# Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space

Yong Wei  School of Mathematical Sciences, University of Science and
Technology of China, Hefei 230026, P.R. China<EMAIL_ADDRESS>,  Bo Yang
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R.
China<EMAIL_ADDRESS>and  Tailong Zhou  School of Mathematics, Sichuan
University, Chengdu 610065, Sichuan, P.R. China<EMAIL_ADDRESS>
###### Abstract.

We consider the volume preserving flow of smooth, closed and convex
hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1}$ with speed given by a
general nonhomogeneous function of the Gauss curvature. For a large class of
speed functions, we prove that the solution of the flow remains convex, exists
for all positive time $t\in[0,\infty)$ and converges to a geodesic sphere
exponentially as $t\to\infty$ in the smooth topology. A key step is to show
the $L^{1}$ oscillation decay of the Gauss curvature to its average along a
subsequence of times going to the infinity, which combined with an argument
using the hyperbolic curvature measure theory implies the Hausdorff
convergence.

###### Key words and phrases:

Gauss Curvature flow, Nonhomogeneous, Hyperbolic space, Convex hypersurface.

###### 2020 Mathematics Subject Classification:

53E10; 53C42

##  1\. Introduction

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

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

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

$\phi(t)=\frac{1}{|M_{t}|}\int_{M_{t}}f(K)\,\mathrm{d}\mu_{t}$ (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). We assume that the function
$f:[0,\infty)\to\mathbb{R}$ satisfies the following conditions:

######  Assumption 1.1.

$f:[0,+\infty)\to\mathbb{R}$ is continuous and $C^{2}$ differentiable in
$(0,+\infty)$, and satisfies the following conditions:

  1. (1)

$f(x)>0$, $f^{\prime}(x)>0$ for $x>0$;

  2. (2)

$\lim\limits_{x\to{+\infty}}f(x)=+\infty$;

  3. (3)

$\exists\Theta>0$ such that $f(x)\leq\Theta f^{\prime}(x)x$ for $x>0$;

  4. (4)

$xf^{\prime\prime}(x)+f^{\prime}(x)\geq 0$ for $x>0$.

######  Remark 1.2.

A particular example of functions satisfying Assumption 1.1 is

$f(x)=x^{\alpha},\quad\alpha>0.$

In this case, the flow (1.1) is the classical volume preserving $\alpha$-Gauss
curvature flow (1.5) in $\mathbb{H}^{n+1}$, which was considered by the
authors in [WYZ2022]. Other natural examples include the linear combination of
powers

$f(x)=\sum_{i=1}^{d}a_{i}x^{k_{i}},\quad a_{i}>0,\quad k_{i}>0$

and

$\displaystyle f(x)=$ $\displaystyle\mathrm{e}^{x}-1,$ $\displaystyle f(x)=$
$\displaystyle x^{\alpha}\ln(x+1),\quad\alpha>0.$

Moreover, any positive strictly increasing convex $C^{2}$ function $f$ with
$f(0)=0$ satisfies Assumption 1.1.

Before describing our main result, we recall the following definitions of
convexity of hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1}$.

######  Definition 1.3.

  1. (1)

A smooth closed hypersurface $M$ in $\mathbb{H}^{n+1}$ is called convex if all
of its principal curvatures $\kappa_{i},i=1,\cdots,n$ are positive everywhere
on $M$. We also say that a hypersurface is weakly convex if all principal
curvatures $\kappa_{i}\geq 0$.

  2. (2)

A smooth closed hypersurface $M$ in $\mathbb{H}^{n+1}$ is called $h$-convex
(also called horospherically convex), if all of its principal curvatures
$\kappa_{i}\geq 1,i=1,\cdots,n$ everywhere on $M$.

  3. (3)

We say that a smooth closed hypersurface $M$ in $\mathbb{H}^{n+1}$ has
positive sectional curvatures if its sectional curvatures
$K(e_{i},e_{j})=\kappa_{i}\kappa_{j}-1>0$ for all $i\neq j\in\\{1,\dots,n\\}$.
This condition is weaker than $h$-convexity but is stronger than the convexity
$\kappa_{i}>0,i=1,\cdots,n.$

Our main result of this paper is the following convergence result for the flow
(1.1) with convex initial hypersurface.

######  Theorem 1.4.

Let $X_{0}:M^{n}\to\mathbb{H}^{n+1}$ be a smooth embedding such that
$M_{0}=X_{0}(M)$ is a closed convex hypersurface in $\mathbb{H}^{n+1}$. Then
the volume preserving flow (1.1) with $f$ satisfying Assumption 1.1 has a
unique smooth convex solution $M_{t}$ for all time $t\in[0,\infty)$, and as
$t\to\infty$ the solution $M_{t}$ converges smoothly and exponentially 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 [Hui87] 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 flow (1.3) converges smoothly to a round
sphere. There are further generalizations of the flow (1.3) for convex
hypersurfaces in $\mathbb{R}^{n+1}$ with $H$ replaced by more general
curvature functions including powers of the $k$th mean curvature
$\sigma_{k}(\kappa),k=1,\cdots,n$. See [AW21, BeSin18, BeSin18b, CS10, Mc04,
Mc05, Mc17, Sine15] for instance. In particular, Bertini and Sinestrari
[BeSin18] considered the volume preserving non-homogeneous mean curvature flow
of convex hypersurfaces in $\mathbb{R}^{n+1}$

$\frac{\partial}{\partial t}X(x,t)=(\phi(t)-f(H))\nu(x,t),$ (1.4)

for a large class of nonhomogeneous function $f(H)$ of the mean curvature with
$f$ satisfying some conditions.

The analogue of the flow (1.3) in the hyperbolic space $\mathbb{H}^{n+1}$ was
first studied by Cabezas-Rivas and Miquel[Cab-Miq2007] in 2007 assuming that
the initial hypersurface is $h$-convex. The nonhomogeneous version (1.4) in
$\mathbb{H}^{n+1}$ was considered by Bertini and Pipoli [Be-Pip2016] also
assuming $h$-convexity initially. Generalizations of the flow (1.3) in
hyperbolic space with $H$ replaced by more general curvature functions were
studied in [BenChenWei, BenWei, GLW-CAG, Mak2012, WX]. In most cases, the
$h$-convexity is assumed in order to prove the smooth convergence of the
solution to geodesic spheres. The main reasons are that the $h$-convexity is
convenient for the analysis of the curvature evolution equations and also
$h$-convexity implies that the outer radius of the enclosed domain is
uniformly controlled by its inner radius. In [BenChenWei], the first author
with Andrews and Chen proved the smooth convergence of volume preserving $k$th
mean curvature flows in $\mathbb{H}^{n+1}(n\geq 2)$ for initial hypersurfaces
with positive sectional curvatures. This condition is weaker than
$h$-convexity but still stronger than the convexity.

Recently, the authors [WYZ2022] considered the volume preserving
$\alpha$-Gauss curvature flow

$\frac{\partial}{\partial t}X(x,t)=(\phi(t)-K^{\alpha})\nu(x,t)$ (1.5)

for convex hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1}$. We showed
that for any $\alpha>0$, the flow (1.5) evolves any smooth, closed convex
hypersurface in $\mathbb{H}^{n+1}$ to a geodesic sphere. The key ingredient we
used in [WYZ2022] is the projection method via the Klein model of the
hyperbolic space which was described earlier by the first author and Andrews
in [BenWei]. Based on this we treated the flow 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 continuity argument then implies that the flow exists
for all positive time. Moreover, for the convergence result we used the
curvature measure theory for convex bodies in $\mathbb{H}^{n+1}$ to show the
Hausdorff convergence and an Alexandrov reflection argument to control the
center of the inner ball of the evolving domains. The curve case of (1.5) was
also treated earlier by the first and second authors in [WY2022]. The results
in [WY2022, WYZ2022] are the first results on non-local type volume preserving
curvature flows in the hyperbolic space with only convexity required on the
initial hypersurface.

Our Theorem 1.4 in this paper generalizes the results [WY2022, WYZ2022] on
volume preserving $\alpha$-Gauss curvature flow (1.5) to volume preserving
nonhomogeneous Gauss curvature flow (1.1) for a large class of nonhomogeneous
functions. In this paper, instead of projecting the flow (1.1) to the
Euclidean space as in [WYZ2022], we prove the curvature lower bound directly
by analyzing an auxiliary function (see §3.2 for details)

$Q(p,t)=\log{\mathfrak{b}}+A\rho,$ (1.6)

where $\mathfrak{b}$ is the largest reciprocal of the principal curvatures
$\kappa_{i},i=1,\cdots,n$, and $\rho$ is the radial function of the evolving
hypersurface $M_{t}$ on the time interval when $M_{t}$ is star-shaped with
respect to some point $p_{0}$. This avoids the use of more complicated
projection method. To show the upper bound on the Gauss curvature, we employ
Tso’s method as in [WYZ2022]. Since we only assumed convexity and the speed
function $f$ is not necessarily homogeneous, the analysis of $f$ and the terms
involving global term $\phi(t)$ need to be carefully treated.

For the long time existence of the flow, we need to show that the solutions
remain smooth as long as the curvatures are bounded. For general non-
homogeneous function $f$, the flow (1.1) may not be concave with respect to
spacial second derivatives, we can not apply Krylov-Safonov’s result to derive
the $C^{2,\alpha}$ estimate of the solution. Instead, we apply a result in
[And2004, Theorem 6] and view $f(K)$ as an increasing function of a concave
operator $K^{1/n}$. To study the asymptotical behavior of $M_{t}$ as
$t\to\infty$, we note that the a prior estimates we obtained depend on the
time and may degenerate as time $t\to\infty$. To overcome this difficulty, as
in [WYZ2022] we employ the curvature measure theory for convex bodies in
hyperbolic space: For general nonhomogeneous function, we still have the
monotonicity of the quermassintegral $\mathcal{A}_{n-1}(\Omega_{t})$ of the
evolving domain $\Omega_{t}$. This together with the long time existence of
the flow implies that

$\int_{M_{t_{i}}}{|K-\bar{K}|d\mu_{t_{i}}}\to
0,\quad\text{as}\,\,t_{i}\to\infty.$ (1.7)

along a subsequence of times $t_{i}\to\infty$, where $\bar{K}$ denotes the
average integral of $K$. Then we can argue as in [WYZ2022] to conclude the
Hausdorff convergence using curvature measure theory and Alexandrov reflection
method, and then improve it to the smooth convergence.

