# A Flow approach to the prescribed Gaussian curvature problem in
$\mathbb{H}^{n+1}$

Haizhong Li  Department of Mathematical Sciences, Tsinghua University, Beijing
100084, P.R. China<EMAIL_ADDRESS>and  RUIJIA ZHANG  Department of
Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the following prescribed Gaussian curvature problem

$K=\frac{\tilde{f}(\theta)}{\phi(\rho)^{\alpha-2}\sqrt{\phi(\rho)^{2}+|\overline{\nabla}\rho|^{2}}},$

a generalization of the Alexandrov problem ($\alpha=n+1$) in hyperbolic space,
where $\tilde{f}$ is a smooth positive function on $\mathbb{S}^{n}$, $\rho$ is
the radial function of the hypersurface, $\phi(\rho)=\sinh\rho$ and $K$ is the
Gauss curvature. By a flow approach, we obtain the existence and uniqueness of
solutions to the above equations when $\alpha\geq n+1$. Our argument provides
a parabolic proof in smooth category for the Alexandrov problem in
$\mathbb{H}^{n+1}$. We also consider the cases $2<\alpha\leq n+1$ under the
evenness assumption of $\tilde{f}$ and prove the existence of solutions to the
above equations.

###### Key words and phrases:

Curvature flow, Monotonicity, Asymptotic behaviour, Hyperbolic space

###### 2010 Mathematics Subject Classification:

35K55, 53E10

##  1\. introduction

The Alexandrov problem proposed by A. D. Alexandrov [A43] is of great
significance to the study of convex bodies. It quests for the existence of
closed convex hypersurfaces with prescribed volume element of the Gaussian
image in the Euclidean space. Let $\hat{M}^{n}$ be the boundary of some convex
domains containing a neighborhood of the origin in $\mathbb{R}^{n+1}$, which
can be written as a radial graph over $\mathbb{S}^{n}$, i.e.,
$\hat{M}^{n}=\\{R(\theta)=r(\theta)\theta|\theta\in\mathbb{S}^{n}\\}$ with
induced metric $\hat{g}_{ij}$. Denote by
$\hat{\nu}(Y):\hat{M}^{n}\rightarrow\mathbb{S}^{n}$ the generalized Gauss map.
For smooth $\hat{M}^{n}$, $\hat{\nu}(Y)$ is the unit outward normal vector at
$Y\in\hat{M}^{n}$. The Alexandrov problem is to reconstruct $\hat{M}^{n}$ by
given integral Gaussian curvature

$\displaystyle\mu(\omega)=|\nu(R(\omega))|$

for a nonnegative completely additive function $\mu$ on the set of Borel
subsets $\omega$ of $\mathbb{S}^{n}$. Furthermore, if $\hat{M}^{n}$ is at
least $C^{2}$, then

$\displaystyle|\nu(R(\omega))|=\int_{R(\omega)}\hat{K}\mathrm{d}v_{\hat{M}^{n}}=\int_{\omega}\hat{K}\sqrt{\det(\hat{g}_{ij})}\mathrm{d}\theta_{\mathbb{S}^{n}}$

where $\hat{K}$ is the Gauss curvature of $\hat{M}^{n}$ and
$\mathrm{d}\theta_{\mathbb{S}^{n}}$ is the standard measure on
$\mathbb{S}^{n}$. If $\mu$ is given by integrating a function, we write it as

$\displaystyle\mu(\omega)=\int_{\omega}\tilde{f}\mathrm{d}\theta_{\mathbb{S}^{n}}.$

Then the Alexandrov problem can be reduced to the following fully nonlinear
partial differential equation

$\displaystyle\hat{K}=\frac{\tilde{f}}{r^{n-1}\sqrt{r^{2}+|\bar{\nabla}r|^{2}}}.$

The existence of regular solutions to this equation was solved by Pogorelev
[P73] for surfaces and Oliker [O83] for higher dimensional cases. For more
related interesting studies of the Alexandrov problem in $\mathbb{R}^{n+1}$,
one can refer to [GL97, T90].

Naturally, similar prescribed Gaussian curvature problems of hypersurface
$M^{n}$ in $\mathbb{H}^{n+1}$ were studied in [O83, O89], where the given
function defined on $M^{n}$ need constrained conditions to ensure $C^{0}$
estimate. Recently, Yang [Y20] studied the Alexandrov problem in the
hyperbolic space. He considered the hypersurface $M^{n}$ in $\mathbb{H}^{n+1}$
whose Gauss curvature measures were prescribed via a radial map. Let $M^{n}$
be the boundary of some convex body in $\mathbb{H}^{n+1}$ enclosing the
origin. we can parametrize it as a graph of the radial function
$\rho(\theta)$, such that
$M^{n}=\\{R(\theta)=(\rho(\theta),\theta):\rho:\mathbb{S}^{n}\rightarrow\mathbb{R}^{+},\theta\in\mathbb{S}^{n}\\}$.
If $M^{n}$ is at least $C^{2}$, similar to the Euclidean space, then we can
define the prescribed Gaussian curvature measure problem by

$\displaystyle\int_{R(\omega)}K\mathrm{d}v_{M^{n}}=\int_{\omega}\tilde{f}\mathrm{d}\theta_{\mathbb{S}^{n}}$

where $\tilde{f}$ is a given positive function on $\mathbb{S}^{n}$. By the
coordinate transformation, we write the both sides of the above integrals on
any Borel set $\omega$ of $\mathbb{S}^{n}$ as

$\displaystyle\int_{\omega}K\sqrt{\det(g_{ij})}\mathrm{d}\theta_{\mathbb{S}^{n}}=\int_{\omega}\tilde{f}\mathrm{d}\theta_{\mathbb{S}^{n}}.$

Then this curvature measure problem is reduced to the following fully
nonlinear PDE

$\displaystyle
K=\frac{\tilde{f}(\theta)}{\phi(\rho)^{n-1}\sqrt{\phi(\rho)^{2}+|\overline{\nabla}\rho|^{2}}}\quad{\rm
on}\ \mathbb{S}^{n}.$ (1.1)

Yang proved the existence and uniqueness of (1.1) with the condition
$\inf\tilde{f}>1$. In his study [Y20], he mentioned that whether the solution
to (1.1) exists with $\tilde{f}$ endowed with other geometric conditions is
still an open question. Note that the condition $\inf\tilde{f}>1$ in [Y20] is
only used in the $C^{0}$ estimate. In the studies of prescribed curvature
measure problem, $C^{0}$ estimates are difficult in most cases, but are also
more geometric. In Theorem 1.2 and Corollary 1.2, we weaken the condition of
$\tilde{f}$ and prove the existence of solutions to (1.1) under the evenness
assumption by deriving a delicate $C^{0}$ estimate. Besides, in this paper we
consider the following more general prescribed Gaussian curvature problems
which correspond to the following fully nonlinear PDE

$\displaystyle
K=\frac{\tilde{f}(\theta)}{\phi(\rho)^{\alpha-2}\sqrt{\phi(\rho)^{2}+|\overline{\nabla}\rho|^{2}}}\quad{\rm
on}\ \mathbb{S}^{n}.$ (1.2)

Motivated by the flow studied by Li-Sheng-Wang [LSW20a] in the Euclidean
space, we provide a curvature flow approach to (1.2) in hyperbolic space.
Write $f(\theta)=\tilde{f}(\theta)^{-1}$. Let $M_{0}$ be a smooth closed
uniformly convex hypersurface in $\mathbb{H}^{n+1}$ enclosing the origin. In
this paper, we study the following sort of flow

$\left\\{\begin{aligned} \frac{\partial}{\partial
t}X(x,t)=&-\phi(\rho)^{\alpha}f(\theta)K(x,t)\nu(x,t)+V(x,t),\\\
X(\cdot,0)=&X_{0}(\cdot),\end{aligned}\right.$ (1.3)

where $\alpha\geq n+1$ is a constant. Here we regard $\mathbb{H}^{n+1}$ as a
warped product space. Any point $X\in\mathbb{H}^{n+1}$ can be parametrized by
$X=(\rho,\theta)\in\mathbb{R}^{+}\times\mathbb{S}^{n}$. Then $f$ is a smooth
positive function defined on $\mathbb{S}^{n}$, $\phi(\rho)=\sinh\rho$, $K$ is
the Gauss curvature of the flow hypersurface $M_{t}$, $\nu$ is the unit
outward normal at $X(x,t)$ and $V=\sinh\rho\ \partial_{\rho}$ is a conformal
Killing vector field on $\mathbb{H}^{n+1}$. Equivalently, up to a tangential
diffeomorphism the flow (1.3) can be written as follows:

$\displaystyle\partial_{t}X=\left(-\phi(\rho)^{\alpha}f(\theta)K(x,t)+u(x,t)\right)\nu(x,t),$
(1.4)

where $u=\langle V,\nu\rangle$ is the support function of $M_{t}$. Note that
the following elliptic equation

$\displaystyle\phi(\rho)^{\alpha}K=\tilde{f}(\theta)u\quad{\rm on}\
\mathbb{S}^{n}$ (1.5)

(that is exactly (1.2)) remains invariant under (1.3).

In this paper, we prove the following results.

######  Theorem 1.1.

Let $M_{0}$ be a smooth, closed, uniformly convex hypersurface in hyperbolic
space $\mathbb{H}^{n+1}$ enclosing the origin. Suppose that $f$ is a smooth
positive function on $\mathbb{S}^{n}$. If

  * (i)

$\alpha>n+1$ or

  * (ii)

$\alpha=n+1$ and $f<1$,

then the flow (1.3) has a unique smooth uniformly convex solution $M_{t}$ for
all time $t>0$. When $t\rightarrow\infty$, $M_{t}$ converges smoothly to the
unique smooth solution of (1.2).

######  Corollary 1.1.

Suppose that $\tilde{f}$ is a smooth positive function on $\mathbb{S}^{n}$.
Then there is a unique smooth uniformly convex hypersurface $M^{n}$ in
$\mathbb{H}^{n+1}$, such that it satisfies (1.2) under one of the following
two assumptions,

  * (i)

$\alpha>n+1$ or,

  * (ii)

$\alpha=n+1$ and $\inf\tilde{f}>1$.

######  Remark 1.1.

Case $(ii)$ in Corollary 1.1 was proved by Fengrui Yang in [Y20, Theorem 6].

######  Theorem 1.2.

Let $M_{0}$ be a smooth, closed, uniformly convex and origin-symmetric
hypersurface in hyperbolic space $\mathbb{H}^{n+1}$. Assume $\alpha=n+1$, and
$f$ is a smooth positive even function on $\mathbb{S}^{n}$ satisfying
$\int_{\mathbb{S}^{n}}f(\theta)^{-1}\mathrm{d}\theta_{\mathbb{S}^{n}}>|S^{n}|$.
Then the flow (1.3) has a smooth uniformly convex solution for all time $t>0$,
and converges smoothly to the unique smooth even solution of (1.1).

