compute that

$\displaystyle\bar{\nabla}_{1}\bar{\nabla}_{1}(\partial_{t}s)=$
$\displaystyle\phi(t)\bar{\nabla}_{1}\bar{\nabla}_{1}A-(K^{Y})^{\alpha}\bar{\nabla}_{1}\bar{\nabla}_{1}B-2\alpha(K^{Y})^{\alpha-1}\bar{\nabla}_{1}K^{Y}\bar{\nabla}_{1}B$
$\displaystyle-\alpha(K^{Y})^{\alpha-1}\bar{\nabla}_{1}\bar{\nabla}_{1}K^{Y}B-\alpha(\alpha-1)(K^{Y})^{\alpha-2}(\bar{\nabla}_{1}K^{Y})^{2}B$
$\displaystyle=$
$\displaystyle\phi(t)\bar{\nabla}_{1}\bar{\nabla}_{1}A-(K^{Y})^{\alpha}\Big{(}\bar{\nabla}_{1}\bar{\nabla}_{1}B-2\alpha\mathfrak{r}^{k\ell}\bar{\nabla}_{1}\mathfrak{r}_{k\ell}\bar{\nabla}_{1}B$
$\displaystyle-\alpha
B\mathfrak{r}^{k\ell}\bar{\nabla}_{1}\bar{\nabla}_{1}\mathfrak{r}_{k\ell}+\alpha^{2}B\big{(}\sum_{k,\ell}\mathfrak{r}^{k\ell}\bar{\nabla}_{1}\mathfrak{r}_{k\ell}\big{)}^{2}+\alpha
B\mathfrak{r}^{kq}\mathfrak{r}^{p\ell}\bar{\nabla}_{1}\mathfrak{r}_{pq}\bar{\nabla}_{1}\mathfrak{r}_{k\ell}\Big{)}$
$\displaystyle=$
$\displaystyle\phi(t)\Big{(}A_{s_{k}}\bar{\nabla}_{k}{\mathfrak{r}_{11}}+A_{s}\mathfrak{r}_{11}+2A_{ss_{1}}s_{1}\mathfrak{r}_{11}+A_{s_{1}s_{1}}{\mathfrak{r}_{11}^{2}}-2A_{s_{1}s_{1}}\mathfrak{r}_{11}s$
$\displaystyle-
A_{s}s+A_{ss}{s_{1}}^{2}-2A_{ss_{1}}ss_{1}+A_{s_{1}s_{1}}s^{2}-A_{s_{1}}s_{1}\Big{)}$
$\displaystyle-(K^{Y})^{\alpha}\Big{(}B_{s_{k}}\bar{\nabla}_{k}{\mathfrak{r}_{11}}+B_{s}\mathfrak{r}_{11}+2B_{ss_{1}}s_{1}\mathfrak{r}_{11}+B_{s_{1}s_{1}}{\mathfrak{r}_{11}^{2}}-2B_{s_{1}s_{1}}\mathfrak{r}_{11}s$
$\displaystyle-
B_{s}s+B_{ss}{s_{1}}^{2}-2B_{ss_{1}}ss_{1}+B_{s_{1}s_{1}}s^{2}-B_{s_{1}}s_{1}$
$\displaystyle-2\alpha\mathfrak{r}^{k\ell}\bar{\nabla}_{1}\mathfrak{r}_{k\ell}(B_{s_{1}}\mathfrak{r}_{11}+B_{s}s_{1}-B_{s_{1}}s)-\alpha
B\mathfrak{r}^{k\ell}\bar{\nabla}_{1}\bar{\nabla}_{1}{\mathfrak{r}_{k\ell}}$
$\displaystyle+{\alpha}^{2}B(\sum_{k,\ell}\mathfrak{r}^{k\ell}\bar{\nabla}_{1}\mathfrak{r}_{k\ell})^{2}+\alpha
B\mathfrak{r}^{ik}\mathfrak{r}^{j\ell}\bar{\nabla}_{1}{\mathfrak{r}_{ij}}\bar{\nabla}_{1}{\mathfrak{r}_{k\ell}}\Big{)}.$
(3.21)

Let $F^{k\ell}=\alpha B(K^{Y})^{\alpha}\mathfrak{r}^{k\ell}$. Taking
derivatives of (3.7),

$\displaystyle\frac{\partial}{\partial
t}\mathfrak{r}_{11}-F^{k\ell}\bar{\nabla}_{k}\bar{\nabla}_{\ell}\mathfrak{r}_{11}=$
$\displaystyle\bar{\nabla}_{1}\bar{\nabla}_{1}(\partial_{t}s)+\partial_{t}s-F^{k\ell}\bar{\nabla}_{k}\bar{\nabla}_{\ell}\mathfrak{r}_{11}.$
(3.22)

Substituting (3.13), (3.16), (3) into (3.22), using the fact that
$(\mathfrak{r}_{k\ell})$ is diagonal at the point we are considering and
rearranging the terms, we obtain the equation (3.2). ∎

We also calculated the evolution equation of
$r^{2}=|Y|^{2}=s^{2}+|\bar{\nabla}s|^{2}$.

######  Lemma 3.3.

Let $s(z,t)$ be a smooth solution of the flow (3.13). If we choose a local
orthonormal frame $e_{1},\cdots,e_{n}$ around the point $z\in\mathbb{S}^{n}$,
then we have:

$\displaystyle\frac{\partial}{\partial
t}r^{2}-F^{k\ell}\bar{\nabla}_{k}\bar{\nabla}_{\ell}r^{2}$ $\displaystyle=$
$\displaystyle
2\phi(t)\Big{(}sA+A_{s}|\bar{\nabla}s|^{2}+A_{s_{k}}s_{i}\mathfrak{r}_{ki}-A_{s_{k}}s_{k}s\Big{)}$
$\displaystyle-$ $\displaystyle
2(K^{Y})^{\alpha}\Big{(}(1-n\alpha)sB+B_{s}|\bar{\nabla}s|^{2}+B_{s_{k}}s_{i}\mathfrak{r}_{ki}-B_{s_{k}}s_{k}s+\alpha
B\sum_{k}{\mathfrak{r}_{kk}}\Big{)},$ (3.23)

where $F^{k\ell}=\alpha B(K^{Y})^{\alpha}\mathfrak{r}^{k\ell}$.

###### Proof.

Since $r^{2}=s^{2}+|\bar{\nabla}s|^{2}$, using (3.13), (3.17) and (3.19), we
have

$\displaystyle\partial_{t}r^{2}=$ $\displaystyle 2ss_{t}+2s_{i}s_{it}$
$\displaystyle=$ $\displaystyle
2s(A\phi(t)-B(K^{Y})^{\alpha})+2s_{i}\Big{(}\phi(t)\bar{\nabla}_{i}A-(K^{Y})^{\alpha}\bar{\nabla}_{i}B+\alpha
B(K^{Y})^{\alpha}\mathfrak{r}^{k\ell}\bar{\nabla}_{i}\mathfrak{r}_{k\ell}\Big{)}$
$\displaystyle=$ $\displaystyle
2\phi(t)\left(sA+A_{s}|\bar{\nabla}s|^{2}+A_{s_{k}}s_{i}\mathfrak{r}_{ki}-A_{s_{k}}s_{i}\sigma_{ki}s\right)$
$\displaystyle-2(K^{Y})^{\alpha}\left(sB+B_{s}|\bar{\nabla}s|^{2}+B_{s_{k}}s_{i}\mathfrak{r}_{ki}-B_{s_{k}}s_{i}\sigma_{ki}s-\alpha
B\mathfrak{r}^{k\ell}\bar{\nabla}_{i}\mathfrak{r}_{k\ell}s_{i}\right),$

and

$\displaystyle\bar{\nabla}_{k}\bar{\nabla}_{\ell}r^{2}=$ $\displaystyle
2\bar{\nabla}_{k}\left(ss_{\ell}+s_{i}s_{i\ell}\right)$ $\displaystyle=$
$\displaystyle 2s_{k}s_{\ell}+2ss_{k\ell}+2s_{ik}s_{i\ell}+2s_{i}s_{ik\ell}$
$\displaystyle=$ $\displaystyle
2\mathfrak{r}_{ik}\mathfrak{r}_{i\ell}-2s\mathfrak{r}_{k\ell}+2s_{i}\bar{\nabla}_{i}\mathfrak{r}_{k\ell}.$

Combining the above two equations gives the equation (3.3). ∎

We conclude the section with following remarks.

######  Remark 3.4.

(i). The projection method via Klein model has also been used recently by Chen
and Huang [10] to study the contracting Gauss curvature flows in the
hyperbolic space.

(ii). Denote $\mathcal{W}^{X}$ the Weingarten matrix of
$X(M^{n})\subset\mathbb{H}^{n+1}$. Then (3.2) and (3.3) imply that the inverse
matrix $\mathcal{W}_{X}^{-1}$ of $\mathcal{W}^{X}$ satisfies

$\displaystyle(\mathcal{W}_{X}^{-1})_{ij}=$
$\displaystyle(h^{-1}_{X})^{jk}g_{ki}^{X}$ $\displaystyle=$
$\displaystyle(h^{-1}_{Y})^{kj}\left(g_{ki}^{Y}+\frac{\langle
Y,\partial_{i}Y\rangle\langle
Y,\partial_{k}Y\rangle}{(1-|Y|^{2})}\right)\sqrt{\frac{1-\langle
N,Y\rangle^{2}}{1-|Y|^{2}}}.$ (3.24)

We can also express (3.4) using $s,\bar{\nabla}s$ and $\mathfrak{r}_{ij}$ as
follows:

$\displaystyle(\mathcal{W}_{X}^{-1})_{ij}=$
$\displaystyle\left(\sigma^{jq}+\frac{\langle\sigma^{ja}\bar{\nabla}_{a}s,\sigma^{qb}\bar{\nabla}_{b}s\rangle}{1-s^{2}-|\bar{\nabla}s|^{2}}\right)\mathfrak{r}_{qi}\sqrt{\frac{1-s^{2}}{1-s^{2}-|\bar{\nabla}s|^{2}}}.$
(3.25)

Since the eigenvalues of $(\mathcal{W}_{X}^{-1})_{ij}$ correspond to the
reciprocal of the principal curvatures $\kappa_{1},\cdots,\kappa_{n}$ of
$X(M^{n},t)\subset\mathbb{H}^{n+1}$, the above equation means that in order to
estimate the lower bound of $\kappa_{i}$, it suffices to estimate the upper
bound of $\mathfrak{r}_{ij}$ together with the $C^{0},C^{1}$ estimates of $s$.
This observation will be used in the next section. Moreover, we believe that
(3.25) would also be useful for studying general fully nonlinear curvature
flows in the hyperbolic space via the projection method described in this
section.

##  4\. A priori estimate

In this section, we first review the $C^{0}$ and $C^{1}$ estimates of the
solution $M_{t}$ to the flow (1.1), which can be proved using a similar
argument as in [1, 2]. Then we derive a positive lower bound on the principal
curvatures $\kappa_{i}$ of the solution $M_{t}$, and prove the upper bound of
the Gauss curvature $K$. Both of them may depend on the time $t$.

###  4.1. $C^{0}$ and $C^{1}$ estimates

Let $M_{0}$ be a smooth, closed and convex hypersurface in the hyperbolic
space. The flow (1.1) has a unique smooth solution $M_{t}$ for at least a
short time and $M_{t}$ is convex on a possibly shorter time interval. Without
loss of generality, we suppose that $M_{t}$ a smooth convex solution to the
flow (1.1) starting from $M_{0}$ on a maximum time interval $[0,T)$, where
$T\leq\infty$.