Throughout the proof, the presence of the nonhomogeneous speed function $f(K)$
introduces some technical difficulties. Assumption 1.1 is crucial for proving
the a prior estimates of the flow (1.1). In fact, the flow (1.1) is parabolic
and has short-time existence due to item (1); items (3) and (4) are used to
prove the time-dependent lower bound on the principal curvatures (see §3.2);
and items (2) - (3) are used to derive the time-dependent upper bound on the
Gauss curvature of the flow (see §3.3).

######  Remark 1.5.

  1. (1)

We remark that nonhomogeneous curvature flows have been considered previously
in the literature for the other type of curvature flows. See [AleSin10,
Guan23, LiLv20] for the contracting curvature flow case and [ChoT97, ChoT98,
Pip22] for the expanding curvature flow case.

  2. (2)

The non-local type volume preserving curvature flows have also been studied in
other aspects and in other ambient spaces. We refer the readers to [AndWei20,
Cab23] for surveys of the volume preserving type curvature flows and their
geometric applications. Some other recent interesting works include [CSV20]
for the volume preserving fractional mean curvature flow in the Euclidean
space, [Lam23] for singularity analysis of volume preserving mean curvature
flow in the Euclidean space and [CaSche23] for a new non-local volume
preserving mean curvature flow in the sphere.

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) and the quermassintegrals in hyperbolic space.
In $\S$3.1, we give the $C^{0}$ and $C^{1}$ estimates of the flow (1.1). In
$\S$3.2, we prove the time-dependent positive lower bound for the principal
curvatures along the flow (1.1), by analyzing the auxiliary function (1.6). In
§3.3 we prove a time-dependent upper bound on the Gauss curvature $K$. 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$4. In §5, 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 §6, we complete the proof of Theorem 1.4.

######  Acknowledgments.

The research was supported by National Key R and D Program of China
2021YFA1001800 and 2020YFA0713100, the Fundamental Research Funds for the
Central Universities and Shuimu Tsinghua Scholar Program (No. 2023SM102). The
authors would like thank Ruijia Zhang for discussions.

##  2\. Preliminaries

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

###  2.1. Hyperbolic space

The hyperbolic space $\mathbb{H}^{n+1},n\geq 1$ 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. The eigenvalues of the Weingarten tensor
$h_{i}^{j}:=g^{jk}h_{ki}$ are the principal curvatures
$\kappa=(\kappa_{1},\cdots,\kappa_{n})$. 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 the 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 the point
$p_{0}\in\Omega$ 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.[GL15, §4])

$\displaystyle g_{ij}$
$\displaystyle=\rho_{i}\rho_{j}+\sinh^{2}\rho\sigma_{ij},$ (2.1)
$\displaystyle h_{ij}$
$\displaystyle=\frac{1}{\sqrt{\sinh^{2}\rho+|\bar{\nabla}\rho|^{2}}}(-(\sinh\rho)\rho_{ij}+2(\cosh\rho)\rho_{i}\rho_{j}+\sinh^{2}\rho\cosh\rho\sigma_{ij}),$
(2.2) $\displaystyle\nu$
$\displaystyle=\frac{1}{\sqrt{1+|\bar{\nabla}\rho|^{2}/\sinh^{2}\rho}}\left(1,-\frac{\rho_{1}}{\sinh^{2}\rho},\cdots,-\frac{\rho_{n}}{\sinh^{2}\rho}\right),$
(2.3) $\displaystyle u$
$\displaystyle=\frac{\sinh^{2}\rho}{\sqrt{\sinh^{2}\rho+|\bar{\nabla}\rho|^{2}}},$
(2.4)

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

$K=\frac{\det h_{ij}}{\det
g_{ij}}=\frac{\det(-\sinh\rho\rho_{ij}+2\cosh\rho\rho_{i}\rho_{j}+\sinh^{2}\rho\cosh\rho\sigma_{ij})}{(\sinh^{2}\rho+|\bar{\nabla}\rho|^{2})^{\frac{n+2}{2}}(\sinh\rho)^{2(n-1)}}.$
(2.5)

###  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
[BenWei]) for the induced metric $g_{ij}$, the area element $d{\mu_{t}}$ and
the speed function $f(K)$:

$\displaystyle\frac{\partial}{\partial t}g_{ij}$
$\displaystyle=2\left(\phi(t)-f(K)\right)h_{ij},$ (2.6)
$\displaystyle\frac{\partial}{\partial t}d\mu_{t}$
$\displaystyle=H\left(\phi(t)-f(K)\right)d\mu_{t},$ (2.7)
$\displaystyle\frac{\partial}{\partial t}f(K)$
$\displaystyle=f^{\prime}\dot{K}^{ij}\left(\nabla_{i}\nabla_{j}{f}+(f-\phi(t))(h_{i}^{k}h_{k}^{j}-\delta_{i}^{j})\right),$
(2.8)

where $f^{\prime}=f^{\prime}(K)$ is the derivative of $f$, $\dot{K}^{ij}$
denote the derivatives with respect to components of the second fundamental
form, and $\nabla$ denotes the Levi-Civita connection on $M_{t}$ with respect
to the induced metric $g_{ij}$.

Let $M_{t}$ be a smooth convex solution to the flow (1.1) on the time interval
$[0,T)$ and assume that $\\{b_{m}^{n}\\}$ is the inverse matrix of the
Weingarten matrix $\\{h_{i}^{j}\\}$. The following lemma gives parabolic type
evolution equations of $h_{i}^{j}$ and $b_{m}^{n}$.

######  Lemma 2.1.

Along the flow (1.1), the Weingarten matrix $h_{i}^{j}$ of $M_{t}$ evolves by

$\displaystyle\partial_{t}{h_{i}^{j}}-f^{\prime}\dot{K}^{k\ell}\nabla_{k}\nabla_{\ell}{h_{i}^{j}}=$
$\displaystyle(f^{\prime\prime}\dot{K}^{k\ell}\dot{K}^{pq}+f^{\prime}\ddot{K}^{k\ell,pq})\nabla_{i}{h_{k\ell}\nabla^{j}{h_{pq}}}$
$\displaystyle+f^{\prime}(HK+\sigma_{n-1}(\kappa))h_{i}^{j}-nf^{\prime}K(h_{i}^{p}h_{p}^{j}+\delta_{i}^{j})$
$\displaystyle+(f-\phi(t))(h_{i}^{k}h_{k}^{j}-\delta_{i}^{j}).$ (2.9)

Furthermore, if $M_{t}$ is a smooth convex solution to the flow (1.1) on the
time interval $[0,T)$ and we assume that $\\{b^{k\ell}\\}$ is the inverse
matrix of the second fundamental form $\\{h_{ij}\\}$ and
$\\{b_{m}^{n}:=g_{ms}b^{sn}\\}$ is the inverse matrix of the Weingarten matrix
$\\{h_{i}^{j}\\}$. Then $b_{m}^{n}$ evolves by

$\displaystyle\partial_{t}b_{m}^{n}-f^{\prime}Kb^{k\ell}\nabla_{k}\nabla_{\ell}{b_{m}^{n}}=$
$\displaystyle-f^{\prime}Kb^{k\ell}\left(b_{i}^{n}b_{m}^{p}b_{q}^{s}\nabla_{k}{h_{s}^{i}}\nabla_{\ell}{h_{p}^{q}}+b_{i}^{n}b_{m}^{p}b_{q}^{s}\nabla_{k}{h_{p}^{q}}\nabla_{\ell}{h_{s}^{i}}\right)$
$\displaystyle-(f^{\prime\prime}K^{2}+f^{\prime}K)b^{k\ell}b^{pq}b_{m}^{i}b_{j}^{n}\nabla_{i}{h_{k\ell}}\nabla^{j}{h_{pq}}$
$\displaystyle+f^{\prime}Kb^{kp}b^{ql}b_{m}^{i}b_{j}^{n}\nabla_{i}{h_{k\ell}}\nabla^{j}{h_{pq}}$
$\displaystyle-f^{\prime}(HK+\sigma_{n-1}(\kappa))b_{m}^{n}+nf^{\prime}K(b_{m}^{i}b_{i}^{n}+\delta_{m}^{n})$
$\displaystyle-(f-\phi(t))(\delta_{m}^{n}-b_{m}^{i}b_{i}^{n}).$ (2.10)

###### Proof.

By [BenWei, Lemma 2.4], we have

$\displaystyle\partial_{t}{h_{i}^{j}}=$
$\displaystyle\dot{f}^{k\ell}\nabla_{k}\nabla_{\ell}{h_{i}^{j}}+\ddot{f}^{k\ell,pq}\nabla_{i}{h_{k\ell}\nabla^{j}{h_{pq}}}$
$\displaystyle+(\dot{f}^{k\ell}h_{k}^{r}h_{r\ell}+\dot{f}^{k\ell}g_{k\ell})h_{i}^{j}-\dot{f}^{k\ell}h_{k\ell}(h_{i}^{p}h_{p}^{j}+\delta_{i}^{j})$
$\displaystyle+(f-\phi(t))(h_{i}^{k}h_{k}^{j}-\delta_{i}^{j}),$ (2.11)

where $\dot{f}^{k\ell},\ddot{f}^{k\ell,pq}$ denote the first and second
derivatives of $f$ with respect to the components of second fundamental form
$(h_{ij})$. Since $f=f(K)$ is a function of Gauss curvature $K$, we have

$\displaystyle\dot{f}^{k\ell}=$ $\displaystyle
f^{\prime}\dot{K}^{k\ell},\qquad\dot{f}^{k\ell}h_{k\ell}=nf^{\prime}K,$ (2.12)
$\displaystyle\ddot{f}^{k\ell,pq}=$ $\displaystyle
f^{\prime}\ddot{K}^{k\ell,pq}+f^{\prime\prime}\dot{K}^{k\ell}\dot{K}^{pq}$
(2.13)

and

$\displaystyle\dot{f}^{k\ell}h_{k}^{r}h_{r\ell}+\dot{f}^{k\ell}g_{k\ell}=$
$\displaystyle
f^{\prime}(\dot{K}^{k\ell}h_{k}^{r}h_{r\ell}+\dot{K}^{k\ell}g_{k\ell})$
$\displaystyle=$ $\displaystyle f^{\prime}(HK+\sigma_{n-1}(\kappa)).$ (2.14)

Substituting (2.12) - (2.14) into (2.11) gives (2.9).