######  Corollary 1.2.

Suppose $\tilde{f}$ is a smooth positive even function on $\mathbb{S}^{n}$
satisfying
$\int_{\mathbb{S}^{n}}\tilde{f}(\theta)\mathrm{d}\theta_{\mathbb{S}^{n}}>|S^{n}|$.
Then there is a unique smooth uniformly convex and origin-symmetric
hypersurface $M^{n}$ in $\mathbb{H}^{n+1}$, such that it satisfies (1.1).

######  Remark 1.2.

In Corollary 1.2, we weaken the condition of $\tilde{f}$ in Case $(ii)$ of
Theorem 1.2, from $\inf\tilde{f}>1$ to
$\int\tilde{f}(\theta)\mathrm{d}\theta_{\mathbb{S}^{n}}>|S^{n}|$ and obtain
the existence of the even Alexandrov problem in hyperbolic space.

When $2<\alpha\leq n+1$, we consider the following flow

$\left\\{\begin{aligned} \frac{\partial}{\partial
t}X(x,t)=&-\phi(\rho)^{\alpha}f(\theta)K(x,t)\nu(x,t)+\eta(t)V(x,t),\\\
X(\cdot,0)=&X_{0}(\cdot),\end{aligned}\right.$ (1.6)

where
$\eta(t)=\frac{\displaystyle\int_{\mathbb{S}^{n}}\frac{K}{u}\phi^{n+1}\mathrm{d}\theta_{\mathbb{S}^{n}}}{\displaystyle\int_{\mathbb{S}^{n}}\phi^{n+1-\alpha}f^{-1}\mathrm{d}\theta_{\mathbb{S}^{n}}}$
and obtain the existence of the following equation

$\displaystyle\phi(\rho)^{\alpha}K=c\tilde{f}(\theta)u\quad{\rm on}\
\mathbb{S}^{n}.$ (1.7)

######  Theorem 1.3.

Let $M_{0}$ be as in Theorem 1.2. Assume $2<\alpha\leq n+1$, and $f$ is a
smooth positive even function on $\mathbb{S}^{n}$. Then the flow (1.6) has a
smooth uniformly convex solution for all time $t>0$. When
$t\rightarrow\infty$, $M_{t}$ converges smoothly to the smooth solution of
(1.7) for some positive constant $c$ in a subsequence.

######  Corollary 1.3.

Suppose that $\tilde{f}$ is a smooth positive even function on
$\mathbb{S}^{n}$. If $2<\alpha<n+1$, then there is a smooth uniformly convex
and origin-symmetric hypersurface $M^{n}$ in $\mathbb{H}^{n+1}$, such that it
satisfies (1.7) for some positive constant $c$.

Curvature flows in hyperbolic space have been studied extensively in recent
years. In these studies, constrained flows were introduced to prove geometric
inequalities, see, e.g., [ACW18, Hu-Li-Wei2020, LWX14, SX19, WX14].
Convergence results for inverse curvature flows were obtained in [G11, LZ19,
S15, S'15, WWZ20]. Furthermore, volume preserving curvature flows in
hyperbolic space have been studied, see for example [AW18, M12]. All these
flows in hyperbolic space have the same limiting shape in common, i.e., they
all become round. When $f\equiv 1$, case $(i)$ in Theorem 1.1 was proved by
Fang Hong in [F21], in which he proved that the flow (1.3) converges smoothly
to a geodesic sphere. For the first time, we introduce the function $f$ in
flows (1.3), (1.6) and derive the convergence results of solutions to (1.2),
(1.7), which build a bridge between curvature flows in hyperbolic space and
solutions to the elliptic equation. By using the Klein model (see also in
[AW18, CH21, W19]), we project the hyperbolic flow (1.3) to the Euclidean
space and obtain the projection flow (5.14). We discover the monotone function
(5.16) along (5.14) and derive the asymptotic convergence result of (1.3). For
$\alpha\leq n+1$, we design the flow (1.6) and deduce the convergence result
by deriving a delicate $C^{0}$ estimate.

This paper is organized as follows. In Section 2, we collect some properties
of star-shaped hypersurfaces in hyperbolic space, derive some evolution
equations of various geometric quantities along (1.3), (1.6) and show that the
flows can be reduced to a scalar parabolic PDE for the radial function. In
Section 3, we prove $C^{0}$, $C^{1}$ estimates when $\alpha\geq n+1$ and show
that the hypersurface preserves star-shaped along the flow (1.3). In Section
4, we obtain the uniform bound of the Gauss curvature $K$ which implies the
short time existence of the flow (1.3). By using a new auxiliary function, we
obtain the uniform bound of the principal curvatures of $M_{t}$ and establish
the a priori estimates for the long time existence of (1.3). In section 5, we
study the asymptotic behaviour of the flow (1.3) by projecting $M_{t}$ to the
Euclidean space, prove the uniqueness of the solution to (1.2) and complete
the proof of Theorem 1.1. In section 6, we complete the proof of Theorem 1.2
by deriving a delicate $C^{0}$ estimate. In Section 7, we study the normalized
flow (1.6) under the evenness assumption when $2<\alpha\leq n+1$ and complete
the proof of Theorem 1.3.

######  Acknowledgments.

The authors were partially supported by NSFC grant No.11831005 and NSFC grant
No.12126405. The authors would like to thank Professor Xianfeng Wang and
Professor Yong Wei for helpful discussions.

##  2\. Preliminaries

In this paper, we fix a point $o\in\mathbb{H}^{n+1}$ and consider the polar
geodesic coordinates centered at $o$ and regard $\mathbb{H}^{n+1}$ as a warped
product space $[0,+\infty)\times\mathbb{S}^{n}$ equipped with Riemannian
metric

$\displaystyle
g_{\mathbb{H}^{n+1}}=d\rho^{2}+\phi(\rho)^{2}g_{\mathbb{S}^{n}}$

where $\phi(\rho)=\sinh\rho$ and $g_{\mathbb{S}^{n}}$ is the standard metric
on the unit sphere $\mathbb{S}^{n}$. Denote

$\displaystyle\Phi(\rho)=\int_{0}^{\rho}\sinh s\
\mathrm{d}s=\phi^{\prime}(\rho)-1.$ (2.1)

The conformal Killing vector field can be written as
$V=D\Phi=\phi\partial\rho$ and $D^{2}\Phi=\phi^{\prime}g_{\mathbb{S}^{n}}$. In
particular, $DV=\phi^{\prime}g_{\mathbb{H}^{n+1}}$.

###  2.1. Hypersurfaces in hyperbolic space

Let $M^{n}$ be a closed hypersurface in $\mathbb{H}^{n+1}$ and
$\\{x^{1},\cdots,x^{n}\\}$ be a local coordinate system of $M^{n}$. We regard
$\nu$ as the unit outward normal vector field of $M^{n}$. We denote by the
induced metric $g_{ij}=g(X_{i},X_{j})$ and the second fundamental form
$h_{ij}=h(X_{i},X_{j})$ of $M^{n}$, where the second fundamental form is
defined by $h(X,Y)=\langle\nabla_{X}\nu,Y\rangle$ with any two tangent vector
fields $X,Y\in TM^{n}$. The Weingarten matrix is regarded as
$\mathcal{W}=\\{h_{i}{}^{j}\\}=\\{h_{ik}g^{kj}\\}$, where $\\{g^{ij}\\}$ is
the inverse matrix of $\\{g_{ij}\\}$. The principal curvature
$\kappa=(\kappa_{1},\cdots,\kappa_{n})$ of $M^{n}$ are eigenvalues of
$\mathcal{W}$. Let $f(\kappa)$ be a symmetric function of the principal
curvatures $\kappa=(\kappa_{1},\kappa_{2},\cdots,\kappa_{n})$. There exists a
function $\mathcal{F}(\mathcal{W})$ defined on the Weingarten matrix, such
that $F(\mathcal{W})=f(\kappa)$. Since $h_{i}{}^{j}=\sum_{k}h_{ik}g^{kj}$,
$\mathcal{F}$ can be viewed as a function $\hat{\mathcal{F}}(h_{ij},g_{ij})$
defined on the second fundamental form $\\{h_{ij}\\}$ and the metric
$\\{g_{ij}\\}$. In the subsequent article, we denote

$\displaystyle\dot{\mathcal{F}}^{pq}(\mathcal{W}):=$
$\displaystyle\frac{\partial\hat{\mathcal{F}}}{\partial
h_{pq}}(h_{ij},g_{ij}),\quad\ddot{\mathcal{F}}^{pq,rs}(\mathcal{W}):=\frac{\partial^{2}\hat{\mathcal{F}}}{\partial
h_{pq}\partial h_{rs}}(h_{ij},g_{ij}).$

Here we collect some formulas of hypersurface in hyperbolic space (see [GL15,
HL21]).

######  Lemma 2.1.

Let $(M^{n},g)$ be a smooth hypersurface in $\mathbb{H}^{n+1}$. Then we have

$\displaystyle\nabla_{i}\Phi=\langle
V,X_{i}\rangle,\quad\nabla_{j}\nabla_{i}\Phi=\phi^{\prime}g_{ij}-uh_{ij}.$
(2.2)

The support function $u=\langle V,\nu\rangle$ satisfies

$\displaystyle\nabla_{i}u=\langle V,X_{k}\rangle
h_{i}{}^{k},\quad\nabla_{j}\nabla_{i}u=\langle V,\nabla
h_{ij}\rangle+\phi^{\prime}h_{ij}-uh_{i}{}^{k}h_{kj}$ (2.3)

where $\nabla$ is the levi-Civita connection on $M^{n}$ with respect to the
induced metric and $\\{X_{1},\cdots,X_{n}\\}$ is a basis of the tangent space
of $M^{n}$.

Then we have the first and second derivatives of the distance function $\rho$.

######  Corollary 2.1.

$\displaystyle\nabla_{i}\rho=\frac{\langle
V,X_{i}\rangle}{\phi},\quad\nabla_{j}\nabla_{i}\rho=\frac{\phi^{\prime}}{\phi}(g_{ij}-\nabla_{j}\rho\nabla_{i}\rho)-\frac{uh_{ij}}{\phi}.$
(2.4)

###### Proof.

Observe that

$\displaystyle\nabla_{i}\Phi=\phi\nabla_{i}\rho,\quad\nabla_{j}\nabla_{i}\Phi=\phi\nabla_{j}\nabla_{i}\rho+\phi^{\prime}\nabla_{j}\rho\nabla_{i}\rho.$

Combing with (2.2) we get (2.4) by a direct calculation. ∎

###  2.2. Evolution equations

For convenience, we consider the following flow

$\frac{\partial}{\partial t}X(x,t)=-\Theta\nu(x,t)+\tilde{\eta}(t)V$ (2.5)

where $\Theta=\phi(\rho)^{\alpha}f(\theta)K$ and the global term
$\tilde{\eta}(t)$ is a function of time $t$. For $\tilde{\eta}(t)\equiv 1$,
(2.5) is the flow (1.3); for $\tilde{\eta}(t)=\eta(t)$, (2.5) is the flow
(1.6).