Denote by $\Omega_{t}$ as the domain enclosed by $M_{t}$ such that
$\partial\Omega_{t}=M_{t}$. As the velocity of the flow (1.1) only depends on
the curvature which is invariant under reflection with respect to a totally
geodesic hyperplane, we can argue as in [1, §4] using the Alexandrov
reflection method to show that the inner radius and outer radius of
$\Omega_{t}$ are uniformly bounded.

######  Lemma 4.1.

Let $M_{t}$ be the smooth convex solution to the flow (1.1) on the time
interval $t\in[0,T)$. Denote $\rho_{-}(t)$, $\rho_{+}(t)$ be the inner radius
and outer radius of $\Omega_{t}$. Then there exist positive constants $c_{1},\
c_{2}$ depending only on $n$, $\alpha$ and $M_{0}$ such that

$0<c_{1}\leq\rho_{-}(t)\leq\rho_{+}(t)\leq c_{2}$ (4.1)

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

By (4.1), the inner radius of $\Omega_{t}$ is bounded below by a positive
constant $c_{1}$. This implies that for each $t\in[0,T)$ there exists a
geodesic ball $B_{c_{1}}(p_{t})$ of radius $c_{1}$ centered at some point
$p_{t}$ such that $B_{c_{1}}(p_{t})\subset\Omega_{t}$. A similar argument as
in [2, Lemma 4.2] yields the existence of a geodesic ball with fixed center
enclosed by the flow hypersurface on a suitable fixed time interval.

######  Lemma 4.2.

Let $M_{t}$ be the smooth convex solution to the flow (1.1) on the time
interval $[0,T)$. For any $t_{0}\in[0,T)$, let $B_{\rho_{0}}(p_{0})$ be the
inball of $\Omega_{t_{0}}$, where $\rho_{0}=\rho_{-}(t_{0})$. Then

$B_{\rho_{0}/2}(p_{0})\subset\Omega_{t},\ \
t\in[t_{0},\min\\{T,t_{0}+\tau\\})$ (4.2)

for some $\tau$ depending only on $n$, $\alpha$ and $M_{0}$.

Since the projection (3.1) is a diffeomorphism, by the equation (3.2) we know
that if the flow (1.1) has a smooth convex solution on the maximal time
interval $[0,T)$, then so does the projected flow (3.10). Given
$t_{0}\in[0,T)$ and let $B_{\rho_{0}}(p_{0})$ be the inball of
$\Omega_{t_{0}}$, where $\rho_{0}=\rho_{-}(t_{0})$.

Consider the support function
$u(x,t)=\sinh\rho_{p_{0}}\langle\partial_{\rho_{p_{0}}},\nu\rangle$ of $M_{t}$
with respect to the point $p_{0}$, where $\rho_{p_{0}}$ is the distance
function in $\mathbb{H}^{n+1}$ from the point $p_{0}$. Since $M_{t}$ is
convex, by (4.1) and (4.2), we see

$\begin{split}u(x,t)&\geq\sinh\left(\frac{\rho_{0}}{2}\right)\geq\sinh\left(\frac{c_{1}}{2}\right)=:2c,\\\
u(x,t)&\leq\sinh(2c_{2}),\end{split}$ (4.3)

and

$0<\frac{c_{1}}{2}\leq\rho_{p_{0}}(t)\leq 2c_{2}<\infty$ (4.4)

for any $t\in[t_{0},\min\\{T,t_{0}+\tau\\})$. Assume that $p_{0}$ is the
origin of $\mathbb{H}^{n+1}$, we project the flow (1.1) in $\mathbb{H}^{n+1}$
onto $B_{1}(0)\subset\mathbb{R}^{n+1}$ with respect to the original point
$p_{0}$. Then on the time interval $[t_{0},\min\\{T,t_{0}+\tau\\})$, the
Euclidean distance $r$ from the origin satisfies $r=\tanh{\rho_{p_{0}}}$ and
is uniformly bounded from below and above. Under the Gauss map parametrization
(3.6), the support function $s$ of $Y_{t}(M^{n})$ satisfies

$r^{2}=s^{2}+|\bar{\nabla}s|^{2}.$

We conclude with the following $C^{0}$ and $C^{1}$ estimates:

######  Lemma 4.3.

Let $M_{t}$ be the smooth convex solution of the flow (1.1) on the time
interval $[0,T)$. Given $t_{0}\in[0,T)$ and $\tau$ be defined as in Lemma 4.2.
Let $B_{\rho_{0}}(p_{0})$ be the inball of $\Omega_{t_{0}}$ with
$\rho_{0}=\rho_{-}(t_{0})$ and we project $\mathbb{H}^{n+1}$ onto
$B_{1}(0)\subset\mathbb{R}^{n+1}$ with respect to the original point $p_{0}$.
We have the following estimates of the projected flow (3.10):

$0<c_{3}\leq 1-r^{2}\leq 1-s^{2}\leq c_{4}<1,\qquad|s|\leq
c_{5}<1,\qquad|\bar{\nabla}s|\leq c_{6}<1$ (4.5)

for any $t\in[t_{0},\min\\{T,t_{0}+\tau\\})$, where $\\{c_{i}\\}_{i=3,4,5,6}$
are positive constants depending only on $n$, $\alpha$ and $M_{0}$.

###  4.2. Preserving convexity

In this subsection, we prove the lower bound on the principal curvatures of
the solution $M_{t}$ of the flow (1.1).

######  Proposition 4.4.

Let $M_{0}$ be a smooth, closed and convex hypersurface in $\mathbb{H}^{n+1}$
and $M_{t},t\in[0,T)$, be the smooth solution of the flow (1.1) starting from
$M_{0}$. Then there exist constants $\Lambda_{1}$ and $\Lambda_{3}$ depending
only on $n$, $\alpha$ and $M_{0}$ such that the principal curvatures
$\kappa_{i}$ of $M_{t}$ satisfy

$\kappa_{i}\geq\Lambda_{3}^{-1}\Lambda_{1}^{-\frac{2t}{\tau}}$ (4.6)

for all $i\in\\{1,\dots,n\\}$ and $t\in[0,T)$, where $\tau$ is the constant in
Lemma 4.2.

###### Proof.

Since $M_{0}$ is convex, by continuity the solution $M_{t}$ is convex for at
least a short time. Without loss of generality, we assume that the solution is
convex on the time interval $[0,T)$ and aim to derive the estimate (4.6) on
this interval $[0,T)$. In fact, if $T_{1}<T$ is the largest time such that
$M_{t}$ is convex for all $0\leq t<T_{1}$, then the estimate (4.6) implies
that the principal curvatures of $M_{T_{1}}$ has a positive lower bound and
this contradicts with the maximality of $T_{1}$. Therefore, $M_{t}$ remains
convex for all time $t\in[0,T)$ and satisfies the estimate (4.6).

So in the following, we will derive the estimate (4.6) on the interval $[0,T)$
where the solution $M_{t}$ is convex. Given any fixed time $t_{0}\in[0,T)$ and
$\tau$ be defined as in Lemma 4.2. Let $B_{\rho_{0}}(p_{0})$ be the inball of
$\Omega_{t_{0}}$, where $\rho_{0}=\rho_{-}(t_{0})$. We consider the solution
$Y_{t}(M^{n})$ of the projected flow (3.10) on $\mathbb{R}^{n+1}$ with respect
to the original point $p_{0}$. Let $\mathfrak{r}_{ij}=s_{ij}+s\sigma_{ij}$ be
the inverse of the Weingarten matrix of the projected hypersurface $Y_{t}$
with respect to an orthonormal frame on $\mathbb{S}^{n}$, whose eigenvalues
$(\mathfrak{r}_{1},\dots,\mathfrak{r}_{n})$ are the principal radii of
curvature of $Y_{t}$. As the principal curvatures $\kappa_{i}$ of
$M_{t}=X(M^{n},t)$ are the eigenvalues of the Weingarten matrix
$\mathcal{W}_{X}$, by the expression (3.25) and the $C^{0},C^{1}$ estimates in
Lemma 4.3, in order to prove the lower bound of $\kappa_{i}$, it suffices to
prove the upper bound of
$\mathfrak{r}(z,t)=\max_{i=1,\dots,n}{\mathfrak{r}_{i}(z,t)}$.

For this purpose, we consider the function

${G}(z,t):=\log{\mathfrak{r}}+\frac{L}{2}r^{2},\qquad
t\in[t_{0},\min\\{T,t_{0}+\tau\\}),$

where $L>0$ is a large constant to be determined. Suppose that the maximum of
${G}$ on $\mathbb{S}^{n}\times[t_{0},\min\\{T,t_{0}+\tau\\})$ is attained at
$(\bar{z},\bar{t})$. We choose a local orthonormal frame $e_{1},\dots,e_{n}$
around $\bar{z}$ such $\\{\mathfrak{r}_{ij}(z,t)\\}$ is diagonal at
$(\bar{z},\bar{t})$ and
$\mathfrak{r}_{11}(\bar{z},\bar{t})=\mathfrak{r}_{1}(\bar{z},\bar{t})=\mathfrak{r}(\bar{z},\bar{t})$.
Then without loss of generality, we can view the function $G$ as the following
form

$G(z,t)=\log{\mathfrak{r}_{11}(z,t)}+\frac{L}{2}r^{2}.$ (4.7)

If $\bar{t}=t_{0}$, we have

$G(z,t)\leq\max_{z\in\mathbb{S}^{n}}G(z,t_{0})~{}\leq\log\max_{z\in\mathbb{S}^{n}}\mathfrak{r}_{11}(z,t_{0})+\frac{L}{2}(1-c_{3})$
(4.8)

for $(z,t)\in\mathbb{S}^{n}\times[t_{0},\min\\{T,t_{0}+\tau\\})$. In the
following, we assume $\bar{t}>t_{0}$. We shall apply the maximum principle to
the evolution equation of $G$ to derive the upper bound of
$\mathfrak{r}_{11}$.

We have calculated the evolution equations of $\mathfrak{r}_{11}$ and of
$r^{2}$ in Lemma 3.2 and Lemma 3.3. Since $(\bar{z},\bar{t})$ is a maximum
point of $G$, at $(\bar{z},\bar{t})$ there hold

$\displaystyle 0=\bar{\nabla}_{i}G=$
$\displaystyle\frac{\bar{\nabla}_{i}\mathfrak{r}_{11}}{\mathfrak{r}_{11}}+L(ss_{i}+s_{k}s_{ki})=\frac{\bar{\nabla}_{i}\mathfrak{r}_{11}}{\mathfrak{r}_{11}}+Ls_{k}\mathfrak{r}_{ki}$
(4.9)

and

$\displaystyle 0$
$\displaystyle\leq\partial_{t}G-F^{k\ell}\bar{\nabla}_{k}\bar{\nabla}_{\ell}G$
$\displaystyle=\frac{1}{\mathfrak{r}_{11}}(\partial_{t}\mathfrak{r}_{11}-F^{k\ell}\bar{\nabla}_{k}\bar{\nabla}_{\ell}\mathfrak{r}_{11})+\frac{L}{2}(\partial_{t}r^{2}-F^{k\ell}\bar{\nabla}_{k}\bar{\nabla}_{\ell}r^{2})+F^{k\ell}\frac{\bar{\nabla}_{k}\mathfrak{r}_{11}\bar{\nabla}_{\ell}\mathfrak{r}_{11}}{(\mathfrak{r}_{11})^{2}}$
$\displaystyle=:Q_{1}+Q_{2},$ (4.10)

where $Q_{1}$ involves the term $\phi(t)$:

$\displaystyle Q_{1}=$
$\displaystyle\frac{\phi(t)}{\mathfrak{r}_{11}}\Big{(}A_{s_{1}s_{1}}{\mathfrak{r}_{11}}^{2}+2A_{ss_{1}}s_{1}\mathfrak{r}_{11}-2A_{s_{1}s_{1}}\mathfrak{r}_{11}s+A_{s}\mathfrak{r}_{11}+A_{s_{k}}\bar{\nabla}_{k}{\mathfrak{r}_{11}}$
$\displaystyle+A+A_{ss}{s_{1}}^{2}-2A_{ss_{1}}ss_{1}-A_{s}s+A_{s_{1}s_{1}}s^{2}-A_{s_{1}}s_{1}\Big{)}$
$\displaystyle+\phi(t)L\Big{(}sA+A_{s}|\bar{\nabla}s|^{2}+A_{s_{k}}s_{i}\mathfrak{r}_{ki}-A_{s_{k}}s_{i}\sigma_{ki}s\Big{)},$
(4.11)

and $Q_{2}$ involves the term $(K^{Y})^{\alpha}$:

$\displaystyle Q_{2}=$
$\displaystyle-\frac{(K^{Y})^{\alpha}}{{\mathfrak{r}}_{11}}\Big{(}B_{s_{1}s_{1}}{\mathfrak{r}_{11}}^{2}+2B_{ss_{1}}s_{1}\mathfrak{r}_{11}-2B_{s_{1}s_{1}}\mathfrak{r}_{11}s+B_{s}\mathfrak{r}_{11}$
$\displaystyle+B+B_{ss}{s_{1}}^{2}-2B_{ss_{1}}ss_{1}-B_{s}s+B_{s_{1}s_{1}}s^{2}-B_{s_{1}}s_{1}$
$\displaystyle+B_{s_{k}}\bar{\nabla}_{k}{\mathfrak{r}_{11}}+\alpha
B\mathfrak{r}^{kk}\mathfrak{r}^{\ell\ell}(\bar{\nabla}_{1}{\mathfrak{r}_{k\ell}})^{2}+{\alpha}^{2}B(\sum_{k}\mathfrak{r}^{kk}\bar{\nabla}_{1}\mathfrak{r}_{kk})^{2}$
$\displaystyle-2\alpha\mathfrak{r}^{kk}\bar{\nabla}_{1}\mathfrak{r}_{kk}(B_{s_{1}}\mathfrak{r}_{11}+B_{s}s_{1}-B_{s_{1}}s)+\alpha
B\mathfrak{r}^{kk}(\mathfrak{r}_{11}-\mathfrak{r}_{kk})\Big{)}$
$\displaystyle-(K^{Y})^{\alpha}L\Big{(}(1-n\alpha)sB+B_{s}|\bar{\nabla}s|^{2}+B_{s_{k}}s_{i}\mathfrak{r}_{ki}-B_{s_{i}}s_{i}s+\alpha
B\sum_{k}{\mathfrak{r}_{kk}}\Big{)}$
$\displaystyle+F^{k\ell}\frac{\bar{\nabla}_{k}\mathfrak{r}_{11}\bar{\nabla}_{\ell}\mathfrak{r}_{11}}{(\mathfrak{r}_{11})^{2}}.$
(4.12)

In the following, we estimate the terms $Q_{1}$ and $Q_{2}$ separately. First,
we use (4.9) to cancel the term on $\bar{\nabla}_{k}{\mathfrak{r}_{11}}$ in
(4.2) and get

$\displaystyle Q_{1}=$
$\displaystyle\frac{\phi(t)}{\mathfrak{r}_{11}}\Big{(}A_{s_{1}s_{1}}{\mathfrak{r}_{11}}^{2}+2A_{ss_{1}}s_{1}\mathfrak{r}_{11}-2A_{s_{1}s_{1}}\mathfrak{r}_{11}s+A_{s}\mathfrak{r}_{11}$
$\displaystyle+A+A_{ss}{s_{1}}^{2}-2A_{ss_{1}}ss_{1}-A_{s}s+A_{s_{1}s_{1}}s^{2}-A_{s_{1}}s_{1}\Big{)}$
$\displaystyle+\phi(t)L\Big{(}sA+A_{s}|\bar{\nabla}s|^{2}-A_{s_{i}}s_{i}s\Big{)}.$
(4.13)

Since

$A_{s_{1}s_{1}}=-\frac{(1-s^{2})^{\frac{1}{2}}}{(1-{s}^{2}-|\bar{\nabla}{s}|^{2})^{\frac{3}{2}}}\left(1-({s}^{2}+|\bar{\nabla}{s}|^{2})+s_{1}^{2}\right)<0$

and the coefficients such as $A,A_{s},A_{ss_{1}},\cdots$ depend only on $s$
and $\bar{\nabla}s$ which are uniformly bounded on the interval
$[t_{0},\min\\{T,t_{0}+\tau\\})$, we obtain the estimate:

$\displaystyle Q_{1}\leq$
$\displaystyle~{}\frac{\phi(t)}{{\mathfrak{r}}_{11}}\left(a_{1}+(a_{2}+a_{3}L){\mathfrak{r}}_{11}-a_{4}{\mathfrak{r}}_{11}^{2}\right),$
(4.14)

where $a_{i}>0,i=1,2,3,4$ depend only on $n,\alpha$ and $M_{0}$.

To estimate $Q_{2}$, we use (4.9) to kill the terms involving
$\bar{\nabla}_{k}\mathfrak{r}_{11}$ and get

$\displaystyle-\frac{(K^{Y})^{\alpha}}{{\mathfrak{r}}_{11}}\alpha
B\mathfrak{r}^{kk}\mathfrak{r}^{\ell\ell}(\bar{\nabla}_{1}{\mathfrak{r}_{k\ell}})^{2}+F^{k\ell}\frac{\bar{\nabla}_{k}\mathfrak{r}_{11}\bar{\nabla}_{\ell}\mathfrak{r}_{11}}{(\mathfrak{r}_{11})^{2}}$
$\displaystyle\leq$
$\displaystyle-\frac{(K^{Y})^{\alpha}}{{\mathfrak{r}}_{11}}\alpha
B\mathfrak{r}^{kk}\mathfrak{r}^{11}(\bar{\nabla}_{k}{\mathfrak{r}_{11}})^{2}+\alpha
B(K^{Y})^{\alpha}\mathfrak{r}^{kk}\frac{(\bar{\nabla}_{k}\mathfrak{r}_{11})^{2}}{(\mathfrak{r}_{11})^{2}}=0.$

Then

$\displaystyle Q_{2}\leq$
$\displaystyle-\frac{(K^{Y})^{\alpha}}{{\mathfrak{r}}_{11}}\Big{(}B_{s_{1}s_{1}}{\mathfrak{r}_{11}}^{2}+\left(2B_{ss_{1}}s_{1}-2B_{s_{1}s_{1}}s+B_{s}\right)\mathfrak{r}_{11}$
$\displaystyle+B+B_{ss}{s_{1}}^{2}-2B_{ss_{1}}ss_{1}-B_{s}s+B_{s_{1}s_{1}}s^{2}-B_{s_{1}}s_{1}$
$\displaystyle+{\alpha}^{2}B(\sum_{k}\mathfrak{r}^{kk}\bar{\nabla}_{1}\mathfrak{r}_{kk})^{2}-2\alpha\mathfrak{r}^{kk}\bar{\nabla}_{1}\mathfrak{r}_{kk}(B_{s_{1}}\mathfrak{r}_{11}+B_{s}s_{1}-B_{s_{1}}s)\Big{)}$
$\displaystyle-(K^{Y})^{\alpha}L\Big{(}(1-n\alpha)sB+B_{s}|\bar{\nabla}s|^{2}-B_{s_{i}}s_{i}s+\alpha
B\mathfrak{r}_{11}\Big{)}.$ (4.15)

The third line on the right hand side of (4.2) can be estimated using Cauchy-
Schwarz inequality:

$\displaystyle-{\alpha}^{2}B(\sum_{k}\mathfrak{r}^{kk}\bar{\nabla}_{1}\mathfrak{r}_{kk})^{2}+2\alpha\mathfrak{r}^{kk}\bar{\nabla}_{1}\mathfrak{r}_{kk}(B_{s_{1}}\mathfrak{r}_{11}+B_{s}s_{1}-B_{s_{1}}s)$
$\displaystyle\leq$
$\displaystyle\frac{1}{B}\left(B_{s_{1}}\mathfrak{r}_{11}+B_{s}s_{1}-B_{s_{1}}s\right)^{2}.$

Since the coefficients such as $B,B_{s_{i}},B_{s},B_{ss_{i}}$ are bounded and
$B>0$, we obtain the estimate

$\displaystyle Q_{2}\leq$
$\displaystyle\frac{(K^{Y})^{\alpha}}{{\mathfrak{r}}_{11}}\bigg{(}b_{1}+(b_{2}+b_{3}L){\mathfrak{r}}_{11}+(b_{4}-b_{5}L){\mathfrak{r}}_{11}^{2}\bigg{)},$
(4.16)

where $b_{i}>0,i=1,2,3,4,5$ depend only on $n,\alpha$ and $M_{0}$.

Combining the two estimates (4.14) and (4.16) and choosing
$L={2b_{4}}/{b_{5}}$, we can get an uniform upper bound
${\mathfrak{r}}_{11}(\bar{z},\bar{t})\leq\Lambda$, where $\Lambda$ depends
only on $n$, $\alpha$ and $M_{0}$. It follows from

$\displaystyle G(z,t)$
$\displaystyle\leq\max\left\\{G(\bar{z},\bar{t}),\max_{z\in\mathbb{S}^{n}}G(z,t_{0})\right\\}$

that

$\displaystyle\mathfrak{r}_{11}(z,t)$
$\displaystyle\leq\max\left\\{\mathfrak{r}_{11}(\bar{z},\bar{t}),\max_{z\in\mathbb{S}^{n}}\mathfrak{r}_{11}(z,t_{0})\right\\}\exp\Big{(}\frac{L}{2}\big{(}1-c_{3}-r^{2}(z,t)\big{)}\Big{)}$
$\displaystyle\leq\max\left\\{\Lambda,\max_{z\in\mathbb{S}^{n}}\mathfrak{r}_{11}(z,t_{0})\right\\}\exp\Big{(}\frac{L}{2}(c_{4}-c_{3})\Big{)}$
$\displaystyle=\Lambda_{1}\max\left\\{\Lambda,\max_{z\in\mathbb{S}^{n}}\mathfrak{r}_{11}(z,t_{0})\right\\}$
(4.17)

for all $(z,t)\in\mathbb{S}^{n}\times[t_{0},\min\\{T,t_{0}+\tau\\})$, where
$\Lambda_{1}=\exp\big{(}\frac{L}{2}(c_{4}-c_{3})\big{)}\geq 1$ and
$c_{3},c_{4}$ are constants in Lemma 4.3.