To derive equation (2.10), we calculate using
$b^{n}_{m}h_{n}^{k}=\delta_{m}^{k}$ that

$\displaystyle\partial_{t}{b_{m}^{n}}=$ $\displaystyle-
b_{m}^{i}b_{j}^{n}\partial_{t}{h_{i}^{j}}$ $\displaystyle=$
$\displaystyle-f^{\prime}\dot{K}^{k\ell}b_{m}^{i}b_{j}^{n}\nabla_{k}\nabla_{\ell}{h_{i}^{j}}$
$\displaystyle-(f^{\prime\prime}\dot{K}^{k\ell}\dot{K}^{pq}+f^{\prime}\ddot{K}^{k\ell,pq})b_{m}^{i}b_{j}^{n}\nabla_{i}{h_{k\ell}}\nabla^{j}{h_{pq}}$
$\displaystyle-f^{\prime}(HK+\sigma_{n-1})b_{m}^{n}+nf^{\prime}K(b_{m}^{i}b_{i}^{n}+\delta_{m}^{n})$
$\displaystyle-(f-\phi(t))(\delta_{m}^{n}-b_{m}^{i}b_{i}^{n}).$ (2.15)

Note that

$\displaystyle\nabla_{\ell}{b_{m}^{n}}$
$\displaystyle=-b_{i}^{n}b_{m}^{s}\nabla_{\ell}{h_{s}^{i}},$ (2.16)
$\displaystyle\nabla_{k}\nabla_{\ell}b_{m}^{n}$
$\displaystyle=-b_{m}^{i}b_{j}^{n}\nabla_{k}\nabla_{\ell}{h_{i}^{j}}+b_{i}^{n}b_{m}^{p}b_{q}^{s}\nabla_{k}{h_{p}^{q}}\nabla_{\ell}{h_{s}^{i}}$
$\displaystyle\qquad+b_{i}^{n}b_{m}^{p}b_{q}^{s}\nabla_{k}{h_{s}^{i}}\nabla_{\ell}{h_{p}^{q}},$
(2.17) $\displaystyle\dot{K}^{k\ell}$ $\displaystyle=Kb^{k\ell},$ (2.18)
$\displaystyle\ddot{K}^{k\ell,pq}$
$\displaystyle=Kb^{k\ell}b^{pq}-Kb^{kp}b^{q\ell}.$ (2.19)

Combining (2.15)-(2.19) gives (2.10). ∎

On the time interval when $M_{t}$ is star-shaped with respect to some point
$p_{0}$, the radial function $\rho$ evolves by

$\frac{\partial}{\partial
t}\rho=\left(\phi(t)-f(K)\right)\sqrt{1+\frac{|\bar{\nabla}\rho|^{2}}{\sinh^{2}{\rho}}},$
(2.20)

where $K$ is expressed in (2.5) as a function of
$\bar{\nabla}^{2}\rho,\bar{\nabla}\rho$ and $\rho$. 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 Lemma 4.3 in[BenWei])

$\frac{\partial}{\partial
t}{u}=f^{\prime}\dot{K}^{ij}\nabla_{i}\nabla_{j}{u}+\cosh{\rho_{p_{0}}}(x)\Big{(}\phi(t)-f-nKf^{\prime}\Big{)}+f^{\prime}KHu.$
(2.21)

The following lemma will be used to prove the lower bound of the principal
curvatures.

######  Lemma 2.2.

We have

$\nabla_{i}\rho=\langle\partial_{\rho},e_{i}\rangle,\qquad\nabla_{j}\nabla_{i}\rho=\coth\rho\left(g_{ij}-\nabla_{i}\rho\nabla_{j}\rho\right)-\frac{uh_{ij}}{\sinh\rho}.$
(2.22)

###### Proof.

Since $V=\sinh\rho\partial_{\rho}$ is a conformal Killing vector field, i.e.,

$\langle D_{X}(\sinh\rho\partial_{\rho}),Y\rangle=\cosh\rho\langle X,Y\rangle$
(2.23)

for any tangential vector fields $X,Y$ in $\mathbb{H}^{n+1}$, we have (see,
e.g.,[GL15, §2])

$\displaystyle\nabla_{i}\coth\rho=$
$\displaystyle\langle\sinh\rho\partial_{\rho},e_{i}\rangle$
$\displaystyle\nabla_{j}\nabla_{i}\coth\rho=$ $\displaystyle\cosh\rho
g_{ij}-uh_{ij}.$

Observing that

$\displaystyle\nabla_{i}\cosh\rho=$ $\displaystyle\sinh\rho\nabla_{i}\rho,$
$\displaystyle\nabla_{j}\nabla_{i}\coth\rho=$
$\displaystyle\sinh\rho\nabla_{j}\nabla_{i}\rho+\cosh\rho\nabla_{j}\rho\nabla_{\rho},$

we conclude the equation (2.22). ∎

###  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 [Sol05, Definition 2.1] 111Note that the
definition for $\mathcal{A}_{k}$ given here is the same as that for $W_{k+1}$
given in [Sol05] up to a constant. In fact, we have
$\mathcal{A}_{k}=(n+1)\binom{n}{k}W_{k+1}$.):

$\mathcal{A}_{k}(\Omega)=(n-k)\binom{n}{k}\frac{\omega_{k}\cdots\omega_{0}}{\omega_{n-1}\cdots\omega_{n-k-1}}\int_{\mathcal{L}_{k+1}}{\chi(L_{k+1}\cap\Omega)dL_{k+1}}$
(2.24)

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,
$\mathcal{A}_{0}(\Omega)=|\partial\Omega|$. Furthermore, we set

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

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 function 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:

$\displaystyle\mathcal{A}_{1}(\Omega)=$
$\displaystyle\int_{\partial\Omega}{\sigma_{1}}d\mu-n\mathcal{A}_{-1}(\Omega),$
(2.25) $\displaystyle\mathcal{A}_{k}(\Omega)=$
$\displaystyle\int_{\partial\Omega}{\sigma_{k}}d\mu-\frac{n-k+1}{k-1}\mathcal{A}_{k-2}(\Omega),\quad
k=2,\cdots,n.$ (2.26)

The quermassintegrals for smooth domains satisfy a nice variational property
(see [BA97]):

$\frac{d}{dt}\mathcal{A}_{k}(\Omega_{t})=(k+1)\int_{M_{t}}\eta\sigma_{k+1}d\mu_{t},\quad
k=0,\cdots,n-1$ (2.27)

along any normal variation with velocity $\eta$.

The quermassintegrals defined by (2.24) 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

$\mathcal{A}_{k}(E)\leq\mathcal{A}_{k}(F)$ (2.28)

for all $k=0,1,\cdots,n$. Moreover, they are continuous with respect to the
Hausdorff 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
$B_{\lambda}(L):=\\{q\in\mathbb{H}^{n+1}|~{}\mathrm{d}_{\mathbb{H}^{n+1}}(q,L)<\lambda\\}$.

##  3\. A priori estimates

In this section, we first show 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 [BenChenWei, BenWei]. 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 $K$.

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

Let $M_{t}$ be a smooth convex solution to the flow (1.1) on a maximal
existence time interval $[0,T)$. Denote $\Omega_{t}$ the domain enclosed by
$M_{t}$. As the velocity of the flow (1.1) only depends on the curvature and
is invariant under reflection with respect to a totally geodesic hyperplane,
we can argue as in [BenChenWei, §4] using the Alexandrov reflection method to
show that the inner radius and outer radius of $\Omega_{t}$ are uniformly
bounded:

######  Lemma 3.1.

Let $M_{t}$ be a 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 $n$ and $M_{0}$ such that

$0<c_{1}\leq\rho_{-}(t)\leq\rho_{+}(t)\leq c_{2}$ (3.1)

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

By (3.1), the inner radius of $\Omega_{t}$ is bounded below by a positive
constant $c_{1}$. This implies that there exists a geodesic ball of radius
$c_{1}$ contained in $\Omega_{t}$ for each $t\in[0,T)$. The following lemma
says that there exist a geodesic ball with fixed center enclosed by the flow
hypersurface on a suitable fixed time interval. For the proof, we adapt a
similar argument as in [BenWei, Lemma 4.2] but for our nonhomogeneous case we
need the positivity and monotonicity of $f$ as in Assumption 1.1.

######  Lemma 3.2.

Let $M_{t}$ be a 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

$B_{\rho_{0}/2}(p_{0})\subset\Omega_{t},\ \
t\in[t_{0},\min\\{T,t_{0}+\tau\\})$ (3.2)

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

###### Proof.

Given $p_{0}$, we denote by $\rho_{p_{0}}$ the distance function to $p_{0}$ in
$\mathbb{H}^{n+1}$ and by $\partial_{\rho}=\partial_{\rho_{p_{0}}}$ the
gradient vector of $\rho_{p_{0}}$. For any $x\in M_{t}$,

$\displaystyle\frac{\partial}{\partial t}\sinh^{2}\rho_{p_{0}}(x)=$
$\displaystyle
2\langle\sinh\rho_{p_{0}}(x)\partial\rho,\frac{\partial}{\partial
t}(\sinh\rho_{p_{0}}(x)\partial_{\rho})\rangle$ $\displaystyle=$
$\displaystyle
2\sinh\rho_{p_{0}}(x)\cosh\rho_{p_{0}}(x)\Big{(}\phi(t)-f(K(x,t))\Big{)}\langle\partial_{\rho},\nu\rangle,$
(3.3)

where we used the conformal property (2.23) of $\sinh\rho\partial\rho$. It
follows from (3.1) that

$\displaystyle\frac{\partial}{\partial t}\rho_{p_{0}}(x)=$
$\displaystyle\Big{(}\phi(t)-f(K(x,t))\Big{)}\langle\partial\rho,\nu\rangle$
$\displaystyle\geq$
$\displaystyle~{}-f(K(x,t))\langle\partial_{\rho},\nu\rangle,$

since $\phi(t)>0$ and $\langle\partial_{\rho},\nu\rangle>0$ on $M_{t}$.