######  Lemma 2.2.

Along the flow (2.5), we have the following evolution equations (also see
[WWZ20, F21]). The induced metric evolves by

$\displaystyle\frac{\partial}{\partial t}g_{ij}=$ $\displaystyle-2\Theta
h_{ij}+2\phi^{\prime}\tilde{\eta}(t)g_{ij}.$ (2.6)

The support function evolves by

$\displaystyle\frac{\partial}{\partial t}u=$
$\displaystyle-\phi^{\prime}\Theta+\phi^{\prime}\tilde{\eta}(t)u+\langle
V,\nabla\Theta\rangle.$ (2.7)

The second fundamental form evolves by

$\displaystyle\frac{\partial}{\partial
t}h_{i}{}^{j}=\nabla_{i}\nabla^{j}\Theta+\Theta
h_{i}{}^{k}h_{k}{}^{j}-\tilde{\eta}(t)\phi^{\prime}h_{i}{}^{j}+(\tilde{\eta}(t)u-\Theta)\delta_{i}{}^{j}$
(2.8)

where $\nabla$ is the Levi-Civita connection of the induced metric on $M_{t}$.

###### Proof.

By a direct calculation, we have

$\displaystyle\frac{\partial}{\partial t}g_{ij}=$
$\displaystyle{\partial_{t}}\langle\partial_{i}X,\partial_{j}X\rangle$
$\displaystyle=$ $\displaystyle\langle
D_{i}\left(-\Theta\nu+\tilde{\eta}(t)V\right),\partial_{j}X\rangle+\langle\partial_{i}X,D_{j}\left(-\Theta\nu+\tilde{\eta}(t)V\right)\rangle$
$\displaystyle=$ $\displaystyle-\Theta\left(\langle
D_{i}\nu,\partial_{j}X\rangle+\langle\partial_{i}X,D_{j}\nu\rangle\right)+2\tilde{\eta}(t)\phi^{\prime}g_{ij}$
$\displaystyle=$ $\displaystyle-2\Theta
h_{ij}+2\tilde{\eta}(t)\phi^{\prime}g_{ij}.$

Since $\partial_{t}\nu$ is tangential,

$\displaystyle\frac{\partial}{\partial t}\nu=$
$\displaystyle\langle\partial_{t}\nu,\partial_{j}X\rangle g^{il}\partial_{l}X$
(2.9) $\displaystyle=$
$\displaystyle-\langle\nu,\partial_{j}\left(-\Theta\nu+\tilde{\eta}(t)V\right)\rangle
g^{jl}\partial_{l}X$ $\displaystyle=$ $\displaystyle\partial_{j}\Theta
g^{jl}\partial_{l}X=\nabla\Theta.$

Using (2.9), we obtain the evolution of the support function $u$ as follows:

$\displaystyle\frac{\partial}{\partial t}u=$ $\displaystyle\partial_{t}\langle
V,\nu\rangle=\langle-\phi^{\prime}\Theta\nu+\phi^{\prime}\tilde{\eta}(t)V,\nu\rangle+\langle
V,\nabla\Theta\rangle$ $\displaystyle=$
$\displaystyle-\phi^{\prime}\Theta+\phi^{\prime}\tilde{\eta}(t)u+\langle
V,\nabla\Theta\rangle.$

Now we calculate the evolution of $h_{ij}$

$\displaystyle\frac{\partial}{\partial t}h_{ij}=$
$\displaystyle-\partial_{t}\langle D_{\partial_{i}X}\partial_{j}X,\nu\rangle$
$\displaystyle=$ $\displaystyle-\langle
D_{\partial_{i}X}D_{\partial_{j}X}(-\Theta\nu+\tilde{\eta}(t)V),\nu\rangle-R^{\mathbb{H}^{n+1}}(\partial_{i}X,\partial_{t}X,\partial_{j}X,\nu)-\langle
D_{\partial_{i}X}\partial_{j}X,\nabla\Theta\rangle$ $\displaystyle=$
$\displaystyle\partial_{i}\partial_{j}\Theta-\Theta(h^{2})_{ij}+\tilde{\eta}(t)\phi^{\prime}h_{ij}+(\tilde{\eta}(t)u-\Theta)g_{ij}-\langle\nabla_{\partial_{i}X}\partial_{j}X,\nabla\Theta\rangle$
$\displaystyle=$ $\displaystyle\nabla_{i}\nabla_{j}\Theta-\Theta
h_{i}{}^{k}h_{kj}+\tilde{\eta}(t)\phi^{\prime}h_{ij}+(\tilde{\eta}(t)u-\Theta)g_{ij}.$

From (2.6), we have

$\displaystyle\frac{\partial}{\partial
t}g^{ij}=-g^{il}\left(\partial_{t}g_{lm}\right)g^{mj}=2\Theta
h^{ij}-2\tilde{\eta}(t)\phi^{\prime}g^{ij}.$

Thus

$\displaystyle\frac{\partial}{\partial t}h_{i}{}^{j}=$
$\displaystyle\partial_{t}h_{il}g^{lj}+h_{il}\partial_{t}g^{lj}$ (2.10)
$\displaystyle=$ $\displaystyle\nabla_{i}\nabla^{j}\Theta+\Theta
h_{i}{}^{k}h_{k}{}^{j}-\tilde{\eta}(t)\phi^{\prime}h_{i}{}^{j}+(\tilde{\eta}(t)u-\Theta)\delta_{i}{}^{j}.$

∎

###  2.3. Parametrization by radial graph

For a closed star-shaped hypersurface $M^{n}\subset\mathbb{H}^{n+1}$, we can
parametrize it as a graph of the radial function
$\rho(\theta):\mathbb{S}^{n}\to\mathbb{R}$, i.e.,

$\displaystyle
M^{n}=\\{(\rho(\theta),\theta):\rho:\mathbb{S}^{n}\rightarrow\mathbb{R}^{+},\quad\theta\in\mathbb{S}^{n}\\}$

where $\theta=(\theta^{1},\cdots,\theta^{n})$ is a local normal coordinate
system of $\mathbb{S}^{n}$ and $\rho$ is a smooth function on
$\mathbb{S}^{n}$. Let $f_{i}=\overline{\nabla}_{i}f$,
$f_{ij}=\overline{\nabla}^{2}_{ij}f$, where $\overline{\nabla}$ is the Levi-
Civita connection on $\mathbb{S}^{n}$ with respect to the standard metric
$g_{\mathbb{S}^{n}}$.
The tangent space of $M^{n}$ is spanned by (also see [CLW18])

$\displaystyle X_{i}=\rho_{i}\partial_{\rho}+\partial_{\theta_{i}}$ (2.11)

and the unit outward normal vector is

$\displaystyle\nu=\frac{\partial_{\rho}-\frac{\rho^{i}\partial_{\theta_{i}}}{\phi^{2}}}{w},$

where we set

$\displaystyle w=\sqrt{1+\frac{|\overline{\nabla}\rho|^{2}}{\phi^{2}}}.$
(2.12)

Then the support function and the induced metric can be expressed as

$\displaystyle
u=\frac{\phi^{2}}{\sqrt{\phi^{2}+|\overline{\nabla}\rho|^{2}}},$ (2.13)
$\displaystyle g_{ij}=\phi^{2}\delta_{ij}+\rho_{i}\rho_{j},\quad
g^{ij}=\frac{1}{\phi^{2}}(\delta^{ij}-\frac{\rho_{i}\rho_{j}}{\phi^{2}+|\overline{\nabla}\rho|^{2}}).$
(2.14)

The second fundamental form is given by

$\displaystyle
h_{ij}=\frac{-\phi\rho_{ij}+2\phi^{\prime}\rho_{i}\rho_{j}+\phi^{2}\phi^{\prime}\delta_{ij}}{\sqrt{\phi^{2}+|\overline{\nabla}\rho|^{2}}}$
(2.15)

and we have the Weingarten matrix

$\displaystyle
h_{i}{}^{j}=\frac{1}{\phi^{2}\sqrt{\phi^{2}+|\overline{\nabla}\rho|^{2}}}\left(\delta^{jk}-\frac{\rho_{j}\rho_{k}}{\phi^{2}+|\overline{\nabla}\rho|^{2}}\right)\left(-\phi\rho_{ki}+2\phi^{\prime}\rho_{k}\rho_{i}+\phi^{2}\phi^{\prime}\delta_{ki}\right).$
(2.16)

Similar to [LSW20a, p. 901], the flow (2.5) can be written as a scalar
parabolic PDE for the radial function

$\left\\{\begin{aligned} \frac{\partial}{\partial
t}\rho(\theta,t)=&-\phi(\rho)^{\alpha}f(\theta)wK+\tilde{\eta}(t)\phi(\theta,t),\quad\text{for
}(\theta,t)\in\mathbb{S}^{n}\times[0,+\infty),\\\
\rho(\cdot,0)=&\rho_{0}(\cdot),\end{aligned}\right.$ (2.17)

where $w$ is the function defined in (2.12).

##  3\.  $C^{0}$ and $C^{1}$ estimates

In this section, we establish the $C^{0}$ and $C^{1}$ estimates of the flow
(1.3) for the proof of Theorem 1.1. Especially, we show that the flow
hypersurface $M_{t}$ preserves star-shaped along (1.3).

###  3.1.  $C^{0}$ estimate

In this subsection, we will show that the radial function $\rho$ of (1.3) has
uniform bound under the assumption of Theorem 1.1.

######  Lemma 3.1.

Let $\rho(\cdot,t)$ be a smooth, positive, uniformly convex solution to (2.17)
on $\mathbb{S}^{n}\times[0,T)$ provided $\tilde{\eta}(t)\equiv 1$. If
$\alpha>n+1$, or $\alpha=n+1$ with $f<1$, then there is a positive constant
$C$ depending only on $n$, $\max f$, $\inf f$ and the initial hypersurface
$M_{0}$, such that

$\displaystyle\frac{1}{C}\leq\rho(\cdot,t)\leq C,\quad\forall t\in[0,T).$

###### Proof.

Fix time $t$ and suppose that $\rho$ attains its maximum at point $(p_{0},t)$.
At $(p_{0},t)$, we have $|\overline{\nabla}\rho|=0$ and $\rho_{ij}\leq 0$.
From (2.16),

$\displaystyle h_{i}{}^{j}\geq\frac{\phi^{\prime}}{\phi}\delta_{i}{}^{j}.$
(3.1)

Inserting (3.1) into (2.17), we obtain

$\displaystyle\partial_{t}\rho\leq-\phi^{\alpha-n}\phi^{\prime
n}f+\phi=\phi(-\phi^{\alpha-n-1}\phi^{\prime
n}f+1)\leq\phi(-\phi^{\alpha-1}f+1),$ (3.2)

where we use the fact $\frac{\phi^{\prime}}{\phi}\geq 1$. Since $\alpha\geq
n+1$, if $\phi\leq(\min f)^{-\frac{1}{\alpha-1}}$ for all $t\geq 0$, then we
obtain the uniform upper bound of $\rho$. If there is some $t_{0}$, such that
$\phi>(\min f)^{-\frac{1}{\alpha-1}}$, then we obtain
$\rho\leq\max_{\mathbb{S}^{n}}\rho(\cdot,0)$ by (3.2). Thus, $\rho$ obtains
the uniform upper bound with respect to the positive lower bound of $f$ on
$\mathbb{S}^{n}$ and the initial hypersurface.