Note that $t_{0}\in[t_{0}-\frac{\tau}{2},\min\\{T,t_{0}+\frac{\tau}{2}\\})$.
Applying the above argument for the time interval
$[t_{0}-\frac{\tau}{2},\min\\{T,t_{0}+\frac{\tau}{2}\\})$ gives

$\max_{z\in\mathbb{S}^{n}}\mathfrak{r}_{11}(z,t_{0})\leq\Lambda_{1}\max\left\\{\Lambda,\max_{z\in\mathbb{S}^{n}}\mathfrak{r}_{11}(z,t_{0}-\frac{\tau}{2})\right\\}.$
(4.18)

Combining (4.2), (4.18) and the fact that $\Lambda_{1}\geq 1$, we have

$\mathfrak{r}_{11}(z,t)\leq\Lambda_{1}^{2}\max\left\\{\Lambda,\max_{z\in\mathbb{S}^{n}}\mathfrak{r}_{11}(z,t_{0}-\frac{\tau}{2})\right\\}.$

By backward induction on the time interval, we finally get

$\displaystyle\mathfrak{r}_{11}(z,t)$
$\displaystyle\leq\Lambda_{1}^{\left[\frac{2t_{0}}{\tau}\right]+2}\max\\{\Lambda,\max_{z\in\mathbb{S}^{n}}\mathfrak{r}_{11}(z,0)\\}$
$\displaystyle\leq\Lambda_{1}^{\frac{2t}{\tau}+2}\max\\{\Lambda,\max_{z\in\mathbb{S}^{n}}\mathfrak{r}_{11}(z,0)\\}~{}=:\Lambda_{2}\Lambda_{1}^{\frac{2t}{\tau}}$
(4.19)

for all $(z,t)\in\mathbb{S}^{n}\times[t_{0},\min\\{T,t_{0}+\tau\\})$, where
$[\cdot]$ denotes the integer part of a real constant, and $\Lambda_{1}$,
$\Lambda_{2}=\Lambda_{1}^{2}\max\\{\Lambda,\max_{z\in\mathbb{S}^{n}}\mathfrak{r}_{11}(z,0)\\}$
are constants depending only on $n$, $\alpha$ and $M_{0}$. Since $t_{0}$ is
arbitrary, by (3.25) and the $C^{0},C^{1}$ estimates, we conclude that the
principal curvatures $\kappa_{i}$ of the solution $M_{t}$ of the flow (1.1)
satisfy

$\kappa_{i}\geq\Lambda_{3}^{-1}\Lambda_{1}^{-\frac{2t}{\tau}},\qquad
i=1,\dots,n,$

for all time $t\in[0,T)$, where $\Lambda_{3}=C\Lambda_{2}$ for a constant $C$
which estimates the bound on the coefficient of (3.25) involving $s$ and
$\bar{\nabla}s$. ∎

###  4.3. Upper bound of Gauss curvature

Now we use the technique of Tso [29] to prove the upper bound of the Gauss
curvature of the solution $M_{t}$ along the flow (1.1).

######  Proposition 4.5.

Let $M_{t},\ t\in[0,T)$ be the smooth solution of the flow (1.1) starting from
a smooth closed convex hypersurface $M_{0}$. If $T<\infty$, then there is a
constant $C$ depending on $n,\alpha,M_{0}$ and $T$ such that the Gauss
curvature $K$ of $M_{t}$ satisfies

$\max_{M_{t}}K\leq C$

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

###### Proof.

For any given $t_{0}\in[0,T)$, let $B_{\rho_{0}}(p_{0})$ be the inball of
$\Omega_{t_{0}}$, where $\rho_{0}=\rho_{-}(t_{0})$. Consider the support
function
$u(x,t)=\sinh\rho_{p_{0}}(x)\langle{\partial_{\rho_{p_{0}}},\nu}\rangle$ of
$M_{t}$ with respect to point $p_{0}$, where $\rho_{p_{0}}(x)$ is the distance
function in $\mathbb{H}^{n+1}$ from the point $p_{0}$. Since $M_{t}$ is convex
for all $t\in[0,T)$, by (4.3) we have

$2c\leq u\leq\sinh(2c_{2})$ (4.20)

on $M_{t}$ for any $t\in\left[t_{0},\min\\{T,t_{0}+\tau\\}\right)$. We define
the auxiliary function

$W=\frac{K^{\alpha}}{u-c},$

which is well-defined for time
$t\in\left[t_{0},\min\\{T,t_{0}+\tau\\}\right)$. We shall apply the maximum
principle to the evolution equation of $W$ to derive the upper bound of $K$.

Combining (2.4) and (2.5), we compute that along the flow (1.1) $W$ evolves as

$\displaystyle\frac{\partial}{\partial t}{W}=$ $\displaystyle\alpha
K^{\alpha-1}\dot{K}^{ij}(W_{ij}+\frac{2}{u-c}u_{i}W_{j})$
$\displaystyle\quad-\frac{\phi(t)}{u-c}\bigg{(}\alpha
K^{\alpha-1}(HK-\sigma_{n-1}(\kappa))+W\cosh\rho_{{p_{0}}}(x)\bigg{)}$
$\displaystyle\quad+(1+n\alpha)\frac{K^{2\alpha}}{(u-c)^{2}}\cosh\rho_{{p_{0}}}(x)-\frac{c\alpha
K^{2\alpha}}{(u-c)^{2}}H-\alpha WK^{\alpha-1}\sigma_{n-1}(\kappa)$
$\displaystyle\leq$ $\displaystyle~{}\alpha
K^{\alpha-1}\dot{K}^{ij}(W_{ij}+\frac{2}{u-c}u_{i}W_{j})+\underbrace{\frac{\phi(t)}{u-c}\alpha
K^{\alpha-1}\left(\sigma_{n-1}(\kappa)-HK\right)}_{(I)}$
$\displaystyle\qquad+(1+n\alpha)W^{2}\cosh\rho_{{p_{0}}}(x)-\alpha cHW^{2}.$
(4.21)

In the previous work [2, 1], $M_{t}$ is assumed to be either $h$-convex
($\kappa_{i}>1,~{}i=1,\cdots,n$) or positively curved
($\kappa_{i}\kappa_{j}>1$, $\forall~{}i\neq j$), so the terms $(I)$ involving
$\phi(t)$ can be thrown away when we estimate the upper bound of $W$. But in
our case we only have convexity, so we still need to estimate the terms $(I)$
in (4.21) carefully.

Let $\tilde{W}(t)=\max_{M_{t}}W(x,t)$. Noting that $K^{\alpha}=(u-c)W$, by the
definition (1.2) of $\phi(t)$ and the upper bound (4.20) of $u$, we have

$\phi(t)=\frac{1}{|M_{t}|}\int_{M_{t}}K^{\alpha}\,\mathrm{d}t\leq\max_{M_{t}}{K^{\alpha}(\cdot,t)}\leq(\sinh(2c_{2})-c)\tilde{W}.$

By the lower bound on the principal curvatures in Lemma 4.4, we also have

$\displaystyle\sigma_{n-1}(\kappa)$
$\displaystyle=K(\frac{1}{\kappa_{1}}+\cdots\frac{1}{\kappa_{n}})\leq
nK(\min_{1\leq i\leq n}\kappa_{i})^{-1}\leq
nK\Lambda_{3}\Lambda_{1}^{{2T}/{\tau}}.$

It follows that the terms $(I)$ can be estimated as

$(I)\leq~{}n\alpha(\sinh(2c_{2})-c)\Lambda_{3}\Lambda_{1}^{{2T}/{\tau}}\tilde{W}^{2}~{}=:~{}\beta_{1}(T)\tilde{W}^{2},$
(4.22)

where for simplicity of the notations we denote
$\beta_{1}(T)=n\alpha(\sinh(2c_{2})-c)\Lambda_{3}\Lambda_{1}^{{2T}/{\tau}}$,
which depends on the maximal existence time $T<\infty$.

The first term on the last line of (4.21) can be simply estimated from above
by

$(1+n\alpha)W^{2}\cosh\rho_{{p_{0}}}(x)~{}\leq(1+n\alpha)\cosh(2c_{2})\tilde{W}^{2}.$
(4.23)

The last term of (4.21) provides sufficient negative term, since by $H\geq
nK^{1/n}$ there holds:

$\displaystyle-\alpha cHW^{2}\leq$ $\displaystyle-\alpha ncK^{1/n}W^{2}$
$\displaystyle=$ $\displaystyle-\alpha
nc(u-c)^{\frac{1}{n\alpha}}W^{2+\frac{1}{n\alpha}}$ $\displaystyle\leq$
$\displaystyle-\alpha nc^{1+\frac{1}{n\alpha}}W^{2+\frac{1}{n\alpha}}.$ (4.24)

Combining (4.22) - (4.3), we arrive at

$\displaystyle\frac{d}{dt}\tilde{W}\leq$
$\displaystyle\bigg{(}\beta_{1}(T)+(1+n\alpha)\cosh(2c_{2})-\alpha
nc^{1+\frac{1}{n\alpha}}\tilde{W}^{\frac{1}{n\alpha}}\biggr{)}\tilde{W}^{2}.$
(4.25)

The coefficient of the hightest order term on the right hand side of (4.25) is
negative, so the comparison principle implies that $\tilde{W}$ is bounded
above by a positive constant. In fact, whenever

$\tilde{W}\geq\bigg{(}\frac{2}{\alpha
n}\left(\beta_{1}(T)+(1+n\alpha)\cosh(2c_{2})\right)\bigg{)}^{n\alpha}c^{-n\alpha-1}~{}=:\beta_{2}(T),$

we have

$\displaystyle\frac{d}{dt}\tilde{W}\leq$ $\displaystyle-\frac{\alpha
n}{2}c^{1+\frac{1}{n\alpha}}\tilde{W}^{2+\frac{1}{n\alpha}}.$

Therefore,

$\tilde{W}(t)\leq\max\left\\{\bigg{(}W^{-1-\frac{1}{n\alpha}}(t_{0})+\frac{n\alpha+1}{2}c^{1+\frac{1}{n\alpha}}(t-t_{0})\bigg{)}^{-\frac{n\alpha}{n\alpha+1}},\beta_{2}(T)\right\\}$
(4.26)

for all time $t\in\left[t_{0},\min\\{T,t_{0}+\tau\\}\right)$.

For $t_{0}=0$, we obtain from (4.26) the upper bound

$\tilde{W}(t)\leq\max\left\\{\tilde{W}(0),\beta_{2}(T)\right\\},\quad\forall
t\in[0,\min\\{\tau,T\\})$

and so

$K^{\alpha}\leq\sinh(2c_{2})\max\left\\{\tilde{W}(0),\beta_{2}(T)\right\\},\quad\forall
t\in[0,\min\\{\tau,T\\}).$ (4.27)

Next, for $t_{0}=\tau/2$, the estimate (4.26) implies

$\displaystyle\tilde{W}(t)\leq$
$\displaystyle\max\left\\{(\frac{n\alpha+1}{2}(t-t_{0}))^{-\frac{n\alpha}{n\alpha+1}}c^{-1},\beta_{2}(T)\right\\}$
$\displaystyle\leq$
$\displaystyle~{}\max\left\\{(\frac{(n\alpha+1)\tau}{4})^{-\frac{n\alpha}{n\alpha+1}}c^{-1},\beta_{2}(T)\right\\}$

for $t\in[\tau,\min\\{3\tau/2,T\\})$, and so

$K^{\alpha}\leq\sinh(2c_{2})\max\left\\{(\frac{(n\alpha+1)\tau}{4})^{-\frac{n\alpha}{n\alpha+1}}c^{-1},\beta_{2}(T)\right\\}$
(4.28)

for $t\in[\tau,\min\\{3\tau/2,T\\})$. Repeating the above argument for
$t_{0}=m\tau/2$ ($m\geq 2$), we can get the estimate (4.28) for
$t\in[\frac{(m+1)\tau}{2},\min\\{\frac{(m+2)\tau}{2},T\\})$, which covers the
whole time interval $[0,T)$. Combining (4.27) and (4.28), we complete the
proof of Proposition 4.5. ∎

##  5\. Long time existence

In this section, we prove the long time existence of the flow (1.1).

######  Theorem 5.1.

Let $M_{0}$ be a smooth closed convex hypersurface in $\mathbb{H}^{n+1}$ and
$M_{t}$ be the smooth solution of the flow (1.1) starting from $M_{0}$. Then
$M_{t}$ remains convex and exists for all time $t\in[0,\infty)$.

###### Proof.

We will argue by contradiction. Let $[0,T)$ be the maximum interval such that
the solution of the flow (1.1) exists with $T<\infty$. Then combining
Proposition 4.4 and Proposition 4.5 yields that the principal curvatures
$\kappa=(\kappa_{1},\dots,\kappa_{n})$ of the solution $M_{t}$ satisfy

$0<\underline{\kappa}_{0}\leq\kappa_{i}\leq\overline{\kappa}_{0},\quad
i=1,\dots,n$ (5.1)

for all time $t\in[0,T)$, where the constants
$\underline{\kappa}_{0},\overline{\kappa}_{0}$ depend on $n,\alpha,M_{0}$ and
$T$.