Denote $\rho(t)=\min_{M_{t}}\rho_{p_{0}}(x)$. At the minimum point, we have
$\langle\partial_{\rho},\nu\rangle=1$ and $\kappa_{i}\leq\coth\rho(t)$. Since
$f$ is strictly increasing, we have

$f(K)\leq f(\coth^{n}\rho(t))$

at the minimum point and so

$\frac{d}{dt}\rho(t)\geq-f(\coth^{n}\rho(t)).$ (3.4)

Let $\bar{\rho}(t)$ be the solution of the ODE

$\left\\{\begin{aligned}
\frac{d}{dt}\bar{\rho}(t)=&-f(\coth^{n}\bar{\rho}(t))\\\
\bar{\rho}(t_{0})=&\rho_{0}.\end{aligned}\right.$ (3.5)

Denote $\tau$ as the time such that $\bar{\rho}(t_{0}+\tau)=\rho_{0}/2$. Since
$f$ is positive, $\bar{\rho}(t)$ is strictly decreasing. It follows that the
inverse $t=t(\bar{\rho})$ of the function $\bar{\rho}(t)$ is well-defined and
satisfies

$\frac{dt}{d\bar{\rho}}=-\frac{1}{f(\coth^{n}\bar{\rho})}.$

Integrating this equation gives

$\displaystyle\tau=$ $\displaystyle\int_{t_{0}}^{t_{0}+\tau}dt$
$\displaystyle=$
$\displaystyle-\int_{\rho_{0}}^{\rho_{0}/2}\frac{d\bar{\rho}}{f(\coth^{n}\bar{\rho})}$
$\displaystyle=$
$\displaystyle\int_{\rho_{0}/2}^{\rho_{0}}\frac{ds}{f(\coth^{n}s)},$

which depends only on the bounds of $\rho_{0}$ and not on $t_{0}$.

Since $\rho(t_{0})=\rho_{0}$, by comparing (3.4) and the ODE (3.5), we
conclude that

$\rho(t)\geq\frac{\rho_{0}}{2},\quad\forall~{}t\in[t_{0},\min\\{t_{0}+\tau,T\\}).$

This means that $B_{\rho_{0}/2}(p_{0})\subset\Omega_{t}$ for all
$t\in[t_{0},\min\\{t_{0}+\tau,T\\})$. ∎

Let $M_{t}$ be a 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 $\rho_{0}=\rho_{-}(t_{0})$. 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 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 (3.1) and (3.2), we see

$\begin{split}u(x,t)&\geq\sinh\left(\frac{\rho_{0}}{2}\right)\geq\sinh\left(\frac{c_{1}}{2}\right)=:2c,\\\
u(x,t)&\leq\sinh(2c_{2})\end{split}$ (3.6)

and

$0<\frac{c_{1}}{2}\leq\rho_{p_{0}}(t)\leq 2c_{2}<\infty$ (3.7)

for any $t\in[t_{0},\min\\{T,t_{0}+\tau\\})$. By (2.4), we have the $C^{1}$
estimate on $\rho$,

$|\bar{\nabla}\rho|\leq\sqrt{\sinh^{2}\rho+|\bar{\nabla}\rho|^{2}}=\frac{\sinh^{2}\rho}{u}\leq\frac{\sinh^{2}(2c_{2})}{\sinh\left({c_{1}}/{2}\right)}$
(3.8)

on the time interval $t\in[t_{0},\min\\{T,t_{0}+\tau\\})$, where
$|\bar{\nabla}\rho|$ is the norm of the gradient of $\rho$ with respect to the
round metric on $\mathbb{S}^{n}$.

Moreover, we have

$\displaystyle 1-|\nabla\rho|^{2}=$
$\displaystyle|D\rho|^{2}-|\nabla\rho|^{2}$ $\displaystyle=$
$\displaystyle\langle D\rho,\nu\rangle^{2}=\frac{u^{2}}{\sinh^{2}\rho}$
$\displaystyle\geq$
$\displaystyle\frac{\sinh^{2}(c_{1}/2)}{\sinh^{2}(2c_{2})}~{}=:c_{3}$ (3.9)

holds on $t\in[t_{0},\min\\{T,t_{0}+\tau\\})$, where $D$ and $\nabla$ denote
the Levi-Civita connection on $\mathbb{H}^{n+1}$ and on $M_{t}$ with respect
to the induced metric. The estimate (3.1) will be used crucially in the proof
of Proposition 3.3.

###  3.2. Preserving convexity

In this subsection, we show that the solution $M_{t}$ of the flow (1.1)
preserves the convexity. This follows from the following (time-dependent)
lower bound on the principal curvatures of $M_{t}$.

######  Proposition 3.3.

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
$M_{0}$. If $T<\infty$, then there exist constants $\Lambda_{1}$ and
$\Lambda_{2}$ depending only on $n,M_{0}$ and $\Theta$, such that the
principal curvatures $\kappa_{i}$ of $M_{t}$ satisfy

$\kappa_{i}\geq\Lambda_{2}^{-1}\mathrm{e}^{-\frac{2\Lambda_{1}}{\tau}t}$
(3.10)

for all $i=1,\dots,n$ and $t\in[0,T)$, where $\tau$ is the constant in Lemma
3.2 and $\Theta$ is the constant in item (3) of Assumption 1.1.

###### Proof.

Since $M_{0}$ is convex, by continuity the solution $M_{t}$ is convex for at
least a short time. Let $T_{1}<T$ be the largest time such that $M_{t}$ is
convex for all $0\leq t<T_{1}$. If we can derive the estimate (3.10) on the
interval $[0,T_{1})$, then it implies a contradiction with the maximality of
$T_{1}$ and so that $M_{t}$ is convex on the whole time interval $[0,T)$, and
the estimate (3.10) holds for all $t\in[0,T)$. Therefore, without loss of
generality, we can assume that $M_{t}$ is convex for $t\in[0,T)$ and we need
to show the estimate (3.10) holds for all $t\in[0,T)$.

For any time $t_{0}\in[0,T)$, let $B_{\rho_{0}}(p_{0})$ be the inball of
$\Omega_{t_{0}}$, where $\rho_{0}=\rho_{-}(t_{0})$. Denote

$\mathfrak{b}_{i}:=\frac{1}{\kappa_{i}},\quad\text{and}\quad\mathfrak{b}:=\max_{i=1,\cdots,n}{\mathfrak{b}_{i}}.$

In order to prove the lower bound of the principal curvatures $\kappa_{i}$, it
suffices to prove the upper bound of $\mathfrak{b}$.

We consider the auxiliary function

$Q(p,t):=\log{\mathfrak{b}}+A\rho,\qquad t\in[t_{0},\min\\{T,t_{0}+\tau\\}),$
(3.11)

where $A>0$ is a large constant to be determined and $\tau$ is the constant in
Lemma 3.2. Suppose that the maximum of $Q$ on
$M\times[t_{0},\min\\{T,t_{0}+\tau\\})$ is attained at $(\bar{p},\bar{t})$. We
choose a local orthonormal frame $e_{1},\dots,e_{n}$ around $\bar{p}$ such
that at $(\bar{p},\bar{t})$ we have

$g_{ij}=\delta_{ij},\qquad h_{ij}=\kappa_{i}\delta_{ij}.$

By a rotation, we also assume that

$\mathfrak{b}(\bar{p},\bar{t})=\mathfrak{b}_{1}(\bar{p},\bar{t})=b^{1}_{1}(\bar{p},\bar{t}).$

Let $\xi=(1,0\cdots,0)$ be a contravariant vector field and set

$\lambda=\dfrac{b^{ij}\xi_{i}\xi_{j}}{g^{ij}\xi_{i}\xi_{j}},$

which is well-defined in a neighbourhood of $(\bar{p},\bar{t})$. Defining
$\tilde{Q}(p,t)$ by replacing $\mathfrak{b}$ by $\lambda$ in (3.11), we see
that $\tilde{Q}$ attends its maximum at $(\bar{p},\bar{t})$. Moreover, at
$(\bar{p},\bar{t})$ we have $\partial_{t}\lambda=\partial_{t}b_{1}^{1}$ and
the spatial derivatives also coincide. That is, $\lambda$ satisfies the same
evolution equation as $b_{1}^{1}$ at the point $(\bar{p},\bar{t})$. Therefore,
for the sake of clarity (see a similar argument as in [Ger14, §4]), we can
treat $b_{1}^{1}$ as a scalar function and pretend that $Q$ is defined by

$Q(p,t)=\log{b_{1}^{1}}+A\rho,\qquad t\in[t_{0},\min\\{T,t_{0}+\tau\\}),$
(3.12)

As $\\{b_{m}^{n}\\}$ is diagonal at $(\bar{p},\bar{t})$ and
$b^{1}_{1}(\bar{p},\bar{t})=\mathfrak{b}(\bar{p},\bar{t})$, by (2.10) at the
point $(\bar{p},\bar{t})$, we have

$\displaystyle\partial_{t}b_{1}^{1}-f^{\prime}K^{k\ell}\nabla_{k}\nabla_{\ell}{b_{1}^{1}}$
$\displaystyle=$
$\displaystyle-\mathfrak{b}^{2}(f^{\prime\prime}K^{2}+f^{\prime}K)(b^{kk}\nabla_{1}{h_{kk}})^{2}-\mathfrak{b}^{2}f^{\prime}Kb^{kk}b^{\ell\ell}(\nabla_{1}{h_{k\ell}})^{2}$
$\displaystyle-\mathfrak{b}f^{\prime}(HK+\sigma_{n-1}(\kappa))+nf^{\prime}K(\mathfrak{b}^{2}+1)-(f-\phi(t))(1-\mathfrak{b}^{2}).$
(3.13)

Combining (2.20) and (2.22), we also have

$\displaystyle\partial_{t}\rho-f^{\prime}\dot{K}^{k\ell}\nabla_{k}\nabla_{\ell}\rho=$
$\displaystyle(\phi(t)-f(K))\sqrt{1+\frac{|\bar{\nabla}\rho|^{2}}{\sinh^{2}{\rho}}}+nf^{\prime}K\frac{u}{\sinh\rho}$
$\displaystyle-f^{\prime}\coth{\rho}\left(\sigma_{n-1}(\kappa)-Kb^{k\ell}\nabla_{k}\rho\nabla_{\ell}\rho\right).$
(3.14)

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

$Q(p,t)\leq Q(\bar{p},t_{0})\leq\log\max_{p\in
M}{\mathfrak{b}(p,t_{0})}+2Ac_{2}$ (3.15)

for $(p,t)\in M\times[t_{0},\min\\{T,t_{0}+\tau\\})$. In the following, we
assume $\bar{t}>t_{0}$. We shall apply the maximum principle to the evolution
equation of $Q$.