Suppose $\rho$ attains its spatial minimum at point $(q_{0},t)$. Similarly, at
$(q_{0},t)$, we have

$\displaystyle h_{i}{}^{j}\leq\frac{\phi^{\prime}}{\phi}\delta_{i}{}^{j}$
(3.3)

and

$\displaystyle\partial_{t}\rho\geq-\phi^{\alpha-n}\phi^{\prime
n}f+\phi=\phi(-\phi^{\alpha-n-1}\phi^{\prime n}f+1).$ (3.4)

When $\alpha>n+1$, $\phi^{\alpha-n-1}\phi^{\prime
n}=(\sinh\rho)^{\alpha-n-1}(\cosh\rho)^{n}\rightarrow 0$ as $\rho\rightarrow
0$. When $\alpha=n+1$, $\phi^{\alpha-n-1}\phi^{\prime
n}=(\cosh\rho)^{n}\rightarrow 1$ as $\rho\rightarrow 0$. Hence if
$\phi^{\alpha-n-1}\phi^{\prime n}\geq\frac{1}{\max f}$ for all $t\geq 0$, then
we obtain the uniform lower bound of $\rho$ provided $\alpha>n+1$ or
$\alpha=n+1$ with the assumption $f<1$. If there is some $t_{0}$, such that
$\phi^{\alpha-n-1}\phi^{\prime n}<\frac{1}{\max f}$, we obtain
$\rho\geq\min_{\mathbb{S}^{n}}\rho(\cdot,0)$ by (3.4). Thus $\rho$ has the
uniform lower bound depending on $\min f$ and the initial hypersurface. ∎

###  3.2.  $C^{1}$ estimate

In this subsection, we derive a uniform upper bound of the gradient of $\rho$
by using the approach in [G06, WWZ20].

######  Lemma 3.2.

Let $\rho(\cdot,t)$ be a smooth, positive, uniformly convex solution to (2.17)
on $\mathbb{S}^{n}\times[0,T)$. Based on the results of Lemma 3.1, we have

$|\overline{\nabla}\rho|\leq C,$

where $C$ only depends on the uniform upper and lower bounds of $\rho$.

###### Proof.

Fix some time $t$ and consider the auxiliary function $Q=\log w+\beta\rho$,
where we assume $\beta=-2\tanh(\rho_{\min})$ from Lemma 3.1. At the maximal
point of $Q$, we have $\overline{\nabla}Q=0$, then

$\displaystyle\frac{\gamma_{li}\gamma_{l}}{w^{2}}+\beta\rho_{i}=0,$ (3.5)

where we regard $\gamma(\rho)=\log(1-\frac{2}{e^{\rho}+1})$ and
$\frac{d\gamma}{d\rho}=\frac{1}{\phi}$. Then (2.12), (2.14), (2.15), (2.16)
becomes

$\displaystyle g_{ij}=\phi^{2}(\delta_{ij}+\gamma_{i}\gamma_{j}),\quad
g^{ij}=\frac{1}{\phi^{2}}(\delta^{ij}-\frac{\gamma_{i}\gamma_{j}}{w^{2}}),$
(3.6) $\displaystyle w=\sqrt{1+|\overline{\nabla}\gamma|^{2}},\quad
h_{ij}=\frac{\phi^{\prime}}{\phi w}g_{ij}-\frac{\phi}{w}\gamma_{ij}.$ (3.7)

The Weingarten matrix turns into

$\displaystyle h_{i}{}^{j}=h_{il}g^{lj}=\frac{\phi^{\prime}}{\phi
w}\delta_{i}{}^{j}-\frac{\phi}{w}\gamma_{il}g^{lj}.$ (3.8)

Inserting (3.8) into (3.5) and multiplying $w^{2}\rho^{i}$ on the both sides
of (3.5), we obtain

$\displaystyle\frac{w}{\phi}(\frac{\phi^{\prime}}{\phi
w}\delta_{i}{}^{j}-h_{i}{}^{j})g_{lj}\gamma_{l}\rho^{i}+\beta
w^{2}|\overline{\nabla}\rho|^{2}=0.$ (3.9)

By a direct calculation, from (3.6) and (3.7), we have

$\displaystyle
g_{lj}\gamma_{l}=\phi\rho_{j}+|\overline{\nabla}\rho|^{2}\frac{\rho_{j}}{\phi}=\phi
w^{2}\rho_{j}.$ (3.10)

This together with (3.9) implies that

$\displaystyle(\beta+\frac{\phi^{\prime}}{\phi})|\overline{\nabla}\rho|^{2}=wh_{i}{}^{j}\rho_{j}\rho^{i}.$
(3.11)

Since $\\{h_{i}{}^{j}\\}$ is positive-definite and
$\beta+\frac{\phi^{\prime}}{\phi}<0$, we obtain $\overline{\nabla}\rho=0$ at
the maximal point of $Q$. Thus $Q_{\max}\leq\beta\rho_{\min}<0$. And

$\displaystyle|\overline{\nabla}\rho|\leq\sinh(\rho_{\max})\sqrt{e^{4\rho_{\max}\tanh(\rho_{min})}-1}.$

∎

######  Remark 3.1.

When the hypersurface is uniformly convex in hyperbolic space, the gradient
estimate of $\rho$ follows from the uniform upper and lower bounds of the
radial function $\rho$.

######  Corollary 3.1.

Based on the results of Lemma 3.1, along the flow (1.3), the hypersurface
$M_{t}$ preserves star-shaped and the support function $u$ satisfies

$\displaystyle\frac{1}{C}\leq u\leq C,\quad\forall t\in[0,T).$

for some constant $C>0$, where $C$ only depends on $C_{0}$ estimate.

###### Proof.

Recall (2.13),

$\displaystyle
u=\frac{\phi}{\sqrt{1+\frac{|\overline{\nabla}\rho|^{2}}{\phi^{2}}}}.$

The upper and lower bounds of $u$ follows from Lemma 3.1 and 3.2. Besides, we
have

$\displaystyle\langle\partial_{\rho},\nu\rangle=\frac{u}{\phi}=\frac{1}{w}=\frac{1}{\sqrt{1+\frac{|\overline{\nabla}\rho|^{2}}{\phi^{2}}}}\geq\frac{1}{C^{\prime}}$

for some $C^{\prime}>0$ depends on $\max|\overline{\nabla}\rho|$ and
$\min\rho$. Thus the hypersurface preserves star-shaped along the flow (2.5).
∎

######  Remark 3.2.

When the hypersurface is uniformly convex in hyperbolic space, the upper and
lower bounds of the support function of $u$ follow from the uniform bound of
the radial function $\rho$.

##  4\. $C^{2}$ estimates

In this section, let us assume that we have already obtained the uniform upper
and lower bounds of the radial function $\rho$ and the global term
$\tilde{\eta}(t)$ along the flow (2.5). From Lemma 3.2 and Corollary 3.1, the
uniform bound of $\rho$ implies the upper and lower bounds of $w$ and the
support function $u$. We shall establish the $C^{2}$ estimate under this
assumption.

###  4.1. The bounds of $K$

In this subsection, we show that $K$ is bounded from above and below along
(2.5).

######  Lemma 4.1.

Along (2.5), there is a constant $c>0$ depending on $M_{0}$, $\alpha$, $n$ and
the uniform bounds of $\rho$, $\tilde{\eta}(t)$ and $f$, such that

$K\geq c.$

###### Proof.

First, we calculate the evolution equation of $\Theta=\phi^{\alpha}fK$

$\displaystyle\frac{\partial}{\partial t}\Theta=$
$\displaystyle\phi(\rho)^{\alpha}f\partial_{t}K+Kf\partial_{t}\left(\phi(\rho)^{\alpha}\right)$
(4.1) $\displaystyle=$
$\displaystyle\alpha\phi^{\alpha-1}\phi^{\prime}\partial_{t}\rho
fK+\phi^{\alpha}f\frac{\partial K}{\partial
h_{i}{}^{j}}\partial_{t}h_{i}{}^{j}.$

Differentiating $\langle V,V\rangle=\phi^{2}$, we obtain

$\displaystyle\phi\phi^{\prime}\partial_{t}\rho=\langle\partial_{t}V,V\rangle=\phi^{\prime}\langle\partial_{t}X,V\rangle.$
(4.2)

By (2.5), we have

$\displaystyle\partial_{t}\rho=\frac{\langle\partial_{t}X,V\rangle}{\phi}=\frac{-\Theta
u+\tilde{\eta}(t)\phi^{2}}{\phi}=-\frac{\Theta}{w}+\tilde{\eta}(t)\phi.$ (4.3)

Inserting (2.8) and (4.3) into (4.1), we get

$\displaystyle\frac{\partial}{\partial t}\Theta$ (4.4) $\displaystyle=$
$\displaystyle-\alpha\phi^{\alpha-1}\phi^{\prime}\frac{\Theta}{w}fK+\alpha\tilde{\eta}(t)\phi^{\prime}\Theta+\phi^{\alpha}f\frac{\partial
K}{\partial h_{i}{}^{j}}\left(\nabla_{i}\nabla^{j}\Theta+\Theta
h_{i}{}^{k}h_{k}{}^{j}-\phi^{\prime}\tilde{\eta}(t)h_{i}{}^{j}+\left(\tilde{\eta}(t)u-\Theta\right)\delta_{i}{}^{j}\right)$
$\displaystyle=$
$\displaystyle\phi^{\alpha}f\dot{K}^{ij}\Theta_{ij}-\frac{\alpha\Theta^{2}\phi^{\prime}}{w\phi}+(\alpha-n)\tilde{\eta}(t)\phi^{\prime}\Theta+(\tilde{\eta}(t)u-\Theta)\phi^{\alpha}f\sigma_{n-1}+\Theta^{2}H,$

where we use $\frac{\partial K}{\partial
h_{i}{}^{j}}\delta_{i}{}^{j}=\sigma_{n-1}$ and $\frac{\partial K}{\partial
h_{i}{}^{j}}h_{i}{}^{k}h_{k}{}^{j}=KH$ in the last equality. At the spatial
minimum point of $\Theta$ on $M_{t}$, we have
$\phi^{\alpha}f\dot{K}^{ij}\Theta_{ij}\geq 0$. We can assume
$\Theta<\inf\limits_{[0,T)}\tilde{\eta}(t)\min\frac{u}{2}$ from Corollary 3.1
without loss of generality. Hence,