To prove the long time existence of the solution $M_{t}$ of the flow (1.1), we
need to derive the higher order regularity estimates. Recall that in $\S$3, up
to a tangential diffeomorphism, the flow equation (1.1) is equivalent the
following scalar parabolic equation

$\displaystyle\frac{\partial}{\partial t}s=$ $\displaystyle
A\phi(t)-B(K^{Y})^{\alpha}=~{}A\phi(t)-B(\det\mathfrak{r})^{-\alpha}$ (5.2)

on the sphere $\mathbb{S}^{n}$, where $\mathfrak{r}_{ij}=s_{ij}+s\sigma_{ij}$
and $A$, $B$ are functions defined in (3.12) which depend only on $s$ and
$\bar{\nabla}s$. Combing Lemma 4.3 and the curvature estimate (5.1) gives the
$C^{2}$ estimates of $s(z,t)$. Denote the right hand side of (5.2) by
$G(\bar{\nabla}^{2}s,\bar{\nabla}s,s,z,t)$. Then the derivatives of $G$ with
respect to the first argument are given by

$\dot{G}^{ij}=n\alpha
B(\det\mathfrak{r})^{-\alpha-\frac{1}{n}}\frac{\partial(\det\mathfrak{r})^{\frac{1}{n}}}{\partial
s_{ij}}$ (5.3)

and

$\begin{split}\ddot{G}^{ij,k\ell}&=n\alpha
B(\det\mathfrak{r})^{-\alpha-\frac{1}{n}}\frac{\partial^{2}(\det\mathfrak{r})^{\frac{1}{n}}}{\partial
s_{ij}\partial s_{k\ell}}\\\
&-n\alpha(n\alpha+1)B(\det\mathfrak{r})^{-\alpha-\frac{2}{n}}\frac{\partial(\det\mathfrak{r})^{\frac{1}{n}}}{\partial
s_{ij}}\frac{\partial(\det\mathfrak{r})^{\frac{1}{n}}}{\partial
s_{k\ell}}.\end{split}$ (5.4)

The estimates we have established imply the existence of a constant
$C=C(T)>0$, such that $1/CI\leq(\dot{G^{ij}})\leq CI$, that is, the operator
$G$ is uniformly elliptic on the finite time interval $[0,T)$. Since
$(\det\mathfrak{r})^{\frac{1}{n}}$ is concave with respect to $s_{ij}$, we see
that $G$ is also a concave operator. We can apply Theorem 1.1 in [28] to
obtain a $C^{2,\gamma}$ estimate on $s$, for a suitable $\gamma\in(0,1)$. See
also the arguments in [9, 23] for the $C^{2,\gamma}$ estimate of the solutions
to volume preserving curvature flows. Then by the parabolic Schauder theory
(see [21]), we can deduce all higher order regularity estimates of $s$ on
$[0,T)$ and a standard continuation argument then shows that $T=+\infty$. ∎

######  Remark 5.2.

Note that the curvature estimate (5.1) of the solution $M_{t}$ of the flow
(1.1) depends on time $t$ and may degenerate as time $t\to\infty$. To study
the asymptotical behavior of $M_{t}$ as $t\to\infty$, we still need to get an
uniform curvature estimate which does not depend on time. This will be
obtained in the next two sections.

##  6\. Monotonicity and Hausdorff convergence

In this section, we prove the monotonicity of $\mathcal{A}_{n-1}(\Omega_{t})$,
the subsequential Hausdorff convergence of the solution $M_{t}$ of (1.1) and
the convergence of the center of the inner ball of $\Omega_{t}$ to a fixed
point.

Denote the average integral of the Gauss curvature by

$\bar{K}=\frac{1}{|M_{t}|}\int_{M_{t}}{Kd\mu_{t}}=\frac{\mathcal{A}_{n}(\Omega_{t})+\frac{1}{n-1}\mathcal{A}_{n-2}(\Omega_{t})}{\mathcal{A}_{0}(\Omega_{t})}.$
(6.1)

It follows from the monotonicity (2.10) of quermassintegrals with respect to
inclusion of convex sets and the estimates on inner radius and outer radius in
Lemma 4.1 that $\bar{K}$ is uniformly bounded from above and below by positive
constants depending only on $n$, $\alpha$ and $M_{0}$.

###  6.1. Monotonicity for $\mathcal{A}_{n-1}$

We first show the following monotonicity of $\mathcal{A}_{n-1}(\Omega_{t})$
along the flow (1.1), which will be useful in proving the subsequential
Hausdorff convergence of $M_{t}$.

######  Lemma 6.1.

Let $M_{t}$ be a smooth convex solution of the volume preserving flow (1.1).
Denote by $\Omega_{t}$ the domain enclosed by $M_{t}$. Then
$\mathcal{A}_{n-1}(\Omega_{t})$ is monotone decreasing in time $t$, and is
strictly decreasing unless $\Omega_{t}$ is a geodesic ball.

###### Proof.

From the evolution equation (2.9) for the quermassintegrals of $\Omega_{t}$,
we have

$\frac{d}{dt}\mathcal{A}_{n-1}(\Omega_{t})=n\int_{M_{t}}{K(\phi(t)-K^{\alpha})d\mu_{t}}.$

Since $\phi(t)$ is defined as in (1.2), we have

$\displaystyle\frac{d}{dt}\mathcal{A}_{n-1}(\Omega_{t})$
$\displaystyle=\frac{n}{|M_{t}|}\left(\int_{M_{t}}Kd\mu_{t}\int_{M_{t}}K^{\alpha}d\mu_{t}-|M_{t}|\int_{M_{t}}K^{\alpha+1}d\mu_{t}\right)$
$\displaystyle=-n\int_{M_{t}}{(K-\bar{K})(K^{\alpha}-{\bar{K}}^{\alpha})d\mu_{t}}\leq
0.$ (6.2)

Note that equality holds in (6.1) if and only if $K$ is a constant on $M_{t}$,
which means $M_{t}$ is a geodesic sphere by the Alexandrov type theorem for
hypersurfaces with constant Gauss curvature in the hyperbolic space (see
[24]). ∎

###  6.2. Subsequential Hausdorff convergence

In this subsection, we prove that there exists a sequence of times
$t_{i}\to\infty$ such that the solution $M_{t_{i}}$ converges in Hausdorff
sense to a geodesic sphere. We first prove the following estimate:

######  Lemma 6.2.

Let $M_{0}$ be a smooth closed and convex hypersurface in $\mathbb{H}^{n+1}$
and $M_{t}$ be the smooth solution of the flow (1.1) starting from $M_{0}$.
Then there exists a sequence of times $\\{t_{i}\\},t_{i}\to\infty$, such that

$\int_{M_{t_{i}}}{|K-\bar{K}|d\mu_{t_{i}}}\to
0,\quad\text{as}\,\,t_{i}\to\infty.$ (6.3)

###### Proof.

Applying the monotonicity in Lemma 6.1 and the long time existence of the flow
(1.1), we have

$n\int_{0}^{\infty}\int_{M_{t}}{(K-\bar{K})(K^{\alpha}-{\bar{K}}^{\alpha})d\mu_{t}}dt\leq{\mathcal{A}_{n-1}}(\Omega_{0})<\infty.$

Therefore there exists a sequence of times $t_{i}\to\infty$ such that

$\int_{M_{t_{i}}}{(K-\bar{K})(K^{\alpha}-{\bar{K}}^{\alpha})d\mu_{t_{i}}}\to
0.$ (6.4)

If $\alpha\geq 1$, we have

$\displaystyle(K-\bar{K})(K^{\alpha}-{\bar{K}}^{\alpha})$
$\displaystyle=\alpha\int_{0}^{1}{((1-s)\bar{K}+sK)^{\alpha-1}ds}\cdot(K-\bar{K})^{2}$
$\displaystyle\geq\alpha\int_{0}^{1}{(1-s)^{\alpha-1}ds}\bar{K}^{\alpha-1}(K-\bar{K})^{2}$
$\displaystyle\geq C(K-\bar{K})^{2},$

where we used the fact that $\bar{K}$ is uniformly bounded and $\alpha\geq 1$.
Therefore, (6.4) implies that

$\int_{M_{t_{i}}}{(K-\bar{K})^{2}d\mu_{t_{i}}}\to
0,\quad\text{as}\,\,i\to\infty$

for a sequence of times $t_{i}\to\infty$ and the estimate (6.3) follows by the
Hölder inequality.

We next prove the estimate (6.3) for $0<\alpha<1$. In this case, we have

$\displaystyle\int_{M_{t_{i}}}|K-\bar{K}|d\mu_{t_{i}}$ $\displaystyle=$
$\displaystyle\int_{M_{t_{i}}}|K-\bar{K}|^{1/2}|K-\bar{K}|^{1/2}d\mu_{t_{i}}$
$\displaystyle=$
$\displaystyle\int_{M_{t_{i}}}\Big{(}\frac{1}{\alpha}\int_{0}^{1}{((1-s)\bar{K}^{\alpha}+sK^{\alpha})^{\frac{1}{\alpha}-1}ds}\Big{)}^{1/2}\cdot\Big{|}(K^{\alpha}-\bar{K}^{\alpha})(K-\bar{K})\Big{|}^{1/2}d\mu_{t_{i}}$
$\displaystyle\leq$
$\displaystyle\Big{(}\underbrace{\int_{M_{t_{i}}}\frac{1}{\alpha}\int_{0}^{1}{((1-s)\bar{K}^{\alpha}+sK^{\alpha})^{\frac{1}{\alpha}-1}ds}d\mu_{t_{i}}}_{(I)}\Big{)}^{1/2}\Big{(}\underbrace{\int_{M_{t_{i}}}(K-\bar{K})(K^{\alpha}-{\bar{K}}^{\alpha})d\mu_{t_{i}}}_{(II)}\Big{)}^{1/2}.$

The second term $(II)$ tends to zero as $t_{i}\to\infty$ by (6.4). We show
that the first term $(I)$ is bounded for any $0<\alpha<1$. In fact, if
$\alpha\in[\frac{1}{2},1)$, we have $1<1/\alpha\leq 2$. This implies that

$\displaystyle(I)=$
$\displaystyle\frac{1}{\alpha}\int_{M_{t_{i}}}\int_{0}^{1}{((1-s)\bar{K}^{\alpha}+sK^{\alpha})^{\frac{1}{\alpha}-1}ds}d\mu_{t_{i}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\alpha}\int_{M_{t_{i}}}(\bar{K}^{\alpha}+K^{\alpha})^{\frac{1}{\alpha}-1}d\mu_{t_{i}}$
$\displaystyle=$
$\displaystyle\frac{1}{\alpha}\int_{M_{t_{i}}}\frac{\bar{K}^{\alpha}+K^{\alpha}}{(\bar{K}^{\alpha}+K^{\alpha})^{2-\frac{1}{\alpha}}}d\mu_{t_{i}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\alpha\bar{K}^{2\alpha-1}}\int_{M_{t_{i}}}(\bar{K}^{\alpha}+K^{\alpha})d\mu_{t_{i}}$
$\displaystyle\leq$
$\displaystyle\frac{2}{\alpha}\bar{K}^{1-\alpha}\mathcal{A}_{0}(\Omega_{t_{i}})\leq~{}C$

is uniformly bounded from above, where we used the Hölder inequality in the
fourth inequality to get the estimate