Since $(\bar{p},\bar{t})$ is a maximum point of $Q$, at $(\bar{p},\bar{t})$
there hold

$0=\nabla_{i}Q=\frac{\nabla_{i}b_{11}}{\mathfrak{b}}+A\nabla_{i}\rho$ (3.16)

and

$\displaystyle 0\leq$
$\displaystyle\partial_{t}Q-f^{\prime}\dot{K}^{k\ell}\nabla_{k}\nabla_{\ell}{Q}$
$\displaystyle=$
$\displaystyle\frac{1}{\mathfrak{b}}\left(\partial_{t}b_{1}^{1}-f^{\prime}\dot{K}^{k\ell}\nabla_{k}\nabla_{\ell}b_{1}^{1}\right)+\frac{f^{\prime}}{\mathfrak{b}^{2}}\dot{K}^{k\ell}\nabla_{k}b_{11}\nabla_{\ell}b_{11}$
$\displaystyle+A\left(\partial_{t}\rho-f^{\prime}\dot{K}^{k\ell}\nabla_{k}\nabla_{\ell}{\rho}\right)$
$\displaystyle=$
$\displaystyle-\mathfrak{b}(f^{\prime\prime}K^{2}+f^{\prime}K)(b^{kk}\nabla_{1}{h_{kk}})^{2}-\mathfrak{b}f^{\prime}Kb^{kk}b^{\ell\ell}(\nabla_{1}{h_{k\ell}})^{2}$
$\displaystyle-f^{\prime}(HK+\sigma_{n-1}(\kappa))+nf^{\prime}K(\mathfrak{b}+\frac{1}{\mathfrak{b}})-(f-\phi(t))(\frac{1}{\mathfrak{b}}-\mathfrak{b})$
$\displaystyle+\frac{f^{\prime}K}{\mathfrak{b}^{2}}b^{kk}(\nabla_{k}b_{11})^{2}+A(\phi(t)-f)\sqrt{1+\frac{|\bar{\nabla}\rho|^{2}}{\sinh^{2}{\rho}}}$
$\displaystyle-
Af^{\prime}\coth{\rho}\left(\sigma_{n-1}(\kappa)-Kb^{kk}(\nabla_{k}\rho)^{2}\right)+nAf^{\prime}K\frac{u}{\sinh\rho}$
$\displaystyle=$ $\displaystyle:Q_{1}+Q_{2},$ (3.17)

where $Q_{1}$ denote the terms involving $\phi(t)$:

$Q_{1}=\phi(t)\left(\frac{1}{\mathfrak{b}}-\mathfrak{b}+A\sqrt{1+\frac{|\bar{\nabla}\rho|^{2}}{\sinh^{2}{\rho}}}\right),$
(3.18)

and $Q_{2}$ denote the remaining terms:

$\displaystyle Q_{2}=$
$\displaystyle-\mathfrak{b}(f^{\prime\prime}K^{2}+f^{\prime}K)(b^{kk}\nabla_{1}{h_{kk}})^{2}$
$\displaystyle-\mathfrak{b}f^{\prime}Kb^{kk}b^{\ell\ell}(\nabla_{1}{h_{k\ell}})^{2}+\frac{f^{\prime}K}{\mathfrak{b}^{2}}b^{kk}(\nabla_{k}b_{11})^{2}$
$\displaystyle-f^{\prime}(HK+\sigma_{n-1}(\kappa))+nf^{\prime}K(\mathfrak{b}+\frac{1}{\mathfrak{b}})-f(\frac{1}{\mathfrak{b}}-\mathfrak{b})$
$\displaystyle-
Af^{\prime}\coth{\rho}\left(\sigma_{n-1}(\kappa)-Kb^{kk}(\nabla_{k}\rho)^{2}\right)$
$\displaystyle-
Af\sqrt{1+\frac{|\bar{\nabla}\rho|^{2}}{\sinh^{2}{\rho}}}+nAf^{\prime}K\frac{u}{\sinh\rho}.$
(3.19)

Estimate of $Q_{1}$: By (2.4) and (3.6), (3.7), we see that

$\sqrt{1+\frac{|\bar{\nabla}\rho|^{2}}{\sinh^{2}{\rho}}}=\frac{\sinh\rho}{u}\leq\frac{\sinh(2c_{2})}{\sinh(c_{1}/2)}~{}=:c_{4}.$

Then, if

$\mathfrak{b}>\frac{Ac_{4}+\sqrt{A^{2}c_{4}^{2}+4}}{2},$ (3.20)

we have $Q_{1}<0$.

Estimate of $Q_{2}$: Firstly, by item (4) in Assumption 1.1, we have

$\displaystyle-\mathfrak{b}(f^{\prime\prime}K^{2}+f^{\prime}K)(b^{kk}\nabla_{1}{h_{kk}})^{2}\leq~{}0.$
(3.21)

Using (2.16) and (2.18), we also have

$\displaystyle-\mathfrak{b}f^{\prime}Kb^{kk}b^{\ell\ell}(\nabla_{1}{h_{k\ell}})^{2}+\frac{f^{\prime}K}{\mathfrak{b}^{2}}b^{kk}(\nabla_{k}b_{11})^{2}$
$\displaystyle=$
$\displaystyle-\mathfrak{b}f^{\prime}Kb^{kk}b^{\ell\ell}(\nabla_{1}{h_{k\ell}})^{2}+f^{\prime}\mathfrak{b}^{2}Kb^{kk}(\nabla_{k}h_{11})^{2}$
$\displaystyle\leq$ $\displaystyle 0,$ (3.22)

where the inequality is obtained by discarding the terms with $\ell\neq 1$.
For the fourth line of (3.19),

$\displaystyle\text{Line 4 of }\eqref{eq-Q_2}\leq$ $\displaystyle-
Af^{\prime}\coth{\rho}\left(\sigma_{n-1}(\kappa)-K\left(\sum_{k}{b^{kk}}\right)|\nabla\rho|^{2}\right)$
$\displaystyle=$ $\displaystyle-
Af^{\prime}\coth{\rho}~{}\sigma_{n-1}(\kappa)\left(1-|\nabla\rho|^{2}\right)$
$\displaystyle\leq$ $\displaystyle-
c_{3}Af^{\prime}\coth{\rho}~{}\sigma_{n-1}(\kappa)$ $\displaystyle\leq$
$\displaystyle-c_{3}Af^{\prime}\sigma_{n-1}(\kappa)$ (3.23)

where we used the estimate (3.1) and that $c_{3}$ is the constant in (3.1)
which depends only on $n$ and $M_{0}$.

Substituting (3.21)-(3.23) into (3.19), we have

$\displaystyle Q_{2}\leq$
$\displaystyle-f^{\prime}(HK+\sigma_{n-1}(\kappa))+nf^{\prime}K(\mathfrak{b}+\frac{1}{\mathfrak{b}})-f(\frac{1}{\mathfrak{b}}-\mathfrak{b})$
$\displaystyle-
c_{3}Af^{\prime}~{}\sigma_{n-1}(\kappa)-Af\sqrt{1+\frac{|\bar{\nabla}\rho|^{2}}{\sinh^{2}{\rho}}}+nAf^{\prime}K\frac{u}{\sinh\rho}$
$\displaystyle\leq$
$\displaystyle-f^{\prime}\sigma_{n-1}(\kappa)\left(1+c_{3}A\right)+(nf^{\prime}K+f)\mathfrak{b}$
$\displaystyle-f^{\prime}HK+\frac{nf^{\prime}K}{\mathfrak{b}}+nAf^{\prime}K\frac{u}{\sinh\rho}$
$\displaystyle=$
$\displaystyle-\frac{f^{\prime}K}{\mathfrak{b}}\biggl{[}(1+c_{3}A)\left(\sum_{k}b^{kk}\right)\mathfrak{b}-\left(n+\frac{f}{f^{\prime}K}\right)\mathfrak{b}^{2}$
$\displaystyle\qquad-\frac{nAu}{\sinh\rho}\mathfrak{b}+\mathfrak{b}H-n\biggr{]},$
(3.24)

where we have thrown away the negative terms $-\frac{f}{\mathfrak{b}}$ and
$-Af\sqrt{1+\frac{|\bar{\nabla}\rho|^{2}}{\sinh^{2}{\rho}}}$ in the second
inequality of (3.24). Since

$\sum_{k}{b^{kk}}>\mathfrak{b},\quad\frac{f}{f^{\prime}K}\leq\Theta,\quad\frac{u}{\sinh\rho}\leq
1,\quad H\geq\frac{n}{\mathfrak{b}},$ (3.25)

where $\Theta$ is the constant in item (3) in Assumption 1.1, substituting
(3.25) into (3.24) gives the following estimates

$Q_{2}\leq-f^{\prime}K\Big{(}(1+c_{3}A-n-\Theta)\mathfrak{b}-nA\Big{)}.$
(3.26)

Choosing

$A=\frac{n+\Theta}{c_{3}},$ (3.27)

which depends only on $n,M_{0}$ and $\Theta$, we see that if

$\mathfrak{b}>\frac{n(n+\Theta)}{c_{3}}$ (3.28)

then $Q_{2}<0$.

Combing (3.17) with (3.20) and (3.28), we know that if

$\mathfrak{b}>\max\left\\{\frac{Ac_{4}+\sqrt{A^{2}c_{4}^{2}+4}}{2},\frac{n(n+\Theta)}{c_{3}}\right\\}=:a_{1},$

where $A=\frac{n+\Theta}{c_{3}}$, then at the point $(\bar{p},\bar{t})$, we
have

$0\leq\partial_{t}Q-f^{\prime}\dot{K}^{k\ell}\nabla_{k}\nabla_{\ell}{Q}=Q_{1}+Q_{2}<0,$

which leads to a contradiction. Therefore we have

$\mathfrak{b}(\bar{p},\bar{t})\leq~{}a_{1}.$ (3.29)

Since $(\bar{p},\bar{t})$ is a maximum point of the function $Q$, Combining
(3.15) with (3.29), we have

$Q(p,t)\leq\max\left\\{\log\max_{p\in M}{\mathfrak{b}(p,t_{0})}+2Ac_{2},\log
a_{1}+2Ac_{2}\right\\}$ (3.30)

for $(p,t)\in M\times[t_{0},\min\\{T,t_{0}+\tau\\})$. Hence we have

$\displaystyle\mathfrak{b}(p,t)\leq$ $\displaystyle\max\left\\{\max_{p\in
M}{\mathfrak{b}(p,t_{0})},a_{1}\right\\}\mathrm{e}^{A(2c_{2}-\frac{c_{1}}{2})}$
$\displaystyle=$ $\displaystyle:\max\left\\{\max_{p\in
M}{\mathfrak{b}(p,t_{0})},a_{1}\right\\}\mathrm{e}^{\Lambda_{1}}$ (3.31)

for $(p,t)\in M\times[t_{0},\min\\{T,t_{0}+\tau\\})$, where
$\Lambda_{1}=\frac{n+\Theta}{a_{1}}(2c_{2}-\frac{c_{1}}{2})$ depending only on
$n,M_{0}$ and $\Theta$.