$\displaystyle\frac{d}{dt}\Theta_{\min}(t)\geq$
$\displaystyle-\frac{\alpha\Theta_{\min}^{2}\phi^{\prime}}{w\phi}+(\alpha-n)\tilde{\eta}(t)\phi^{\prime}\Theta_{\min}+\inf_{[0,T)}\tilde{\eta}(t)\frac{u_{\min}}{2}\phi^{\alpha}f\Theta_{\min}^{\frac{n-1}{n}}$
$\displaystyle\geq$ $\displaystyle-
c_{1}\Theta_{\min}^{2}(t)-c_{2}\Theta_{\min}+c_{3}\Theta_{\min}^{\frac{n-1}{n}}$

for some constant $c_{1},c_{2},c_{3}>0$, where all of them depend on $\alpha$,
$n$ and the uniform upper and lower bounds of $\rho$, $\tilde{\eta}(t)$ and
$f$. Hence there is a positive constant $c_{4}$ depending only on
$c_{1},c_{2},c_{3}$, such that if $\Theta_{\min}(t)\in(0,c_{4})$, we have
$\frac{d}{dt}\Theta_{\min}(t)>0$. Therefore
$\Theta_{\min}(t)\geq\min\\{\Theta_{\min}(0),c_{4},\inf\tilde{\eta}(t)\min\frac{u}{2}\\}$.
Since $\rho$ and $f$ are bounded from above, $K=\phi^{-\alpha}f^{-1}\Theta$ is
bounded from below by some positive constant $c$. ∎

######  Lemma 4.2.

Along (2.5), there is a constant $C>0$ depending only on $M_{0}$, $\alpha$,
$n$ and the uniform bounds of $\rho$, $\tilde{\eta}(t)$, $\min f$ and
$\|f\|_{C^{1}}$, such that

$K\leq C.$

###### Proof.

Let $Q=\log\Theta-\log(u-a)$, where $a=\frac{1}{2}\inf_{M\times[0,T)}u$.
Recall (2.7),

$\displaystyle\frac{\partial}{\partial t}u=$
$\displaystyle-\phi^{\prime}\Theta+\phi^{\prime}\tilde{\eta}(t)u+\langle
V,\nabla\Theta\rangle$ (4.5) $\displaystyle=$
$\displaystyle-\phi^{\prime}(\Theta-\tilde{\eta}(t)u)+\alpha\phi^{\prime}\frac{\Theta}{\phi}\langle
V,\nabla\rho\rangle+\frac{\langle V,\nabla
f\rangle\Theta}{f}+\frac{\langle\nabla K,V\rangle\Theta}{K}$ $\displaystyle=$
$\displaystyle-\phi^{\prime}(\Theta-\tilde{\eta}(t)u)+\alpha\phi^{\prime}|\nabla\rho|^{2}\Theta+\langle
V,\nabla\log f\rangle\Theta+\langle\nabla\log K,V\rangle\Theta.$

Combining with (2.3), we obtain the evolution of $u$

$\displaystyle\partial_{t}u=\phi^{\alpha}f\dot{K}^{ij}u_{ij}-\phi^{\prime}\left((n+1)\Theta-\tilde{\eta}(t)u\right)+\alpha\phi^{\prime}|\nabla\rho|^{2}\Theta+\frac{\langle
V,\nabla f\rangle}{f}\Theta+u\phi^{\alpha}f\dot{K}^{ij}h_{i}{}^{k}h_{kj},$
(4.6)

where we use the Codazzi equation in the equality. $|\nabla f(X)|$ can be
estimated as

$\displaystyle|\nabla f(X)|\leq
C(\min\rho,\|\rho\|_{C^{1}}(\mathbb{S}^{n}),\|f\|_{C^{1}}(\mathbb{S}^{n})).$
(4.7)

At the spatial maximum point $(p_{0},t)$ of $Q$ on $M_{t}$, we have

$\displaystyle\frac{\nabla\Theta}{\Theta}=\frac{\nabla u}{u-a}.$ (4.8)

Inserting (4.6) and (4.8) into (4.4), we obtain

$\displaystyle\partial_{t}Q=$
$\displaystyle\frac{\partial_{t}\Theta}{\Theta}-\frac{\partial_{t}u}{u-a}$
$\displaystyle=$
$\displaystyle\frac{\dot{K}^{ij}\Theta_{ij}}{K}-\frac{\alpha\Theta\phi^{\prime}}{w\phi}+(\alpha-n)\tilde{\eta}(t)\phi^{\prime}+(\tilde{\eta}(t)u-\Theta)\frac{\sigma_{n-1}}{K}+\Theta
H$
$\displaystyle-\frac{\phi^{\alpha}f\dot{K}^{ij}u_{ij}}{u-a}+\phi^{\prime}\frac{(n+1)\Theta-\tilde{\eta}(t)u}{u-a}-\alpha\phi^{\prime}|\nabla\rho|^{2}\frac{\Theta}{u-a}-\frac{\langle
V,\nabla f\rangle}{f}\frac{\Theta}{u-a}-\frac{u}{u-a}\Theta H$
$\displaystyle\leq$
$\displaystyle\phi^{\alpha}f\dot{K}^{ij}Q_{ij}+(\alpha-n)\tilde{\eta}(t)\phi^{\prime}+\phi^{\prime}\frac{(n+1)\Theta}{u-a}+\frac{\phi\Theta|\nabla
f|}{f(u-a)}-n(C_{n}^{k})^{-\frac{1}{n}}\phi^{\alpha}fK^{\frac{1+n}{n}}\frac{a}{u-a},$

where we assume $\Theta>\max\eta\max_{M\times[0,T)}u$ from Corollary 3.1
without loss of generality. Here we also use the Newton-Maclaurin inequality
$H\geq nK^{\frac{1}{n}}$ and the Cauchy-Schwartz inequality $\langle V,\nabla
f\rangle\leq\phi|\nabla f|$ to obtain the last inequality. Then we get

$\displaystyle\partial_{t}Q\leq$ $\displaystyle C_{1}+C_{2}\Theta-
C_{3}\Theta^{\frac{n+1}{n}}$ (4.9)

for some $C_{1},C_{2},C_{3}>0$ depending on $\alpha$, $n$, $\min f$ and
$\|f\|_{C^{1}}$, the uniform bounds of $\rho$ and $\tilde{\eta}(t)$. Besides,
there is a constant $C_{4}$ depending on the uniform upper and lower bounds of
$u$, such that

$\displaystyle\frac{1}{C_{4}}e^{Q}\leq\Theta=(u-a)e^{Q}\leq C_{4}e^{Q}.$
(4.10)

We have

$\displaystyle\partial_{t}Q\leq$ $\displaystyle
C_{1}+C_{2}C_{4}e^{Q}-C_{3}(C_{4})^{-\frac{n+1}{n}}e^{\frac{n+1}{n}Q}.$

Therefore,
$Q\leq\max\\{C_{5},Q_{\max}(0),\max\eta\max\limits_{M\times[0,T)}u\\}$ where
$C_{5}$ is a positive constant depending on $C_{1}$, $C_{2}$, $C_{3}$ and
$C_{4}$. Hence $K=\phi^{-\alpha}f^{-1}(u-a)e^{Q}$ is bounded from above by
some positive constant $C$ depending only on $M_{0}$, $\alpha$, $n$ $\min f$,
$\|f\|_{C^{1}}$ and the uniform bounds of $\rho$ and $\tilde{\eta}(t)$. ∎

In the proof of Lemma 4.1 and Lemma 4.2 we also obtain the uniform bounds of
$\Theta$.

######  Corollary 4.1.

Along (2.5), $\Theta=\phi^{\alpha}fK$ is uniformly bounded, i.e.,

$\displaystyle\frac{1}{C}<\Theta(x,t)<C$

for some constant $C>0$ that only depends on $M_{0}$, $\alpha$, $n$ $\min f$,
$\|f\|_{C^{1}}$ and the uniform bounds of $\rho$ and $\tilde{\eta}(t)$.

We established the $C^{0}$, $C^{1}$ estimates in Section 3 along (1.3) for the
proof of Theorem 1.1. Due to Lemma 4.1 and Lemma 4.2, we have the following
Corollary.

######  Corollary 4.2.

Let $M_{0}$ be as in Theorem 1.1. When $\alpha>n+1$ or $\alpha=n+1$ with $\max
f<1$, there is a positive constant $C$ depending only on $M_{0}$, $\alpha$,
$n$, $\min f$ and $\|f\|_{C^{1}}$ along the flow (1.3), such that

$\displaystyle\frac{1}{C}\leq K\leq C.$ (4.11)

###  4.2. The bound of principal curvatures

In this subsection, we shall show the principal curvatures of $M_{t}$ is
uniformly bounded along the flow (2.5) under the assumption in section 4.

######  Lemma 4.3.

Along (2.5), there is a positive constant $C$ depending on $M_{0}$, $\alpha$,
$n$, $\min f$, $\|f\|_{C^{2}}(\mathbb{S}^{n})$ and the uniform bounds of
$\rho$, $\tilde{\eta}$, such that the principal curvatures satisfy

$\displaystyle\frac{1}{C}\leq\kappa_{i}(\cdot,t)\leq C,\quad\forall\ t\
\in[0,T)\ {\rm and}\ i=1,2,\cdots,n.$

###### Proof.

Denote by $\lambda(x,t)$ the maximal principal radii at $X(x,t)$. Let $A$ be a
positive constant to be determined later. Denote by
$Q^{\prime}=\log\lambda(x,t)+A\rho$. Fix an arbitrary time $T_{0}\in(0,T)$.
Assume that $Q^{\prime}$ attains its maximum at $(x_{0},t_{0})$ provided
$t\in[0,T_{0})$. We then introduce a normal coordinate system
$\\{\partial_{i}\\}$ around $(x_{0},t_{0})$, such that
$\nabla_{\partial_{i}X}{}\partial_{j}X(x_{0},t_{0})=0$ for all
$i,j=1,2\cdots,n$ and
$h_{ij}(x_{0},t_{0})=\kappa_{i}(x_{0},t_{0})\delta_{ij}$. Further, we can
choose $\partial_{1}|_{(x_{0},t_{0})}$ as the eigenvector with respect to
$\lambda(x_{0},t_{0})$, i.e.,
$\lambda(x_{0},t_{0})=\tilde{h}^{11}(x_{0},t_{0})$. Assume that
$\\{\tilde{h}^{ij}\\}$ is the inverse matrix of $\\{h_{ij}\\}$. Clearly

$\displaystyle\lambda(x,t)=\max\\{\tilde{h}^{ij}(x,t)\xi_{i}\xi_{j}|g^{ij}(x,t)\xi_{i}\xi_{j}=1\\}$
(4.12)

For the continuity, using this coordinate system we consider the auxiliary
equation $Q=\log\upsilon+A\rho$, where
$\upsilon=\frac{\tilde{h}^{11}}{g^{11}}$. Note that
$\upsilon(x,t)\leq\lambda(x,t)$ from (4.12) and
$\upsilon(x_{0},t_{0})=\lambda(x_{0},t_{0})$. Thus, $Q(x,t)\leq
Q(x_{0},t_{0})$ for all $t\in[0,T_{0})$. Now we can calculate the derivatives
of $\upsilon$ at $(x_{0},t_{0})$ as follows:

$\partial_{t}\upsilon=-(\tilde{h}^{11})^{2}\partial_{t}h_{11}+\tilde{h}^{11}\partial_{t}g_{11}=-\left(\tilde{h}^{11}\right)^{2}\partial_{t}h_{1}{}^{1},$
(4.13) $\displaystyle\nabla_{i}\upsilon=$
$\displaystyle\frac{\partial\tilde{h}_{1}{}^{1}}{\partial
h_{p}{}^{q}}\nabla_{i}h_{p}{}^{q}=-\tilde{h}_{1}{}^{p}\tilde{h}_{q}{}^{1}\nabla_{i}h_{p}{}^{q},\quad\nabla_{i}\upsilon(x_{0},t_{0})=-(\tilde{h}^{11})^{2}\nabla_{i}h_{11}.$
(4.14)

Then

$\displaystyle\nabla_{j}\nabla_{i}\upsilon=$
$\displaystyle\nabla_{j}(-\tilde{h}_{1}{}^{p}\tilde{h}_{q}{}^{1}\nabla_{i}h_{p}{}^{q})=-\nabla_{j}\tilde{h}_{1}{}^{p}\tilde{h}_{q}{}^{1}\nabla_{i}h_{p}{}^{q}-\tilde{h}_{1}{}^{p}\nabla_{j}\tilde{h}_{q}{}^{1}\nabla_{i}h_{p}{}^{q}-\tilde{h}_{1}{}^{p}\tilde{h}_{q}{}^{1}\nabla_{j}\nabla_{i}h_{p}{}^{q}$
$\displaystyle=$
$\displaystyle\tilde{h}_{1}{}^{r}\tilde{h}_{l}{}^{p}\tilde{h}_{q}{}^{1}\nabla_{j}h_{r}{}^{l}\nabla_{i}h_{p}{}^{q}+\tilde{h}_{1}{}^{p}\tilde{h}_{r}{}^{1}\tilde{h}_{q}{}^{s}\nabla_{j}h_{s}{}^{r}\nabla_{i}h_{p}{}^{q}-\tilde{h}_{1}{}^{p}\tilde{h}_{q}{}^{1}\nabla_{j}\nabla_{i}h_{p}{}^{q}$

and

$\displaystyle\nabla_{j}\nabla_{i}\upsilon(x_{0},t_{0})=$
$\displaystyle-(\tilde{h}^{11})^{2}\nabla_{j}\nabla_{i}h_{11}+2(\tilde{h}^{11})^{2}\tilde{h}^{pp}\nabla_{i}h_{1p}\nabla_{j}h_{1p}.$
(4.15)

Here we denote by $\\{\tilde{h}_{i}{}^{j}\\}$ the inverse matrix of
$\\{h_{i}{}^{j}\\}$, i.e.,
$\\{\tilde{h}_{i}{}^{j}\\}=\\{\tilde{h}^{ik}g_{kj}\\}$. Then We calculate the
first term in (2.8)

$\displaystyle\nabla_{j}\nabla_{i}\Theta=$
$\displaystyle\phi^{\alpha}f\nabla_{j}\nabla_{i}K+K\nabla_{j}\nabla_{i}(\phi^{\alpha}f)+\nabla_{i}(\phi^{\alpha}f)\nabla_{j}K+\nabla_{j}(\phi^{\alpha}f)\nabla_{i}K$
(4.16) $\displaystyle=$
$\displaystyle\phi^{\alpha}f\dot{K}^{pq}h_{pqij}+\phi^{\alpha}f\ddot{K}^{pq,rs}h_{pqi}h_{rsj}+fK\nabla_{j}\nabla_{i}(\phi^{\alpha})+K\phi^{\alpha}\nabla_{j}\nabla_{i}f+K\nabla_{i}\phi^{\alpha}\nabla_{j}f$
$\displaystyle+K\nabla_{j}\phi^{\alpha}\nabla_{i}f+f\nabla_{i}(\phi^{\alpha})\nabla_{j}K+f\nabla_{j}(\phi^{\alpha})\nabla_{i}K+\phi^{\alpha}\nabla_{i}f\nabla_{j}K+\phi^{\alpha}\nabla_{j}f\nabla_{i}K.$

Due to the Codazzi equation and Ricci identity, we have

$\displaystyle\dot{K}^{pq}h_{pqij}=$
$\displaystyle\dot{K}^{pq}h_{piqj}=\dot{K}^{pq}\left(h_{pijq}+h_{lp}R_{liqj}+h_{li}R_{lpqj}\right)$
(4.17) $\displaystyle=$
$\displaystyle\dot{K}^{pq}(h_{ijpq}+h_{lp}h_{lq}h_{ij}-h_{lp}h_{lj}h_{iq}+h_{li}h_{lq}h_{pj}-h_{li}h_{lj}h_{pq}$
$\displaystyle-
h_{pq}\delta_{ij}+h_{jp}\delta_{iq}-h_{qi}\delta_{pj}+h_{ij}\delta_{pq})$
$\displaystyle=$
$\displaystyle\dot{K}^{pq}h_{ijpq}+KHh_{ij}-nKh_{il}h_{lj}-nK\delta_{ij}+\dot{K}^{pp}h_{ij}.$

It’s direct to calculate

$\displaystyle\ddot{K}^{pq,rs}h_{pqi}h_{rsj}=$
$\displaystyle\frac{\partial(K\tilde{h}^{pq})}{\partial
h_{rs}}h_{pqi}h_{rsj}=K\tilde{h}^{pq}\tilde{h}^{rs}h_{pqi}h_{rsj}-K\tilde{h}^{pr}\tilde{h}^{qs}h_{pqi}h_{rsj}.$
(4.18)

Hence by (4.16) at $(x_{0},t_{0})$,

$\displaystyle\nabla_{j}\nabla_{i}\Theta$ (4.19) $\displaystyle=$
$\displaystyle\phi^{\alpha}f\dot{K}^{pq}h_{ijpq}+\phi^{\alpha}fKHh_{ij}-n\phi^{\alpha}fKh_{il}h_{lj}-n\phi^{\alpha}fK\delta{ij}+\phi^{\alpha}f\dot{K}^{pp}h_{ij}+\phi^{\alpha}f\frac{\nabla_{i}K\nabla_{j}K}{K}$
$\displaystyle-\phi^{\alpha}fK\tilde{h}^{pp}\tilde{h}^{qq}h_{pqi}h_{pqj}+\alpha(\alpha-1)\phi^{\alpha-2}\phi^{\prime
2}fK\nabla_{i}\rho\nabla_{j}\rho+\alpha\phi^{\alpha-1}\phi^{\prime}fK\nabla_{j}\nabla_{i}\rho$
$\displaystyle+\phi^{\alpha}K\nabla_{j}\nabla_{i}f+\alpha\phi^{\alpha-1}\phi^{\prime}K\nabla_{i}\rho\nabla_{j}f+\alpha\phi^{\alpha-1}\phi^{\prime}K\nabla_{j}\rho\nabla_{i}f$
$\displaystyle+\alpha\phi^{\alpha-1}\phi^{\prime}f\nabla_{i}\rho\nabla_{j}K+\alpha\phi^{\alpha-1}\phi^{\prime}f\nabla_{j}\rho\nabla_{i}K+\phi^{\alpha}\nabla_{i}f\nabla_{j}K+\phi^{\alpha}\nabla_{j}f\nabla_{i}K.$

Direct computation gives

$\displaystyle\nabla_{j}\nabla_{i}Q=$
$\displaystyle\frac{\nabla_{j}\nabla_{i}\upsilon}{\upsilon}-\frac{\nabla_{i}\upsilon\nabla_{j}\upsilon}{\upsilon^{2}}+A\nabla_{j}\nabla_{i}\rho$
(4.20) $\displaystyle=$
$\displaystyle-\tilde{h}^{11}\nabla_{j}\nabla_{i}h_{11}+2\tilde{h}^{11}\tilde{h}^{pp}\nabla_{i}h_{1p}\nabla_{j}h_{1p}-(\tilde{h}^{11})^{2}\nabla_{i}h_{11}\nabla_{j}h_{11}+A\nabla_{j}\nabla_{i}\rho.$

Recall (2.8) and (4.3). By (4.13) and (4.19), we obtain

$\displaystyle\partial_{t}Q=$
$\displaystyle\frac{\partial_{t}\tilde{h}^{11}}{\tilde{h}^{11}}+A\partial_{t}\rho$
(4.21) $\displaystyle=$
$\displaystyle-\tilde{h}^{11}\left(\nabla_{1}\nabla^{1}\Theta+\Theta
h_{11}^{2}-\phi^{\prime}\tilde{\eta}(t)h_{11}+(\tilde{\eta}(t)u-\Theta)\right)-A\frac{\Theta}{w}+A\tilde{\eta}(t)\phi$
$\displaystyle=$
$\displaystyle-\tilde{h}^{11}[\phi^{\alpha}f\dot{K}^{pq}h_{11pq}+\phi^{\alpha}fKHh_{11}-n\phi^{\alpha}fK(h_{11})^{2}-n\phi^{\alpha}fK+\phi^{\alpha}f\dot{K}^{pp}h_{11}$
$\displaystyle+\phi^{\alpha}f\frac{\nabla_{1}K\nabla_{1}K}{K}-\phi^{\alpha}fK\tilde{h}^{pp}\tilde{h}^{qq}(h_{pq1})^{2}+\alpha(\alpha-1)\phi^{\alpha-2}\phi^{\prime
2}(\nabla_{1}\rho)^{2}fK+\alpha\phi^{\alpha-1}\phi^{\prime}fK\nabla_{1}\nabla_{1}\rho$
$\displaystyle+\phi^{\alpha}K\nabla_{1}\nabla_{1}f+2\alpha\phi^{\alpha-1}\phi^{\prime}K\nabla_{1}\rho\nabla_{1}f+2\alpha\phi^{\alpha-1}\phi^{\prime}f\nabla_{1}\rho\nabla_{1}K+2\phi^{\alpha}\nabla_{1}f\nabla_{1}K]$
$\displaystyle-\Theta
h_{11}+\phi^{\prime}\tilde{\eta}(t)-(\tilde{\eta}(t)u-\Theta)\tilde{h}^{11}-A\frac{\Theta}{w}+A\tilde{\eta}(t)\phi.$