$\int_{M_{t_{i}}}K^{\alpha}d\mu_{t_{i}}\leq\left(\int_{M_{t_{i}}}Kd\mu_{t_{i}}\right)^{\alpha}\mathcal{A}_{0}(\Omega_{t_{i}})^{1-\alpha}=\bar{K}^{\alpha}\mathcal{A}_{0}(\Omega_{t_{i}})$

for $\alpha<1$ and that
$\mathcal{A}_{0}(\Omega_{t_{i}})\leq\mathcal{A}_{0}(B_{c_{2}}(0))\leq C$ which
is due to that the outer radius of $\Omega_{t_{i}}$ is bounded from above by
$c_{2}$. We repeat the argument for $\alpha\in[\frac{1}{k+1},\frac{1}{k})$
with $k=2,3,\cdots$, where we have $(k+1)\alpha\geq 1$ and $0<k\alpha<1$. Then

$\displaystyle(I)\leq$
$\displaystyle\frac{1}{\alpha}\int_{M_{t_{i}}}(\bar{K}^{\alpha}+K^{\alpha})^{\frac{1}{\alpha}-1}d\mu_{t_{i}}$
$\displaystyle=$
$\displaystyle\frac{1}{\alpha}\int_{M_{t_{i}}}\frac{(\bar{K}^{\alpha}+K^{\alpha})^{k}}{(\bar{K}^{\alpha}+K^{\alpha})^{k+1-\frac{1}{\alpha}}}d\mu_{t_{i}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\alpha\bar{K}^{(k+1)\alpha-1}}\int_{M_{t_{i}}}2^{k-1}(\bar{K}^{k\alpha}+K^{k\alpha})d\mu_{t_{i}}$
$\displaystyle\leq$
$\displaystyle\frac{2^{k}}{\alpha}\bar{K}^{1-\alpha}\mathcal{A}_{0}(\Omega_{t_{i}})\leq~{}C$

is uniformly bounded, where in the third inequality we used the Hölder
inequality again as $k\alpha<1$. Therefore for any $0<\alpha<1$, the term
$(I)$ is uniformly bounded from above and thus the estimate (6.3) follows in
this case. ∎

Note that the curvature estimate (5.1) of the solution $M_{t}$ of the flow
(1.1) depends on time $t$ and may degenerate as time $t\to\infty$. So we can
not conclude from (6.3) that the solution $M_{t_{i}}$ converges to a geodesic
sphere as $t_{i}\to\infty$, as we do not have an uniform regularity estimate
to guarantee the existence of a smooth limit of $M_{t_{i}}$. However, we can
apply the similar idea in the work [3, §6] by the first author and Andrews in
the Euclidean space to prove the following subsequential Hausdorff convergence
result:

######  Lemma 6.3.

Let $M_{0}$ be a smooth, closed convex hypersurface in $\mathbb{H}^{n+1}$ and
$M_{t}$ be the smooth solution of the flow (1.1) starting from $M_{0}$. Then
there exists a sequence of times $\\{t_{i}\\},t_{i}\to\infty$, such that
$M_{t_{i}}$ converges to a geodesic sphere $S_{\rho_{\infty}}(p)$ in Hausdorff
sense as $t_{i}\to\infty$, where $p$ is the center of the sphere and the
radius $\rho_{\infty}$ is determined by the fact that $S_{\rho_{\infty}}(p)$
encloses the same volume of $M_{0}$.

###### Proof.

Let $p_{t}$ be the center of the inball of $\Omega_{t}$ and let
$\varphi_{t}:\mathbb{H}^{n+1}\to\mathbb{H}^{n+1}$ be an isometry carrying
$p_{t}$ to the origin $o\in\mathbb{H}^{n+1}$. Clearly, each
$\varphi_{t}(M_{t})$ is a closed convex hypersurface with an inball centered
at the origin and having inner radius $\rho_{-}(t)\geq c_{1}$. For simplicity
of the notations, we still denote the transformed solution
$\varphi_{t}(M_{t})$ as $M_{t}$.

As in §3, we project $\Omega_{t}\subset\mathbb{H}^{n+1}$ onto
$B_{1}(0)\subset\mathbb{R}^{n+1}$ and get the corresponding
$\tilde{\Omega}_{t}\subset B_{1}(0)\subset\mathbb{R}^{n+1}$. Since the outer
radius of $\Omega_{t}$ is uniformly bounded above and so does
$\tilde{\Omega}_{t}$, the Blaschke selection theorem (see Theorem 1.8.7 of
[26]) implies that there exists a sequence of times $t_{i}$ and a convex body
$\tilde{\Omega}$ such that $\tilde{\Omega}_{t_{i}}$ converges to
$\tilde{\Omega}$ in Hausdorff sense as $t_{i}\to\infty$. Note that the
projection yields an one-to-one correspondence between the convex bodies in
$\mathbb{H}^{n+1}$ and the convex bodies in $B_{1}(0)\subset\mathbb{R}^{n+1}$,
then there exists a convex set $\hat{\Omega}\in\mathcal{K}(\mathbb{H}^{n+1})$
such that $\Omega_{t_{i}}$ converges to $\hat{\Omega}$ in Hausdorff sense as
$t_{i}\to\infty$. As each $\Omega_{t_{i}}$ has inner radius
$\rho_{-}(\Omega_{t_{i}})\geq c_{1}$, the limit convex set $\hat{\Omega}$ has
positive inner radius. Without loss of generality, we may assume that the
sequence $t_{i}$ is the same sequence such that (6.3) holds.

By the continuity of quermassintegrals with respect to the Hausdorff distance
in Lemma 2.1, the Hausdorff convergence of $\Omega_{t_{i}}$ to $\hat{\Omega}$
implies that

$\bar{K}=\frac{\mathcal{A}_{n}(\Omega_{t_{i}})+\frac{1}{n-1}\mathcal{A}_{n-2}(\Omega_{t_{i}})}{\mathcal{A}_{0}(\Omega_{t_{i}})}~{}\to~{}\frac{\mathcal{A}_{n}(\hat{\Omega})+\frac{1}{n-1}\mathcal{A}_{n-2}(\hat{\Omega})}{\mathcal{A}_{0}(\hat{\Omega})}~{}=:c$
(6.5)

as $t_{i}\to\infty$. We will show that $\hat{\Omega}$ satisfies the equation
$\Phi_{0}(\hat{\Omega},\cdot)=c\Phi_{n}(\hat{\Omega},\cdot)$ for the curvature
measures $\Phi_{0}$ and $\Phi_{n}$. In fact, by the weak continuity of the
curvature measures in Theorem 2.2, for any bounded continuous function $f$ on
$\mathbb{H}^{n+1}$ with compact support, we have that
$\int{fd\Phi_{0}(\Omega_{t_{i}})}$ converges to
$\int{fd\Phi_{0}(\hat{\Omega})}$, and $\int{fd\Phi_{n}(\Omega_{t_{i}})}$
converges to $\int{fd\Phi_{n}(\hat{\Omega})}$, as $i\to\infty$. Then

$\displaystyle\left|\int{fd\Phi_{0}(\Omega_{t_{i}})}-c\int{fd\Phi_{n}(\Omega_{t_{i}})}\right|$
$\displaystyle=$
$\displaystyle\left|\int_{M_{t_{i}}}{fKd\mathcal{H}^{n}}-\int_{M_{t_{i}}}{cfd\mathcal{H}^{n}}\right|$
$\displaystyle\leq$
$\displaystyle\sup|f|\int_{M_{t_{i}}}{|K-c|d\mathcal{H}^{n}}$
$\displaystyle\leq$
$\displaystyle\sup|f|\int_{M_{t_{i}}}{|K-\bar{K}|d\mathcal{H}^{n}}+\sup|f|\int_{M_{t_{i}}}{|\bar{K}-c|d\mathcal{H}^{n}}~{}\to~{}0$
(6.6)

by the estimates (6.3) and (6.5).

It follows that
$\int{fd\Phi_{0}(\hat{\Omega})}=c\int{fd\Phi_{n}(\hat{\Omega})}$ for all
bounded continuous functions $f$, and therefore
$\Phi_{0}(\hat{\Omega},\cdot)=c\Phi_{n}(\hat{\Omega},\cdot)$ as claimed. Then
by Theorem 2.3, $\hat{\Omega}$ is a geodesic ball. This means that
$\varphi_{t_{i}}(\Omega_{t_{i}})$ converges to a geodesic ball of some radius
$\rho_{\infty}$ centered at the origin. The radius is uniquely determined by
the preserving of the volume. Moreover, by an argument of Alexandrov
reflection, the evolving domains $\Omega_{t}$ cannot leave away from the
initial domain $\Omega_{0}$, that is, there exists a geodesic ball
$B_{R}\subset\mathbb{H}^{n+1}$ of radius $R>0$ containing $\Omega_{0}$ such
that $\Omega_{t}\cap B_{R}\neq\emptyset$ for all time $t\in[0,\infty)$, see
Lemma 4.1 in [1]. In particular, this implies that the centers $p_{t_{i}}$ of
inner ball of $\Omega_{t_{i}}$ are located in a compact subset of
$\mathbb{H}^{n+1}$, and then there exists a subsequence of $t_{i}$ (still
denoted by $t_{i}$) such that $p_{t_{i}}$ converges to a limit point
$p\in\mathbb{H}^{n+1}$. This concludes that for this subsequence of times
$t_{i}$, $\Omega_{t_{i}}$ converges to a geodesic ball $B_{\rho_{\infty}}(p)$
centered at the point $p$ without correction of ambient isometry
$\varphi_{t_{i}}$. This completes the proof of Lemma 6.3. ∎

###  6.3. Convergence of the center of the inner ball

The argument in [3] is not sufficient for us to deduce the Hausdorff
convergence of $M_{t}$ to the geodesic sphere for all time $t\to\infty$, as we
do not have the analogue stability estimate as in [26, Eq.(7.124)] for the
hyperbolic case. However, if we denote $p_{t}$ as the center of the inner ball
of $\Omega_{t}$, we can prove that $p_{t}$ converges to the fixed point
$p\in\mathbb{H}^{n+1}$ for all time $t\to\infty$ using the Alexandrov
reflection and the subsequential Hausdorff convergence of $M_{t}$ proved in
Lemma 6.3.

Recall that $p\in\mathbb{H}^{n+1}$ is the center of the limit geodesic sphere
$S_{\rho_{\infty}}(p)$ in Lemma 6.3. Take an arbitrary direction $z\in
T_{p}\mathbb{H}^{n+1}$. Let $\gamma_{z}$ be the normal geodesic line (i.e.
$|\gamma^{\prime}|=1$) through the point $p$ with $\gamma_{z}(0)=p$ and
$\gamma^{\prime}_{z}(0)=z$, and let $H_{z,s}$ be the totally geodesic
hyperplane in $\mathbb{H}^{n+1}$ that is perpendicular to $\gamma_{z}$ at
$\gamma_{z}(s),s\in\mathbb{R}$. We use the notation $H_{z,s}^{+}$ and
$H_{z,s}^{-}$ for the half-spaces in $\mathbb{H}^{n+1}$ determined by
$H_{z,s}$ as follows:

$H_{z,s}^{+}:=\bigcup_{s^{\prime}\geq s}H_{z,s^{\prime}},\qquad
H_{z,s}^{-}:=\bigcup_{s^{\prime}\leq s}H_{z,s^{\prime}}.$

For a bounded domain $\Omega$ in $\mathbb{H}^{n+1}$, denote

$\Omega_{z}^{+}(s)=\Omega\cap H_{z,s}^{+},\qquad\Omega_{z}^{-}(s)=\Omega\cap
H_{z,s}^{-}.$

The reflection map across $H_{z,s}$ is denoted by $R_{\gamma_{z},s}$. We
define

$\displaystyle S_{\gamma_{z}}^{+}(\Omega)$
$\displaystyle:=\inf\\{s\in\mathbb{R}~{}|~{}R_{\gamma_{z},s}(\Omega_{z}^{+}(s))\subset\Omega_{z}^{-}(s)\\},$
$\displaystyle S_{\gamma_{z}}^{-}(\Omega)$
$\displaystyle:=\sup\\{s\in\mathbb{R}~{}|~{}R_{\gamma_{z},s}(\Omega_{z}^{-}(s))\subset\Omega_{z}^{+}(s)\\}.$

The Alexandrov reflection argument implies that
$S_{\gamma_{z}}^{+}(\Omega_{t})$ is non-increasing in $t$ for each $z$ (see
[2, Lemma 6.1]). By the definitions of $S_{\gamma_{z}}^{+}(\Omega_{t})$ and
$S_{\gamma_{z}}^{-}(\Omega_{t})$, we have $S_{\gamma_{z}}^{-}(\Omega_{t})\leq
S_{\gamma_{z}}^{+}(\Omega_{t})$. Since
$S_{\gamma_{z}}^{-}(\Omega_{t})=-S_{\gamma_{-z}}^{+}(\Omega_{t})$, we also
have that $S_{\gamma_{z}}^{-}(\Omega_{t})$ is non-decreasing in $t$ for each
$z$. Note that the paper [2] deals with the flow with $h$-convex initial
hypersurfaces, the argument in Lemma 6.1 of [2] works for convex solutions as
well. The readers may refer to [11, 12] for more details on the Alexandrov
reflection method.