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

$\max_{p\in M}{\mathfrak{b}(p,t_{0})}\leq\max\left\\{\max_{p\in
M}{\mathfrak{b}(p,t_{0}-\frac{\tau}{2})},a_{1}\right\\}\mathrm{e}^{\Lambda_{1}}.$
(3.32)

Combining (3.2) with (3.32) and the fact that $\mathrm{e}^{\Lambda_{1}}\geq
1$, we have

$\mathfrak{b}(p,t)\leq\max\left\\{\max_{p\in
M}{\mathfrak{b}(p,t_{0}-\frac{\tau}{2})},a_{1}\right\\}\mathrm{e}^{2\Lambda_{1}}.$
(3.33)

By repeating the argument finitely many times, we finally get

$\displaystyle\mathfrak{b}(p,t)$ $\displaystyle\leq\max\left\\{\max_{p\in
M}{\mathfrak{b}(p,0)},a_{1}\right\\}\mathrm{e}^{\left(\left[\frac{2t_{0}}{\tau}\right]+2\right)\Lambda_{1}}$
$\displaystyle\leq\max\left\\{\max_{p\in
M}{\mathfrak{b}(p,0)},a_{1}\right\\}\mathrm{e}^{\left(\frac{2t}{\tau}+2\right)\Lambda_{1}}:=\Lambda_{2}\mathrm{e}^{\frac{2\Lambda_{1}}{\tau}t}$
(3.34)

for all $(p,t)\in M\times[t_{0},\min\\{T,t_{0}+\tau\\})$, where $[\cdot]$
denotes the integer part of a real constant, and
$\Lambda_{1},\Lambda_{2}:=\max\left\\{\max_{p\in
M}{\mathfrak{b}(p,0)},a_{3}\right\\}\mathrm{e}^{2\Lambda_{1}}$ are constants
depending only on $n,M_{0}$ and $\Theta$. Since $t_{0}$ is arbitrary, we
conclude that the principal curvatures $\kappa_{i}$ of the solution $M_{t}$ of
the flow (1.1) satisfy

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

for all time $t\in[0,T)$. This completes the proof of Proposition 3.3. ∎

###  3.3. Upper bound of Gauss curvature

Now we prove the upper bound of the Gauss curvature for the solution $M_{t}$
along the flow (1.1).

######  Proposition 3.4.

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 exists
a constant $C$ depending on $n,M_{0},\Theta$ and $T$ such that the Gauss
curvature $K$ of $M_{t}$ satisfies

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

for any $t\in[0,T)$, where $\Theta$ is the constant in item (3) of Assumption
1.1.

###### Proof.

For any given $t_{0}\in[0,T)$, let $B_{\rho_{0}}(p_{0})$ be the inball of
$\Omega_{t_{0}}$ centered at some point $p_{0}\in\Omega_{t_{0}}$, 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 the point $p_{0}$, where $\rho_{p_{0}}(x)$ is the
distance function in $\mathbb{H}^{n+1}$ to the point $p_{0}$. Since $M_{t}$ is
convex for all $t\in[0,T)$, by (3.6) we have

$2c\leq u\leq\sinh(2c_{2})$ (3.35)

on $M_{t}$ for all $t\in\left[t_{0},\min\\{T,t_{0}+\tau\\}\right)$. We define
the auxiliary function as in [Tso85]

$W=\frac{f(K)}{u-c},$

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

Combining (2.8) and (2.21), we compute that along the flow (1.1) the function
$W$ evolves as

$\displaystyle\frac{d}{dt}{W}=$ $\displaystyle
f^{\prime}\dot{K}^{ij}\left(W_{ij}+\frac{2}{u-c}u_{i}W_{j}\right)$
$\displaystyle-\frac{\phi(t)}{u-c}\Big{(}f^{\prime}(HK-\sigma_{n-1}(\kappa))+W\cosh{\rho_{p_{0}}}(x)\Big{)}$
$\displaystyle+\frac{f}{(u-c)^{2}}(f+nf^{\prime}K)\cosh{\rho_{p_{0}}}(x)$
$\displaystyle-\frac{cf}{(u-c)^{2}}f^{\prime}HK-
Wf^{\prime}\sigma_{n-1}(\kappa)$ $\displaystyle\leq$ $\displaystyle
f^{\prime}\dot{K}^{ij}\left(W_{ij}+\frac{2}{u-c}u_{i}W_{j}\right)+\frac{\phi(t)}{u-c}f^{\prime}\sigma_{n-1}(\kappa)$
$\displaystyle+(1+n\frac{f^{\prime}K}{f})W^{2}\cosh{\rho_{p_{0}}}(x)-c\frac{f^{\prime}K}{f}HW^{2}.$
(3.36)

Let $\widetilde{W}(t)=\max_{M_{t}}W(x,t)$. Noting that $f(K)=(u-c)W$, by the
definition (1.2) and the upper bound (3.35) of $u$, we have:

$\displaystyle\phi(t)=$
$\displaystyle\frac{1}{|M_{t}|}\int_{M_{t}}f(K)d\mu_{t}$ $\displaystyle\leq$
$\displaystyle\max_{M_{t}}f(K(\cdot,t))\leq(\sinh(2c_{2})-c)\widetilde{W}.$

By the lower bound on the principal curvatures in Lemma 3.3, we also have

$\displaystyle\sigma_{n-1}(\kappa)=$ $\displaystyle
K(\frac{1}{\kappa_{1}}+\cdots\frac{1}{\kappa_{n}})$ $\displaystyle\leq$
$\displaystyle nK(\min_{1\leq i\leq n}\kappa_{i})^{-1}\leq
nK\Lambda_{2}\mathrm{e}^{\frac{2\Lambda_{1}}{\tau}T}.$ (3.37)

It follows that

$\frac{\phi(t)}{u-c}f^{\prime}\sigma_{n-1}(\kappa)\leq~{}n(\sinh(2c_{2})-c)\Lambda_{2}\mathrm{e}^{\frac{2\Lambda_{1}T}{\tau}}\frac{f^{\prime}K}{f}\widetilde{W}^{2}.$
(3.38)

Since $H\geq nK^{1/n}$, the last term of (3.3) satisfies

$-c\frac{f^{\prime}K}{f}HW^{2}\leq-
nc\frac{f^{\prime}K^{\frac{n+1}{n}}}{f}W^{2}.$ (3.39)

Substituting (3.38) and (3.39) into (3.3), we arrive at

$\displaystyle\frac{d}{dt}\widetilde{W}\leq$
$\displaystyle\widetilde{W}^{2}\cosh(2c_{2})+n\widetilde{W}^{2}\frac{f^{\prime}K}{f}\Big{(}(\sinh(2c_{2})-c)\Lambda_{2}\mathrm{e}^{\frac{2\Lambda_{1}T}{\tau}}$
$\displaystyle\quad+\cosh(2c_{2})-cK^{1/n}\Big{)}.$ (3.40)

Denote

$\bar{c}=(nc)^{-n}\left((\cosh(2c_{2})+1)\Theta+n(\sinh(2c_{2})-c)\Lambda_{2}\mathrm{e}^{\frac{2\Lambda_{1}T}{\tau}}+n\cosh(2c_{2})\right)^{n},$

which depends on $n,M_{0},\Theta$ and $T<\infty$. Suppose that

$\widetilde{W}(t)\geq\frac{f(\bar{c})}{c}.$

Then for any $x\in M_{t}$ with $W(x,t)=\tilde{W}(t)$, there holds

$\displaystyle f(K(x,t))=$ $\displaystyle W(x,t)(u-c)$ $\displaystyle=$
$\displaystyle\widetilde{W}(t)(u-c)\geq f(\bar{c}).$

As $f$ is strictly increasing, this implies that

$K(x,t)\geq\bar{c}$ (3.41)

for any $x\in M_{t}$ with $W(x,t)=\tilde{W}(t)$. Substituting (3.41) into
(3.3) and noting that $f^{\prime}K/f\geq 1/\Theta$ by item (3) of Assumption
1.1, we have

$\frac{d}{dt}\widetilde{W}(t)\leq-\widetilde{W}^{2}(t)$ (3.42)

whenever $\widetilde{W}(t)\geq{f(\bar{c})}/{c}$. It follows that

$\widetilde{W}(t)\leq\max\left\\{\frac{1}{\widetilde{W}^{-1}(t_{0})+t-t_{0}},\frac{f(\bar{c})}{c}\right\\}$
(3.43)

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

For $t_{0}=0$, we obtain from (3.43) the upper bound

$\widetilde{W}(t)\leq\max\left\\{\widetilde{W}(0),\frac{f(\bar{c})}{c}\right\\},\quad\forall
t\in[0,\min\\{\tau,T\\})$

and so

$\displaystyle f(K)=$ $\displaystyle(u-c)W$ $\displaystyle\leq$
$\displaystyle\sinh(2c_{2})\max\left\\{\widetilde{W}(0),\frac{f(\bar{c})}{c}\right\\},\quad\forall
t\in[0,\min\\{\tau,T\\}).$ (3.44)

Next, for $t_{0}=\tau/2$, the estimate (3.43) implies

$\displaystyle\widetilde{W}(t)$
$\displaystyle\leq\max\left\\{\frac{1}{\widetilde{W}^{-1}(t_{0})+t-t_{0}},\frac{f(\bar{c})}{c}\right\\}$
$\displaystyle\leq\max\left\\{\frac{1}{t-t_{0}},\frac{f(\bar{c})}{c}\right\\}$
$\displaystyle\leq\max\left\\{\frac{2}{\tau},\frac{f(\bar{c})}{c}\right\\}$

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

$f(K)\leq\sinh(2c_{2})\max\left\\{\frac{2}{\tau},\frac{f(\bar{c})}{c}\right\\}$
(3.45)

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 (3.45) for
$t\in[\frac{(m+1)\tau}{2},\min\\{\frac{(m+2)\tau}{2},T\\})$, which covers the
whole time interval $[0,T)$.