Substituting (4.20) into (4.21), we get

$\displaystyle\partial_{t}Q=$
$\displaystyle\phi^{\alpha}f\dot{K}^{ij}\nabla_{j}\nabla_{i}Q-2\phi^{\alpha}fK\tilde{h}^{11}\tilde{h}^{ij}\tilde{h}^{pp}\nabla_{i}h_{1p}\nabla_{j}h_{1p}+\phi^{\alpha}fK\tilde{h}^{ij}(\tilde{h}^{11})^{2}\nabla_{i}h_{11}\nabla_{j}h_{11}$
(4.22)
$\displaystyle-A\phi^{\alpha}f\dot{K}^{ij}\nabla_{j}\nabla_{i}\rho-\phi^{\alpha}fKH+n\phi^{\alpha}fKh_{11}+n\phi^{\alpha}fK\tilde{h}^{11}-\phi^{\alpha}fK\tilde{h}^{pp}-\phi^{\alpha}f\frac{(\nabla_{1}K)^{2}}{K}\tilde{h}^{11}$
$\displaystyle+\phi^{\alpha}fK\tilde{h}^{11}\tilde{h}^{pp}\tilde{h}^{qq}(h_{pq1})^{2}-\alpha(\alpha-1)\phi^{\alpha-2}\phi^{\prime
2}(\nabla_{1}\rho)^{2}fK\tilde{h}^{11}-\alpha\phi^{\alpha-1}\phi^{\prime}fK\tilde{h}^{11}\nabla_{1}\nabla_{1}\rho$
$\displaystyle-\phi^{\alpha}K\tilde{h}^{11}\nabla_{1}\nabla_{1}f-2\alpha\phi^{\alpha-1}\phi^{\prime}K\tilde{h}^{11}\nabla_{1}\rho\nabla_{1}f-2\alpha\phi^{\alpha-1}\phi^{\prime}f\tilde{h}^{11}\nabla_{1}\rho\nabla_{1}K$
$\displaystyle-2\phi^{\alpha}\tilde{h}^{11}\nabla_{1}f\nabla_{1}K-\Theta
h_{11}+\phi^{\prime}\tilde{\eta}(t)-(\tilde{\eta}(t)u-\Theta)\tilde{h}^{11}-A\frac{\Theta}{w}+A\tilde{\eta}(t)\phi.$

Dividing (4.22) by $\Theta$ on both sides, we obtain at $(x_{0},t_{0})$

$\displaystyle\frac{\partial_{t}Q}{\Theta}\leq$
$\displaystyle\tilde{h}^{ij}\nabla_{j}\nabla_{i}Q-A\tilde{h}^{ij}\nabla_{j}\nabla_{i}\rho+nh_{11}+n\tilde{h}^{11}-(\nabla_{1}\log
K)^{2}\tilde{h}^{11}-\alpha(\alpha-1)(\frac{\phi^{\prime}}{\phi})^{2}(\nabla_{1}\rho)^{2}\tilde{h}^{11}$
(4.23)
$\displaystyle-\alpha\frac{\phi^{\prime}}{\phi}\tilde{h}^{11}\nabla_{1}\nabla_{1}\rho-\frac{\nabla_{1}\nabla_{1}f}{f}\tilde{h}^{11}-2\alpha\frac{\phi^{\prime}}{\phi}\tilde{h}^{11}\nabla_{1}\rho\nabla_{1}\log
f-2\alpha\frac{\phi^{\prime}}{\phi}\tilde{h}^{11}\nabla_{1}\rho\nabla_{1}\log
K$ $\displaystyle-2\tilde{h}^{11}\nabla_{1}\log f\nabla_{1}\log
K+\frac{\phi^{\prime}}{\Theta}\tilde{\eta}(t)+\tilde{h}^{11}+A\frac{\tilde{\eta}(t)\phi}{\Theta},$

where we wipe off some nonpositive terms. Recall (2.4), we have at
$(x_{0},t_{0})$

$\displaystyle|\nabla\rho|^{2}=\frac{\sum_{i=1}^{n}\langle
X_{i},V\rangle^{2}}{\phi^{2}}=\frac{|V|^{2}}{\phi^{2}}-\frac{\langle\nu,V\rangle^{2}}{\phi^{2}}\leq
1-(\frac{u}{\phi})^{2}<1-c_{1}$ (4.24)

where $c_{1}>0$ depends on the lower bound of $u$ and the upper bound of
$\rho$. Now substituting (2.4) into (4.23) and using the Cauchy-Schwartz
inequality, we obtain at $(x_{0},t_{0})$

$\displaystyle\frac{\partial_{t}Q}{\Theta}\leq$
$\displaystyle\tilde{h}^{ij}\nabla_{j}\nabla_{i}Q-A\tilde{h}^{ii}\left(\frac{\phi^{\prime}}{\phi}\left(1-(\nabla_{i}\rho)^{2}\right)-\frac{uh_{ii}}{\phi}\right)+nh_{11}+(n+1)\tilde{h}^{11}$
(4.25)
$\displaystyle-\alpha(\alpha-1)(\frac{\phi^{\prime}}{\phi})^{2}(\nabla_{1}\rho)^{2}\tilde{h}^{11}-\alpha\frac{\phi^{\prime}}{\phi}\tilde{h}^{11}\left(\frac{\phi^{\prime}}{\phi}\left(1-(\nabla_{1}\rho)^{2}\right)-\frac{uh_{11}}{\phi}\right)-\frac{\nabla_{1}\nabla_{1}f}{f}\tilde{h}^{11}$
$\displaystyle-2\alpha\frac{\phi^{\prime}}{\phi}\tilde{h}^{11}\nabla_{1}\rho\nabla_{1}\log
f+2\alpha^{2}(\frac{\phi^{\prime}}{\phi})^{2}(\nabla_{1}\rho)^{2}\tilde{h}^{11}+2(\nabla_{1}\log
f)^{2}\tilde{h}^{11}+\frac{\phi^{\prime}}{\Theta}\tilde{\eta}(t)+A\frac{\tilde{\eta}(t)\phi}{\Theta}.$

Using the maximum principle, by (4.24) at $(x_{0},t_{0})$, we have

$\displaystyle\frac{\partial_{t}Q}{\Theta}\leq$ $\displaystyle-
Ac_{1}\tilde{h}^{11}\frac{\phi^{\prime}}{\phi}+A(\frac{nu}{\phi}+\frac{\tilde{\eta}(t)\phi}{\Theta})+nh_{11}+(n+1)\tilde{h}^{11}-\alpha(\alpha-1)(\frac{\phi^{\prime}}{\phi})^{2}(\nabla_{1}\rho)^{2}\tilde{h}^{11}$
(4.26)
$\displaystyle-\alpha(\frac{\phi^{\prime}}{\phi})^{2}(1-\rho_{1}^{2})\tilde{h}^{11}+\alpha\frac{u\phi^{\prime}}{\phi^{2}}-\frac{\nabla_{1}\nabla_{1}f}{f}\tilde{h}^{11}-2\alpha\frac{\phi^{\prime}}{\phi}\tilde{h}^{11}\nabla_{1}\rho\nabla_{1}\log
f$
$\displaystyle+2\alpha^{2}(\frac{\phi^{\prime}}{\phi})^{2}(\nabla_{1}\rho)^{2}\tilde{h}^{11}+2(\nabla_{1}\log
f)^{2}\tilde{h}^{11}+\frac{\phi^{\prime}}{\Theta}\tilde{\eta}(t).$

Write $X\in\mathbb{H}^{n+1}$ as
$X=(\rho,\theta)\in\mathbb{R}^{+}\times\mathbb{S}^{n}$. Then we extend $f$ to
$\left(\mathbb{H}^{n+1},g_{\mathbb{H}^{n+1}}\right)$ as

$\displaystyle f(X)=f(\theta).$ (4.27)

We have by Reilly formula,

$\displaystyle D_{j}D_{i}f=\nabla_{j}\nabla_{i}f+h_{ij}D_{\nu}f.$ (4.28)

where $D$ is the standard Levi-Civita connection with respect to
$g_{\mathbb{H}^{n+1}}$. Then at $(x_{0},t_{0})$, $\nabla_{1}\nabla_{1}f$ can
be further estimated as

$\displaystyle|\nabla_{1}\nabla_{1}f(x_{0},t_{0})|\leq
C\left(\max\rho,\min\rho,\|f\|_{C^{2}}(\mathbb{S}^{n})\right)(1+h_{11}(x_{0},t_{0})).$
(4.29)

By (4.7) and (4.28), We obtain at $(x_{0},t_{0})$

$\displaystyle
0\leq\frac{\partial_{t}Q}{\Theta}\leq-\left(\frac{Ac_{1}}{2}-c_{2}\right)\tilde{h}^{11}-A\left(\frac{c_{1}\tilde{h}^{11}}{2}-c_{3}\right)+c_{4}+c_{5}\frac{1}{\tilde{h}^{11}}$
(4.30)

for some positive constants $c_{1}$, $c_{2}$, $c_{3}$, $c_{4}$, $c_{5}$
depends on the uniform upper and lower bounds of the $\rho$, $\tilde{\eta}$,
$u$ from Corollary 3.1, $K$ from Lemma 4.1-4.2 and $\alpha$, $n$, $\inf f$,
$\|f\|_{C^{2}}(\mathbb{S}^{n})$. We deduce from (4.30) that if we choose
$A=\frac{2c_{2}}{c_{1}}$, $\lambda(x_{0},t_{0})$ can’t be too large, i.e.,
$\lambda(x_{0},t_{0})$ has a uniform upper bound $C(c_{i},i=1,\cdots,5)$
independent of time $T_{0}$. Hence we have the uniform upper bound of the
principal raddi of $M_{t}$, which means that the principal curvatures are
bounded from below by some positive constant $c_{6}$. Meanwhile, by Lemma 4.2,
one can get

$\displaystyle C\geq K\geq c_{6}^{n-1}\kappa_{\max}$

for some constant $C$ from Lemma 4.2. Hence, the principal curvatures are
bounded from above. This completes the proof of Lemma 4.3. ∎

Now we have obtained the a priori estimates of the flow (1.3). By Lemma 3.1,
Lemma 3.2 and Lemma 4.1, these flows have short time existence. Using the
$C^{2}$ estimate given in Lemma 4.3, due to [And04, Theorem 6], we obtain the
$C^{2,\lambda}$ estimate of the scalar equation (2.17). Then the standard
regularity theory implies the estimates for higher order derivatives. Hence,
we obtain the long time existence and regularity for these flows.

######  Theorem 4.1.

Let $M_{0}$ be be as in Theorem 1.1. Assume that $f$ is a smooth positive
function on $\mathbb{S}^{n}$. If $\alpha>n+1$ or $\alpha=n+1$ and $f$
satisfies $f<1$, then the smooth uniformly convex solution to the flow (1.3)
exists for all time $t\in[0,+\infty)$ and there is a constant
$C_{m,\lambda}>0$ depending on $M_{0}$, $\alpha$, $n$, $m$, $\lambda$, and $f$
such that

$\displaystyle\|\rho\|_{C^{m,\lambda}_{(\theta,t)}(\mathbb{S}^{n}\times[0,+\infty))}\leq
C_{m,\lambda}.$

######  Remark 4.1.

If the Gauss curvature $K$ in flow (1.3) is replaced by the $k$-th mean
curvature $\sigma_{k}$ of the hypersurface, using the similar argument we can
obtain the same long time existence results as in Theorem 4.1.

In other words, there is a subsequence of time $\\{t_{i}\\}$, such that
$\\{M_{t_{i}}\\}$ converge to a smooth positive uniformly convex hypersurface
$M_{\infty}$ in $C^{\infty}$ topology. Next we shall see $M_{\infty}$ is a
solution to (1.2).