We need the following lemma in proving the convergence of the center of the
inner ball.

######  Lemma 6.4.

Let $\Omega$ be a bounded convex domain in $\mathbb{H}^{n+1}$, and
$\gamma_{z},H_{z,s},S_{\gamma_{z}}^{+}(\Omega)$ and
$S_{\gamma_{z}}^{-}(\Omega)$ be defined as above. Denote $p_{0}$ as the center
of an inner ball of $\Omega$ and assume that $p_{0}\in H_{z,s_{0}}$. Then we
have $S_{\gamma_{z}}^{-}(\Omega)\leq s_{0}\leq S_{\gamma_{z}}^{+}(\Omega)$.

###### Proof.

Denote $S_{\gamma_{z}}^{+}(\Omega)$ by $\bar{s}$, then we have
$R_{\gamma_{z},\bar{s}}(\Omega_{z}^{+}(\bar{s}))\subset\Omega_{z}^{-}(\bar{s})$.
Assume that $B_{r_{0}}(p_{0})$ is the inner ball of $\Omega$ centered at
$p_{0}$. First we prove that $s_{0}\leq\bar{s}$. If not (i.e.
$s_{0}>\bar{s}$), then
$B_{r_{0}}(p^{\prime}_{0}):=R_{\gamma_{z},\bar{s}}(B_{r_{0}}(p_{0}))\subset\Omega$
by
$R_{\gamma_{z},\bar{s}}(\Omega_{z}^{+}(\bar{s}))\subset\Omega_{z}^{-}(\bar{s})$
and the assumption that $s_{0}>\bar{s}$. See Figure 2. Here
$p^{\prime}_{0}=R_{\gamma_{z},\bar{s}}(p_{0})\subset H_{z,s^{\prime}_{0}}$
with $s^{\prime}_{0}<\bar{s}<s_{0}$. Then we can deduce that the reflection
ball $B_{r_{0}}(p^{\prime}_{0})\cap\partial\Omega=\emptyset$. It follows that
$B_{r_{0}+\varepsilon}(p^{\prime}_{0})\subset\Omega$ for sufficiently small
$\varepsilon$ and hence $p_{0}$ cannot be the center of an inner ball, which
leads to a contradiction.

$\gamma_{z}(s)$$H_{z,0}$$H_{z,\bar{s}}$$H_{z,s_{0}}$$p_{0}$$p^{\prime}_{0}$$p$$\mathbb{H}^{n+1}$$\Omega$
Figure 2. Center of the inner ball.

The inequality $S_{\gamma_{z}}^{-}(\Omega)\leq s_{0}$ can be checked
similarily as above. This completes the proof of Lemma 6.4. ∎

######  Remark 6.5.

For a bounded domain $\Omega$, the center of the inner ball may not be unique.
Lemma 6.4 says that all of them must satisfy the two-sided bounds.

Then we can prove the following convergence result:

######  Lemma 6.6.

Let $M_{0}$ be a smooth, closed convex hypersurface in $\mathbb{H}^{n+1}$ and
$M_{t}$ be the smooth solution of the flow (1.1) starting from $M_{0}$. Denote
the enclosed domain of $M_{t}$ by $\Omega_{t}$. Take an arbitrary direction
$z\in T_{p}{\mathbb{H}^{n+1}}$ and let
$\gamma_{z},S_{\gamma_{z}}^{-}(\Omega_{t}),S_{\gamma_{z}}^{+}(\Omega_{t})$ be
defined as above. Then along the flow (1.1), we have

$\lim_{t\to\infty}S_{\gamma_{z}}^{-}(\Omega_{t})=\lim_{t\to\infty}S_{\gamma_{z}}^{+}(\Omega_{t})=0.$
(6.7)

As a consequence, if we set $p_{t}$ as the center of an inner ball of
$\Omega_{t}$, then we have $d(p_{t},p)\to 0$ as $t\to\infty$.

###### Proof.

Recall that in Lemma 6.3, we have proved that there exists a sequence of times
$t_{i}\to\infty$, such that $M_{t_{i}}$ converges to a geodesic sphere
$S_{\rho_{\infty}}(p)=\partial B_{\rho_{\infty}}(p)$ in Hausdorff sense as
$t_{i}\to\infty$, where $\rho_{\infty}$ is the radius determined by
$\Omega_{0}$ and $p$ is the center of the ball. Then there exists a sequence
of $d_{i}\rightarrow 0$ with $d_{i}<\rho_{\infty}$ such that

$M_{t_{i}}\subset B_{\rho_{\infty}+d_{i}}(p)/B_{\rho_{\infty}-d_{i}}(p).$

Hence by the monotonicity of $S_{\gamma_{z}}^{+}(\Omega_{t})$, if we can show
that $S_{\gamma_{z}}^{+}(\Omega_{t_{i}})\leq C\sqrt{d_{i}}$ for some
$C=C(\rho_{\infty})>0$, then we have $S_{\gamma_{z}}^{+}(\Omega_{t})\leq
C\sqrt{d_{i}},\ \forall t\geq t_{i}$.

Denote the geodesic segment starting from $A_{1}$ to $A_{2}$ by
$\overline{A_{1}A_{2}}$, and the length by $|\overline{A_{1}A_{2}}|$. Let $P,\
P^{\prime}$ be two totally geodesic hyperplane which is perpendicular to
$\gamma_{z}$ and passes through $p$ and some $p^{\prime}$ respectively on
$\gamma_{z}$. Let $E=P\cap\partial B_{\rho_{\infty}-d_{i}}(p)$ and
$E^{\prime}=P^{\prime}\cap\partial B_{\rho_{\infty}+d_{i}}(p)$. By a simple
continuity argument, there exists $P^{\prime}$ such that $\forall
e^{\prime}\in E^{\prime}$, the geodesic line starting form $e^{\prime}$ and is
perpendicular to $P^{\prime}$ would pass through some $e\in E$, see figure 3.

$e^{\prime}\in E^{\prime}$$e\in
E$$p$$\gamma_{z}$$P$$P^{\prime}$$H_{z,s}^{-}$$H_{z,s}^{+}$$l$$a$$b$$A$$r$$p^{\prime}$$R_{\gamma_{z},s}(A)$$s$$\mathbb{H}^{n+1}$$\Omega$
Figure 3. $R_{\gamma_{z},s}(A)\in\Omega_{z}^{-}(s)$

Furthermore, we can pick $P^{\prime}$ such that

$d(P,P^{\prime})=|\overline{pp^{\prime}}|\leq O(\sqrt{d_{i}}).$ (6.8)

To see this, by the law of sines and cosines in hyperbolic space, denote
$\angle A_{1}A_{2}A_{3}$ as the angle between $\overline{A_{1}A_{2}}$ and
$\overline{A_{2}A_{3}}$, we have

$\left\\{\begin{aligned}
&\cosh|\overline{pp^{\prime}}|\cosh|\overline{e^{\prime}p^{\prime}}|=\cosh(\rho_{\infty}+d_{i}),\\\
&\cosh|\overline{e^{\prime}p^{\prime}}|\cosh|\overline{ee^{\prime}}|=\cosh(\rho_{\infty}-d_{i})\cosh|\overline{pp^{\prime}}|\left(=\cosh|\overline{ep^{\prime}}|\right)\\\
&\frac{\sinh|\overline{e^{\prime}p^{\prime}}|}{\sinh(\rho_{\infty}+d_{i})}=\sin\angle
e^{\prime}pp^{\prime}=\cos\angle
epe^{\prime}=\frac{\cosh(\rho_{\infty}-d_{i})\cosh(\rho_{\infty}+d_{i})-\cosh|\overline{ee^{\prime}}|}{\sinh(\rho_{\infty}-d_{i})\sinh(\rho_{\infty}+d_{i})}.\end{aligned}\right.$
(6.9)

Solving (6.9), we get

$\sinh(\rho_{\infty}-d_{i})\left(1+\sinh^{2}|\overline{e^{\prime}p^{\prime}}|\right)=\cosh(\rho_{\infty}-d_{i})\cosh(\rho_{\infty}+d_{i})\sinh|\overline{e^{\prime}p^{\prime}}|.$

We see that there is a solution with
$\sinh|\overline{e^{\prime}p^{\prime}}|=\sinh\rho_{\infty}+O(d_{i})$ and hence
$\cosh|\overline{pp^{\prime}}|=1+O(d_{i}),\
|\overline{pp^{\prime}}|=O(\sqrt{d_{i}})$.

Next, we claim that
$S_{\gamma_{z}}^{+}(\Omega_{t_{i}})\leq|\overline{pp^{\prime}}|$, which yields
$\lim_{t\to\infty}S_{\gamma_{z}}^{+}(\Omega_{t})\leq 0$. To see this, we have
to prove that for $s:=|\overline{pp^{\prime}}|$,
$R_{\gamma_{z},s}(\Omega^{+}_{t_{i}}(s))\subset\Omega^{-}_{t_{i}}(s)$.

Note that $\Omega^{+}_{t_{i}}(s)$ is contained in the cylinder region whose
boundary corresponds to the union of normal geodesic lines through
$e^{\prime}\in E^{\prime}$ in direction perpendicular to $P^{\prime}$. Hence
$\forall A\in\Omega_{t_{i},z}^{+}(s)$, the geodesic line which goes through
$A$ and is perpendicular to $P^{\prime}$, would intersect $\partial
B_{\rho_{\infty}+d_{i}}(p)\cap H_{z,s}^{+}$, $P^{\prime}$ and $\partial
B_{\rho_{\infty}-d_{i}}(p)\cap H_{z,s}^{-}$ at some points $r$, $b$, $l$
respectively. Let $a$ be the closest point on $\overline{rl}$ to $p$, see
figure 3. By the convexity of $\Omega_{t_{i}}$ and that
$B_{\rho_{\infty}-d_{i}}(p)\subset\Omega_{t_{i}}\subset
B_{\rho_{\infty}+d_{i}}(p)$, we have

$\overline{lb}\subset\Omega_{t_{i},z}^{-}(s)$

and $|\overline{bA}|\leq|\overline{br}|$. For our purpose, if we can show

$|\overline{lb}|>|\overline{br}|,$

then the reflection argument follows.