Combining (3.3), (3.45) and the fact that $f$ is an increasing function with
$\lim\limits_{x\to\infty}f(x)=+\infty$ (see items (1) and (2) in Assumption
1.1), we obtain the upper bound $K\leq C$ for a constant $C$ depending on
$n,M_{0},\Theta$ and $T$. This completes the proof of Proposition 3.4. ∎

##  4\. Long time existence

In this section, we prove the long time existence of the flow (1.1). We need
to show that the solution remains smooth as long as the curvatures are
bounded. To prove this, we need the following result which is a special case
of Theorem 6 in [And2004] (see also [AleSin10b, Theorem 3.2]).

######  Theorem 4.1 ([And2004]).

Let $\Omega$ be a domain in $\mathbb{R}^{n}$. Let $u\in
C^{4}(\Omega\times[0,T))$ be a function satisfying

$\frac{\partial}{\partial t}u=F(D^{2}u,Du,u,x,t),$

where $F$ is $C^{2}$ and is elliptic, i.e., $\lambda
I\leq(\dot{F}^{ij})\leq\Lambda I$ for some constants $\Lambda>\lambda>0$.
Suppose that $F$ can be written as $F=\varphi(G(D^{2}u,Du,u,x,t))$, where $G$
is concave with respect to $D^{2}u$ and $\varphi$ is an increasing function on
the range of $G$. Then in any relatively compact
$\Omega^{\prime}\subset\Omega$ and for any $\tau\in(0,T)$ we have

$\|u\|_{C^{2,\beta}(\Omega^{\prime}\times(\tau,T))}\leq C,$

where $\beta\in(0,1)$ depends on $n,\lambda$ and $\Lambda$, and $C$ depends on
$\lambda,\Lambda,\|u\|_{C^{2}(\Omega\times[0,T))}$,
$\tau,\mathrm{dist}(\Omega^{\prime},\partial\Omega)$ and the bounds on the
first and second derivatives of $G$.

The advantage of the above theorem is that it allows to relax the concavity
hypothesis of the usual regularity theorem for fully nonlinear parabolic
equation.

######  Theorem 4.2.

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 $M_{0}$ with
$\phi(t)$ given by (1.2) and the speed function satisfies the assumption 1.1.
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 maximal interval such that
the solution of the flow (1.1) exists with $T<\infty$. Then combining
Proposition 3.3 and Proposition 3.4 yields that the principal curvatures
$\kappa=(\kappa_{1},\dots,\kappa_{n})$ of $M_{t}$ satisfy

$0<\underline{\kappa}_{0}\leq\kappa_{i}\leq\overline{\kappa}_{0},\quad
i=1,\dots,n$ (4.1)

for all time $t\in[0,T)$, where the constants
$\underline{\kappa}_{0},\overline{\kappa}_{0}$ depend on $n,M_{0},\Theta$ 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 up to a
tangential diffeomorphism, the flow equation (1.1) is equivalent to the
following scalar parabolic equation

$\frac{\partial}{\partial
t}\rho=\left(\phi(t)-f(K)\right)\sqrt{1+\frac{|\bar{\nabla}\rho|^{2}}{\sinh^{2}{\rho}}},$
(4.2)

of the radial graph function $\rho$ over $\mathbb{S}^{n}$, where
$K=K(\bar{\nabla}^{2}\rho,\bar{\nabla}\rho,\rho)$ is expressed in (2.5) and
$\bar{\nabla}$ denotes the Levi-Civita connection with respect to the round
metric on $\mathbb{S}^{n}$. Denote the right hand side of (4.2) by
$F[\bar{\nabla}^{2}\rho,\bar{\nabla}\rho,\rho,t]$

For any $t_{0}\in[0,T)$, we consider the solution of $M_{t}$ in the time
interval $[t_{0},\min\\{t_{0}+\tau,T\\})$. Since only $f(K)$ in
$F[\bar{\nabla}^{2}\rho,\bar{\nabla}\rho,\rho,t]$ depends on
$\bar{\nabla}^{2}\rho$, we calculate that

$\displaystyle\dot{F}^{ij}=$ $\displaystyle-f^{\prime}\frac{\partial
K}{\partial\rho_{ij}}\sqrt{1+\frac{|\bar{\nabla}\rho|^{2}}{\sinh^{2}{\rho}}}$
$\displaystyle=$
$\displaystyle\frac{f^{\prime}\sinh\rho}{(\sinh^{2}\rho+|\bar{\nabla}\rho|^{2})^{\frac{n+2}{2}}(\sinh\rho)^{2(n-1)}}\frac{\partial\det
h_{ij}}{\partial
h_{ij}}\sqrt{1+\frac{|\bar{\nabla}\rho|^{2}}{\sinh^{2}{\rho}}},$ (4.3)

where $h_{ij}$ is expressed in (2.2). The $C^{0},C^{1}$ estimates obtained in
(3.7), (3.8) and the curvature bound (4.1) implies that $F$ is elliptic, i.e.,
$\lambda I\leq(\dot{F}^{ij})\leq\Lambda I$ for some constants
$\Lambda>\lambda>0$ depending on $n,M_{0},\Theta$ and $T$. Moreover, since
it’s well known that $K^{1/n}$ is a concave operator with respect to second
derivatives, and $f(K)$ is a strictly increasing function of $K^{1/n}$, by
Theorem 4.1 we derive a $C^{2,\alpha}$ estimate on $\rho$, see also the
arguments in [CS10, Mc05] for the $C^{2,\gamma}$ estimate of the solutions to
volume preserving curvature flows. Then by the parabolic Schauder theory (see
[Lie96]), we can deduce all higher order regularity estimates of $\rho$ on
$[t_{0},\min\\{t_{0}+\tau,T\\})$. As $t_{0}$ is arbitrary, we can obtain the
smoothness of the flow for all time $t\in[0,T)$ and a standard continuation
argument then shows that $T=+\infty$. ∎

######  Remark 4.3.

Note that the curvature estimate (4.1) 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.

##  5\. Hausdorff convergence

In this section, we prove the monotonicity of the quermassintegral
$\mathcal{A}_{n-1}(\Omega_{t})$, 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

$\bar{K}=\frac{1}{|M_{t}|}\int_{M_{t}}{Kd\mu_{t}}=\frac{\mathcal{A}_{n}(\Omega_{t})+\frac{1}{n-1}\mathcal{A}_{n-2}(\Omega_{t})}{\mathcal{A}_{0}(\Omega_{t})},$
(5.1)

where the second equality is due to (2.26). It follows from the monotonicity
(2.28) of quermassintegrals with respect to inclusion of convex sets and the
estimates on inner radius and outer radius in Lemma 3.1 that there exists
uniform positive constants $m_{1},m_{2},a$ and $b$ depending only on $n$ and
$M_{0}$, such that

$0<m_{1}\leq|M_{t}|=A_{0}(\Omega_{t})\leq m_{2},\quad a\leq\bar{K}\leq b.$
(5.2)

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

######  Lemma 5.1.

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

###### Proof.

From the evolution equation (2.27) for the quermassintegrals of $\Omega_{t}$,
we have

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

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

$\displaystyle\frac{d}{dt}\mathcal{A}_{n-1}(\Omega_{t})$
$\displaystyle=\frac{n}{|M_{t}|}\left(\int_{M_{t}}Kd\mu_{t}\int_{M_{t}}f(K)d\mu_{t}-|M_{t}|\int_{M_{t}}Kf(K)d\mu_{t}\right)$
$\displaystyle=n\int_{M_{t}}{f(K)(\bar{K}-K)\,d\mu_{t}}$
$\displaystyle=-n\int_{M_{t}}{(f(K)-f(\bar{K})(K-\bar{K}))}\,d\mu_{t}\leq 0$
(5.3)

due to the assumption that $f$ is strictly increasing. Moreover, equality
holds in (5.3) at some time $t$ 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[MS91]). ∎

###  5.2. Subsequential Hausdorff convergence

We first prove the following estimate on the $L^{1}$ oscillation decay of
Gauss curvature:

######  Lemma 5.2.

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

$\int_{M_{t_{i}}}{|K-\bar{K}|d\mu_{t_{i}}}\to
0,\quad\text{as}\,\,t_{i}\to\infty.$ (5.4)

###### Proof.

By the evolution equation (5.3) and the long time existence of the flow (1.1),
we have

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

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

$\int_{M_{t_{i}}}{(f(K)-f(\bar{K}))(K-{\bar{K}})d\mu_{t_{i}}}\to 0.$ (5.5)

Denote the subset $Y_{t}\subset M_{t}$ as $Y_{t}=\\{p\in
M_{t}|K(p)=\bar{K}\\}$. Then we have

$\displaystyle\int_{M_{t}}{|K-\bar{K}|}\,d\mu_{t}$
$\displaystyle=\int_{M_{t}\setminus
Y_{t}}{\frac{|K-\bar{K}|^{\frac{1}{2}}}{|f(K)-f(\bar{K})|^{\frac{1}{2}}}|K-\bar{K}|^{\frac{1}{2}}|f(K)-f(\bar{K})|^{\frac{1}{2}}}\,d\mu_{t}$
$\displaystyle\leq\underbrace{\left(\int_{M_{t}\setminus
Y_{t}}{\frac{|K-\bar{K}|}{|f(K)-f(\bar{K})|}}\,d\mu_{t}\right)^{\frac{1}{2}}}_{(I)}\left(\int_{M_{t}}{(f(K)-f(\bar{K}))(K-{\bar{K}})d\mu_{t}}\right)^{\frac{1}{2}}.$
(5.6)

Next, we show that the term $(I)$ of (5.6) is uniformly bounded from above. We
divide the set $M_{t}\setminus Y_{t}$ into three disjoint subsets