Denote $\beta=|\overline{bp^{\prime}}|$. Again by the law of sines and cosines
in hyperbolic space, one can calculate

$\tanh|\overline{pp^{\prime}}|=\tanh|\overline{ab}|\cosh\beta,$

hence $|\overline{ab}|$ is monotonously decreasing in $\beta$. It’s obvious
that $|\overline{ap}|$ is monotonously increasing in $\beta$, thus

$|\overline{la}|-|\overline{ar}|=\text{arccosh}\left(\frac{\cosh(\rho_{\infty}-d_{i})}{\cosh|\overline{ap}|}\right)-\text{arccosh}\left(\frac{\cosh(\rho_{\infty}+d_{i})}{\cosh|\overline{ap}|}\right)$

is monotonously decreasing in $\beta$. Then

$|\overline{lb}|-|\overline{br}|=(|\overline{la}|-|\overline{ar}|)+2|\overline{ab}|\geq\left(|\overline{lb}|-|\overline{br}|\right)\bigg{|}_{\beta=|\overline{p^{\prime}e^{\prime}}|}=|\overline{ee^{\prime}}|>0.$
(6.10)

The same argument can also show
$\lim_{t\to\infty}S_{\gamma_{z}}^{-}(\Omega_{t})\geq 0$. Then

$0\leq\lim_{t\to\infty}S_{\gamma_{z}}^{-}(\Omega_{t_{i}})\leq\lim_{t\to\infty}S_{\gamma_{z}}^{+}(\Omega_{t_{i}})\leq
0,$

which finishes our proof. ∎

##  7\. Proofs of Theorem 1.2 and Corollary 1.3

In this section, we complete the proofs of Theorem 1.2 and Corollary 1.3.

###  7.1. Proof of Theorem 1.2

Firstly, we prove the following uniform estimate for the principal curvatures
of $M_{t}$ along the flow (1.1).

######  Lemma 7.1.

Let $M_{0}$ be a smooth, closed and convex hypersurface in
$\mathbb{H}^{n+1}(n\geq 2)$, and $M_{t}$ be the smooth solution of the flow
(1.1) starting from $M_{0}$. Then there exists constants $\underline{\kappa}$,
$\overline{\kappa}$ depending only on $n$, $\alpha$ and $M_{0}$ such that the
principal curvatures $\kappa_{i}$ of $M_{t}$ satisfy:

$\underline{\kappa}\leq\kappa_{i}\leq\overline{\kappa},\qquad i=1,\dots,n$
(7.1)

for all time $t\in[0,+\infty)$.

###### Proof.

By Lemma 6.6, the center $p_{t}$ of an inner ball of $\Omega_{t}$ converges to
a fixed point $p$ as $t\to\infty$. Since the inner radius of $\Omega_{t}$ has
a positive lower bound $\rho_{-}(t)>c_{1}$, there exists a sufficiently large
time $t^{*}$, depending on $c_{1}$ and hence depending only on $n$, $\alpha$
and $M_{0}$, such that $d(p_{t},p)<{c_{1}}/{4}$ for $t\geq t^{*}$. Then we
have:

$B_{c_{1}/4}(p)\subset\Omega_{t},\qquad\forall~{}t\geq t^{*}.$ (7.2)

Applying Proposition 4.4 to the time interval $[0,t^{*})$ and $[t^{*},\infty)$
respectively gives a uniform lower bound for the principal curvatures of
$M_{t}$ for all time $t>0$. In fact, on the time interval $[0,t^{*})$, the
estimate (4.6) implies that the principal curvatures $\kappa_{i}$ of $M_{t}$
satisfy

$k_{i}\geq\Lambda_{3}^{-1}\Lambda_{1}^{-\frac{2t^{*}}{\tau}},\qquad
t\in[0,t^{*}).$ (7.3)

While for time $t\in[t^{*},\infty)$, since $B_{c_{1}/4}(p)\subset\Omega_{t}$
for all time $t\in[t^{*},\infty)$, without loss of generality, we can view the
point $p$ as the origin and project $\Omega_{t}$ onto
$B_{1}(0)\subset\mathbb{R}^{n+1}$ with respect to the point $p$ for all time
$t\geq t^{*}$. By the estimate (4.2) in the proof of Proposition 4.4, we have

$\mathfrak{r}_{11}(z,t)\leq\Lambda_{1}\max\left\\{\Lambda,\max_{z\in\mathbb{S}^{n}}\mathfrak{r}_{11}(z,t^{*})\right\\}$

for all $t\in[t^{*},\infty)$. This together with (7.3) and (3.25) implies that
the principal curvatures of $M_{t}$ are uniformly bounded from below by a
positive constant $\underline{\kappa}$ which depends only on $n$, $\alpha$ and
$M_{0}$.

Since we get the uniform lower bound for the principal curvatures, the uniform
upper bound for the Gauss curvature $K$ follows easily from the proof of
Proposition 4.5. In fact, the upper bound (4.22) for the terms $(I)$ in the
proof of Proposition 4.5 now has a uniform coefficient $\beta_{1}$, which is
independent of time $t$. Therefore there exists a constant
$\overline{\kappa}$, such that $\kappa_{i}\leq\overline{\kappa}$ for all
$i=1,\dots,n$. This completes the proof of Lemma 7.1. ∎

It follows from Lemma 7.1 that the flow (1.1) is uniformly parabolic for all
time $t>0$. Then an argument similar to that in the proof of Theorem 5.1 can
be applied to show that all derivatives of curvatures are uniformly bounded on
$M_{t}$ for all $t>0$. This together with Lemma 6.3 implies there exists a
sequence of times $t_{i}\to\infty$, such that $M_{t_{i}}$ converges smoothly
to a geodesic sphere $S_{\rho_{\infty}}(p)$ as $t_{i}\to\infty$.

The full convergence and the exponential convergence can be obtained by
studying the linearization of the flow (1.1). Fix a sufficiently large time
$t_{i}$, we write $M_{t}$ for $t\geq t_{i}$ as the graph of the radial
function $\rho(\cdot,t)$ over $\mathbb{S}^{n}$ centered at the fixed point
$p\in\mathbb{H}^{n+1}$ and the flow equation (1.1) is equivalent to the
following scalar parabolic PDE

$\left\\{\begin{aligned} \frac{\partial}{\partial
t}\rho&=\left(\phi(t)-K^{\alpha}\right)\sqrt{1+{|\bar{\nabla}\rho|^{2}}/{\sinh^{2}\rho}},\quad
t>t_{i},\\\ \rho(\cdot&,t_{i})=\rho_{t_{i}}(\cdot),\end{aligned}\right.$ (7.4)

for the radial function $\rho$ over the sphere $\mathbb{S}^{n}$, where $K$ is
expressed as a function of $\rho,\bar{\nabla}\rho$ and $\bar{\nabla}^{2}\rho$
via the equation (2.1). Since $M_{t_{i}}$ converges to a geodesic sphere
$S_{\rho_{\infty}}(p)$ as $t_{i}\to\infty$, we can assume that the oscillation
of $\rho(\cdot,{t_{i}})-\rho_{\infty}$ is sufficiently small by choosing
$t_{i}$ large enough. By a direct computation, the linearized equation of the
flow (7.4) about the geodesic sphere $S_{\rho_{\infty}}(p)$ is given by

$\frac{\partial}{\partial
t}\eta=\frac{\alpha\coth^{\alpha-1}{\rho_{\infty}}}{n\sinh^{2}\rho_{\infty}}\left(\bar{\Delta}\eta+n\eta-\frac{n}{|\mathbb{S}^{n}|}\int_{\mathbb{S}^{n}}{\eta\,d\sigma}\right).$
(7.5)

Since the oscillation of $\rho(\cdot,{t_{i}})-\rho_{\infty}$ is already
sufficiently small, it follows exactly as in [8], using [13], that the
solution $\rho(\cdot,t)$ of (7.4) starting at $\rho(\cdot,{t_{i}})$ exists for
all time and converges exponentially to a constant $\rho_{\infty}$. This means
that the hypersurface $\overline{M}_{t}=$ graph $\rho(\cdot,t)$ solves (1.1)
with initial condition $M_{t_{i}}$ and by uniqueness $\overline{M}_{t}$
coincides with $M_{t}$ for $t\geq t_{i}$, and hence the solution $M_{t}$ of
(1.1) with initial condition $M_{0}$ converges exponentially as $t\to\infty$
to the geodesic sphere $S_{\rho_{\infty}}(p)$. This completes the proof of
Theorem 1.2.

###  7.2. Proof of Corollary 1.3

Finally, we give the proof of Corollary 1.3 using the monotonicity of
$\mathcal{A}_{n-1}(\Omega_{t})$ and the convergence result of the flow (1.1).
Firstly, if $\Omega$ is convex, we evolve the boundary $M=\partial\Omega$ by
the flow $\eqref{flow-VMCF}$. Then the inequality (1.5) follows from Theorem
1.2 and the monotonicity in Lemma 6.1 immediately. If equality holds in (1.5)
for such $\Omega$, then equality also holds in (6.1) for all time $t$ which
means that $\Omega$ and $\Omega_{t}$ are all geodesic balls.

In general, for weakly convex $\Omega$ we can approximate $\Omega$ by a family
of convex domains $\Omega_{\varepsilon}$ as $\varepsilon\to 0$. In fact, if we
project the domain $\Omega$ into $B_{1}(0)\subset\mathbb{R}^{n+1}$ as in
$\S$2, the equation (3.2) implies that the image $\hat{\Omega}$ of the
projection is also weakly convex in $\mathbb{R}^{n+1}$. Hence we can
approximate $\hat{\Omega}$ by convex domains (e.g., by using mean curvature
flow). Since the projection is a diffeomorphism, we find a family of convex
domains $\Omega_{\varepsilon}$ that approximate $\Omega$ as $\varepsilon\to
0$. Then the inequality (1.5) for $\Omega$ follows from the one for
$\Omega_{\varepsilon}$ by letting $\varepsilon\to 0$.

To prove the equality case for weakly convex $\Omega$, we employ an argument
previously used in [15]. Suppose that $\Omega$ is a weakly convex domain which
attains the equality of (1.5). Let $M_{+}=\\{x\in M=\partial\Omega,K>0\\}$.
Since there exists at least one point $p$ on a closed hypersurface in
$\mathbb{H}^{n+1}$ such that all the principal curvatures are strictly larger
than 1 at $p$, the subset $M_{+}$ is open and nonempty. We claim that $M_{+}$
is closed as well. In fact, pick any $\eta\in C_{0}^{2}(M_{+})$ compactly
supported in $M_{+}$, let $M_{\varepsilon}$ be a smooth family of variational
hypersurfaces generated by the vector field $V=\eta\nu$. Let
$\Omega_{\varepsilon}$ be the domain enclosed by $M_{\varepsilon}$. It is easy
to show that $M_{\varepsilon}$ is weakly convex when $|\varepsilon|$ is small
enough. Hence

$\mathcal{A}_{n-1}(\Omega_{\varepsilon})\geq\psi_{n}\left(|\Omega_{\varepsilon}|\right)$

holds for sufficiently small $|\varepsilon|$ and with equality holding at
$\varepsilon=0$. Thus

$\displaystyle 0=$
$\displaystyle\frac{d}{d\varepsilon}\bigg{|}_{\varepsilon=0}\Big{(}\mathcal{A}_{n-1}(\Omega_{\varepsilon})-\psi_{n}\left(|\Omega_{\varepsilon}|\right)\Big{)}=n\int_{M}{\left(K-\psi^{\prime}_{n}(|\Omega|)\right)\eta}d\mu.$

Since $\eta\in C_{0}^{2}(M_{+})$ is arbitrary, we have
$K=\psi^{\prime}_{n}(|\Omega|)$ everywhere on $M_{+}$. As this is a closed
condition, we conclude that $M_{+}$ is closed. Therefore $M=M_{+}$ and so
$\Omega$ is a convex domain. Then by the equality case for convex domain, we
conclude that $\Omega$ is a geodesic ball.

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