$M_{t}\setminus Y_{t}=Z_{1}\sqcup Z_{2}\sqcup Z_{3},$

where the subsets $Z_{1},Z_{2}$ and $Z_{3}$ are defined as

$\displaystyle Z_{1}$ $\displaystyle=\\{p\in M_{t}\setminus
Y_{t}|0<K(p)\leq\frac{a}{2}\\},$ (5.7) $\displaystyle Z_{2}$
$\displaystyle=\\{p\in M_{t}\setminus Y_{t}|\frac{a}{2}<K(p)<\frac{3b}{2}\\},$
(5.8) $\displaystyle Z_{3}$ $\displaystyle=\\{p\in M_{t}\setminus
Y_{t}|K(p)\geq\frac{3b}{2}\\}.$ (5.9)

Using (5.2), we calculate as follows:

  * (i)

If $p\in Z_{1}$, since $f$ is an increasing function of $K$, we have

$\frac{|K-\bar{K}|}{|f(K)-f(\bar{K})|}=\frac{\bar{K}-K}{f(\bar{K})-f(K)}\leq\frac{b}{f(a)-f(\frac{a}{2})}$

and hence

$\int_{Z_{1}}{\frac{|K-\bar{K}|}{|f(K)-f(\bar{K})|}}\,d\mu_{t}\leq\frac{m_{2}b}{f(a)-f(\frac{a}{2})}.$
(5.10)

  * (ii)

If $p\in Z_{2}$, then by Lagrange’s mean value theorem, we have

$\displaystyle\frac{|K-\bar{K}|}{|f(K)-f(\bar{K})|}$
$\displaystyle=\frac{1}{f^{\prime}(\xi)},\quad\text{the value $\xi$ is taken
between $K$ and $\bar{K}$}$
$\displaystyle\leq\max_{x\in[\frac{a}{2},\frac{3b}{2}]}\left\\{\frac{1}{f^{\prime}(x)}\right\\}=:c$

and hence

$\int_{Z_{2}}{\frac{|K-\bar{K}|}{|f(K)-f(\bar{K})|}}\,d\mu_{t}\leq m_{2}c.$
(5.11)

  * (iii)

If $p\in Z_{3}$, we have

$\frac{|K-\bar{K}|}{|f(K)-f(\bar{K})|}=\frac{K-\bar{K}}{f(K)-f(\bar{K})}\leq\frac{K}{f(\frac{3b}{2})-f(b)}$

and hence

$\displaystyle\int_{Z_{3}}{\frac{|K-\bar{K}|}{|f(K)-f(\bar{K})|}}\,d\mu_{t}$
$\displaystyle\leq\frac{1}{f(\frac{3b}{2})-f(b)}\int_{M_{t}}{K}\,d\mu_{t}$
$\displaystyle=\frac{|M_{t}|\bar{K}}{f(\frac{3b}{2})-f(b)}$
$\displaystyle\leq\frac{m_{2}b}{f(\frac{3b}{2})-f(b)}.$ (5.12)

Combining (5.10)-((iii)), we conclude that the term $(I)$ of (5.6) is
uniformly bounded from above. Then by (5.5), we complete the proof of Lemma
5.2. ∎

With the estimate (5.4) in hand, we can argue as in [WYZ2022, Lemma 6.3] using
curvature measure theory for convex bodies in $\mathbb{H}^{n+1}$ to get the
subsequential Hausdorff convergence of $M_{t}$ to a geodesic sphere. We state
the result in the following lemma and refer the readers to [WYZ2022, Lemma
6.3] for the proof.

######  Lemma 5.3.

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 $M_{0}$.

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

As in [WYZ2022], since we do not have the analogous stability estimate as in
[RS2014, Eq.(7.124)] for the hyperbolic case, we can not apply the argument in
[AW21] to deduce from Lemma 5.3 the Hausdorff convergence of $M_{t}$ to the
geodesic sphere for all time $t\to\infty$. However, if we denote $p_{t}$ as
the center of the inner ball of $\Omega_{t}$, we can still 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
$M_{t}$ in Lemma 5.3.

Let $p\in\mathbb{H}^{n+1}$ be the center of the limit geodesic sphere
$S_{\rho_{\infty}}(p)$ in Lemma 5.3. 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
[BenWei, 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 [BenWei] deals with the flow with $h$-convex initial
hypersurfaces, the argument in Lemma 6.1 of [BenWei] works for convex
solutions as well. The readers may refer to [Chow97, CG96, Chow23] for more
details on applications of the Alexandrov reflection method in extrinsic
curvature flows.

Then we have the following two results, and the proofs are the same as
[WYZ2022, Lemma 6.4, Lemma 6.6] for the volume preserving $\alpha$-Gauss
curvature flow (1.5) and hence we omit them here.

######  Lemma 5.4.

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 $S_{\gamma_{z}}^{-}(\Omega)\leq s_{0}\leq S_{\gamma_{z}}^{+}(\Omega)$.

######  Lemma 5.5.

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

$\lim_{t\to\infty}S_{\gamma_{z}}^{-}(\Omega_{t})=\lim_{t\to\infty}S_{\gamma_{z}}^{+}(\Omega_{t})=0.$
(5.13)

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 $t\to\infty$, where $p$ is the
center of the limit ball in Lemma 5.3.

##  6\. Smooth convergence

In this section, we complete the proof of Theorem 1.4. Firstly, we prove the
following uniform estimate for the principal curvatures of $M_{t}$ along the
flow (1.1).

######  Lemma 6.1.

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 constants $\underline{\kappa}$, $\overline{\kappa}$
depending only on $n,M_{0}$ and $\Theta$ such that the principal curvatures
$\kappa_{i}$ of $M_{t}$ satisfy:

$\underline{\kappa}\leq\kappa_{i}\leq\overline{\kappa},\qquad i=1,\dots,n$
(6.1)

for all time $t\in[0,+\infty)$, where $\Theta$ is the constant in item (3) of
Assumption 1.1.

###### Proof.

By Lemma 5.5, 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 $\rho_{-}(t)\geq c_{1}$, there exists a sufficiently
large time $t^{*}$, depending on $c_{1}$ and hence depending only on $n$ and
$M_{0}$, such that $d(p_{t},p)<{c_{1}}/{4}$ for $t\geq t^{*}$. Then we have:

$B_{c_{1}/4}(p)\subset\Omega_{t},\qquad\forall~{}t\geq t^{*}.$ (6.2)

Applying Proposition 3.3 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 (3.10) implies that the principal curvatures $\kappa_{i}$ of $M_{t}$
satisfy

$k_{i}\geq\Lambda_{2}^{-1}\mathrm{e}^{-\frac{2\Lambda_{1}}{\tau}t^{*}},\qquad
t\in[0,t^{*}).$ (6.3)

While for time $t\in[t^{*},\infty)$, since $B_{c_{1}/4}(p)\subset\Omega_{t}$
for all time $t\in[t^{*},\infty)$. Then by the estimate (3.2) in the proof of
Proposition 3.3, we have

$\mathfrak{b}(p,t)\leq\max\left\\{\max_{p\in
M}{\mathfrak{b}(p,t^{*})},a_{3}\right\\}\mathrm{e}^{\Lambda_{1}}$

for all $(p,t)\in M\times[t^{*},\infty)$. This together with (6.3) implies
that the principal curvatures of $M_{t}$ are uniformly bounded from below by a
positive constant $\underline{\kappa}$ which depends only on $n,M_{0}$ and
$\Theta$.

Once we have the uniform lower bound for the principal curvatures, the uniform
upper bound for the Gauss curvature $K$ follows easily from the proof of
Proposition 3.4. In fact, the upper bound (3.3) for $\sigma_{n-1}(\kappa)$ in
the proof of Proposition 3.4 now has the form

$\sigma_{n-1}(\kappa)\leq\frac{n}{\underline{\kappa}}K,$

where the coefficient of $K$ does not depend on time $t$.

Therefore, combining the uniform upper bound on $K$ and the lower bound
$\kappa_{i}\geq\underline{\kappa}$, 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 6.1. ∎

It follows from Lemma 6.1 that the flow (1.1) is uniformly parabolic for all
time $t>0$. Then an argument similar to that in the proof of Theorem 4.2 can
be applied to show that all derivatives of curvatures are uniformly bounded on
$M_{t}$ for all $t>0$. This together with Lemma 5.3 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 time convergence and the exponential convergence can be obtained by
studying the linearization of the flow (1.1). For each sufficiently large time
$t_{k}$, we write $M_{t_{k}}$ as the graph of the radial function
$\rho_{t_{k}}(\cdot)$ over $\mathbb{S}^{n}$ centered at $p_{t_{k}}$. For time
$t$ sufficiently close to $t_{k}$, we rewrite the flow equation (1.1) as the
scalar parabolic PDE

$\left\\{\begin{aligned}
\partial_{t}\rho&=\left(\phi(t)-f(K)\right)\sqrt{1+{|\bar{\nabla}\rho|^{2}}/{\sinh^{2}\rho}},\quad
t>t_{k}\\\ \rho(\cdot&,t_{k})=\rho_{t_{k}}(\cdot).\end{aligned}\right.$ (6.4)

Note we can assume the oscillation of $\rho_{t_{k}}-\rho_{\infty}$ is
sufficiently small by choosing $t_{k}$ large enough. By a direct computation,
the linearized equation of the flow (6.4) about the geodesic sphere of radius
$\rho_{\infty}$ is given by

$\frac{\partial}{\partial
t}\eta=\frac{(\coth\rho_{\infty})^{n-1}f^{\prime}}{\sinh^{2}\rho_{\infty}}\left(\bar{\Delta}\eta+n\eta-\frac{n}{|\mathbb{S}^{n}|}\int_{\mathbb{S}^{n}}{\eta\,d\sigma}\right).$
(6.5)

where $f^{\prime}$ is taken the value at the point
$x=(\coth\rho_{\infty})^{n}$.

Since the oscillation of $\rho_{t_{k}}-\rho_{\infty}$ is sufficiently small,
it follows exactly in [Cab-Miq2007], using [Esc98], that the solution
$\rho(\cdot,t)$ of (6.4) starting at $\rho_{t_{k}}(\cdot)$ 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 $M_{t_{k}}$ and by uniqueness $\overline{M}_{t}$ coincides
with $M_{t}$ for $t\geq t_{k}$, and hence the solution $M_{t}$ of (1.1) with
initial condition $M_{0}$ converges exponentially as $t\to\infty$ to the
geodesic sphere of radius $\rho_{\infty}$ (without correction of ambient
isometry). This completes the proof of Theorem 1.4.

## References