##  5\. Proof of Theorem 1.1

In this section, we consider the hyperbolic space as the hyperboloid in
Lorentz space $\mathbb{R}^{n+1,1}$, where

$\displaystyle\mathbb{H}^{n+1}=\\{(x_{1},\cdots,x_{n+1},x_{n+2})\in\mathbb{R}^{n+1,1}|\sum_{i=1}^{n+1}x_{i}^{2}-x_{n+2}^{2}=-1,\
x_{n+2}>0\\}.$

Let $p=(0,\cdots,0,1)$ and $L_{p}$ be the tangent plane of $\mathbb{H}^{n+1}$
at $p$. Denote the projection from $\mathbb{H}^{n+1}$ to $L_{p}$ as

$\displaystyle\pi_{p}:\mathbb{H}^{n+1}$ $\displaystyle\rightarrow L_{p}$
$\displaystyle z=(x_{1},\cdots,x_{n+1},x_{n+2})$
$\displaystyle\mapsto\frac{z}{-\langle
z,p\rangle}=(\frac{x_{1}}{x_{n+2}},\cdots,\frac{x_{n+1}}{x_{n+2}},1).$

Since $\sum_{i=1}^{n+1}\left(\frac{x_{i}}{x_{n+2}}\right)^{2}<1$ on
$\pi_{p}(\mathbb{H}^{n+1})$, $\pi_{p}(\mathbb{H}^{n+1})$ is contained in the
unit ball $B^{n+1}_{p}(1)$ centered at $p$. Recall that $\rho(q)$ is the
geodesic distance between $q$ and $p$ in $\mathbb{H}^{n+1}$. If we regard $r$
as the Euclidean distance from $\pi(q)$ to $p$ in $B^{n+1}_{p}(1)$, we have
the following relation

$\displaystyle r(\theta)=\tanh\rho(\theta).$ (5.1)

Direct computation gives

$\displaystyle\frac{1}{(\cosh\rho)^{2}}=1-r^{2},$ (5.2)
$\displaystyle\sinh\rho=\frac{r}{\sqrt{1-r^{2}}},$ (5.3)

$\displaystyle|\overline{\nabla}\rho|^{2}=\frac{|\overline{\nabla}r|^{2}}{(1-r^{2})^{2}},$
(5.4) $\displaystyle
w=\sqrt{1+\frac{|\overline{\nabla}\rho|^{2}}{\phi^{2}}}=\frac{r}{\hat{u}}\sqrt{\frac{1-\hat{u}^{2}}{1-r^{2}}},$
(5.5) $\displaystyle u=\frac{\hat{u}}{\sqrt{1-\hat{u}^{2}}}.$ (5.6)

Here we can see $u>0$ as long as $1>\hat{u}>0$, which means that
$\pi_{p}(M_{t})$ remains star-shaped as long as $M_{t}$ does. From (5.1), we
have the scalar equation of the radial graph of $\pi_{p}(M_{t})$ under the
flow (2.5)($\tilde{\eta}\equiv 1$),

$\displaystyle\partial_{t}r(\theta,t)=\partial_{\rho}(\tanh\rho)\partial_{t}\rho=\frac{-w\phi(\rho)^{\alpha}f(\theta)K}{(\cosh\rho)^{2}}+\frac{\phi}{(\cosh\rho)^{2}}$
(5.7)

where we denote by $(r(\theta),\theta)$ the radial graph of $\pi_{p}(M_{t})$.
Similar to (2.13)-(2.15), we have in $\hat{M}_{t}=\pi(M_{t})$,

$\displaystyle\hat{\nu}=\frac{r}{\sqrt{r^{2}+|\overline{\nabla}r|^{2}}}(\partial_{r}-\frac{\overline{\nabla}r}{r^{2}}),$
(5.8)
$\displaystyle\hat{u}=\frac{r^{2}}{\sqrt{r^{2}+|\overline{\nabla}r|^{2}}},$
(5.9) $\displaystyle\hat{g}_{ij}=r_{i}r_{j}+r^{2}\delta_{ij}$ (5.10)

and

$\displaystyle\hat{h}_{ij}=\frac{-rr_{ij}+2r_{i}r_{j}+r^{2}\delta_{ij}}{\sqrt{r^{2}+|\overline{\nabla}r|^{2}}}.$
(5.11)

Therefore,

$\displaystyle\hat{K}=\frac{\det{\hat{h}_{ij}}}{\det{\hat{g}_{ij}}}=\frac{\det(-rr_{ij}+2r_{i}r_{j}+r^{2}\delta_{ij})}{(r^{2}+|\overline{\nabla}r|^{2})^{\frac{n+2}{2}}r^{2n-2}}.$
(5.12)

Recall (2.14) and (2.15). Substituting (5.1) in (5.12), we have(see [CH21])

$\displaystyle
K=\hat{K}\left(\frac{(r^{2}+|\overline{\nabla}r|^{2})}{r^{2}+\frac{|\overline{\nabla}r|^{2}}{1-r^{2}}}\right)^{\frac{n+2}{2}}=\hat{K}\left(\frac{1-r^{2}}{1-\hat{u}^{2}}\right)^{\frac{n+2}{2}}.$
(5.13)

Now we calculate the scalar parabolic PDE of the support function $\hat{u}$ of
$\pi_{p}(M_{t})$. Substituting (5.2), (5.3), (5.5) and (5.13) into (5.7), we
have

$\displaystyle\partial_{t}\hat{u}(x,t)=\frac{\hat{u}}{r}\partial_{t}r(\theta,t)$
$\displaystyle=(-\phi(\rho)^{\alpha}f(\theta)Kw+\phi)(1-r^{2})\frac{\hat{u}}{r}$
(5.14)
$\displaystyle=-r^{\alpha}f(\theta)\hat{K}(1-r^{2})^{\frac{n+3-\alpha}{2}}({1-\hat{u}^{2}})^{-\frac{n+1}{2}}+\hat{u}\sqrt{1-r^{2}}.$

Denote by $\psi(r)=r^{\alpha}(1-r^{2})^{\frac{n+2-\alpha}{2}}$ and
$\varphi(\hat{u})=\hat{u}^{-1}({1-\hat{u}^{2}})^{-\frac{n+1}{2}}$. (5.14)
becomes

$\displaystyle\partial_{t}\hat{u}=-\psi(r)\sqrt{1-r^{2}}f(\theta)\varphi(\hat{u})\hat{u}\hat{K}+\hat{u}\sqrt{1-r^{2}}.$
(5.15)

Let $\Psi=\displaystyle\int_{a}^{r}\psi^{-1}(s)s^{n}\mathrm{d}s$,
$\Omega=\displaystyle\int_{a}^{\hat{u}}\varphi(s)\mathrm{d}s$ and

$Q(t)=\displaystyle\int_{\mathbb{S}^{n}}\Psi
f^{-1}\mathrm{d}\theta_{\mathbb{S}^{n}}-\displaystyle\int_{\mathbb{S}^{n}}\Omega\mathrm{d}\sigma_{\mathbb{S}^{n}}.$
(5.16)

where $0<a<1$ some constant to be chosen later for convenience.
$a=\frac{1}{2}\min\limits_{\hat{M_{0}}}r=\frac{1}{2}\min\limits_{\hat{M_{0}}}u$.
We have the following properties of $Q$ along (5.14) (referring to [LL20,
LSW20a]).

######  Lemma 5.1.

Along (5.14), $Q(t)$ is non-decreasing and the equality holds if and only if
$\pi_{p}(M_{t})$ satisfies the following equation

$r^{\alpha}(1-r^{2})^{\frac{n+2-\alpha}{2}}f\hat{K}=\hat{u}(1-\hat{u}^{2})^{\frac{n+1}{2}}.$
(5.17)

###### Proof.

$\displaystyle Q^{\prime}(t)=$
$\displaystyle\partial_{t}\int_{\mathbb{S}^{n}}\Psi
f^{-1}\mathrm{d}\theta_{\mathbb{S}^{n}}-\partial_{t}\int_{\mathbb{S}^{n}}\Omega\mathrm{d}\sigma_{\mathbb{S}^{n}}$
(5.18) $\displaystyle=$
$\displaystyle\int_{\mathbb{S}^{n}}\psi^{-1}(r)r^{n}\partial_{t}rf^{-1}\mathrm{d}\theta_{\mathbb{S}^{n}}-\int_{\mathbb{S}^{n}}\varphi(\hat{u})\partial_{t}\hat{u}\mathrm{d}\sigma_{\mathbb{S}^{n}}.$

Since
$\frac{\partial_{t}r(\theta,t)}{r}=\frac{\partial_{t}\hat{u}(x,t)}{\hat{u}}$
and
$r^{n+1}\mathrm{d}\theta_{\mathbb{S}^{n}}=\frac{\hat{u}}{\hat{K}}\mathrm{d}\sigma_{\mathbb{S}^{n}}$,
(5.18) becomes

$\displaystyle Q^{\prime}(t)=$
$\displaystyle\int_{\mathbb{S}^{n}}\left(\psi^{-1}(r)f^{-1}(\theta)\hat{K}^{-1}-\varphi(\hat{u})\right)\partial_{t}\hat{u}\mathrm{d}\sigma_{\mathbb{S}^{n}}$
(5.19) $\displaystyle=$
$\displaystyle\int_{\mathbb{S}^{n}}\left(\psi^{-1}(r)f^{-1}(\theta)\hat{K}^{-1}-\varphi(\hat{u})\right)^{2}\psi(r)\sqrt{1-r^{2}}f(\theta)\hat{u}\hat{K}\mathrm{d}\sigma_{\mathbb{S}^{n}}$
$\displaystyle\geq$ $\displaystyle 0.$

Here the equality holds if and only if
$\varphi(r)\psi(\hat{u})f(\theta)\hat{K}\equiv 1$. ∎

By Lemma 3.1, we obtain the uniform upper and lower bounds of $\rho$ in the
hyperbolic space. Thus from (5.1), there exists $0<c_{1}<c_{2}<1$ independent
of time, such that $c_{1}<r<c_{2}$ and $c_{1}<\hat{u}<c_{2}$. From (5.16),
$Q(t)$ is uniformly bounded independent of time, i.e., there exists a constant
$C>0$, such that $|Q(t)|\leq C$. Since

$\displaystyle
Q(t)-Q(0)=\int_{0}^{t}\int_{\mathbb{S}^{n}}\left(\psi^{-1}(r)f^{-1}(\theta)\hat{K}^{-1}-\varphi(\hat{u})\right)^{2}\psi(r)\sqrt{1-r^{2}}f(\theta)\hat{u}\hat{K}\mathrm{d}\sigma_{\mathbb{S}^{n}}dt,$
(5.20)

we obtain

$\displaystyle\displaystyle\int_{0}^{\infty}\displaystyle\int_{\mathbb{S}^{n}}\left(\psi^{-1}(r)f^{-1}(\theta)\hat{K}^{-1}-\varphi(\hat{u})\right)^{2}\psi(r)\sqrt{1-r^{2}}f(\theta)\hat{u}\hat{K}\mathrm{d}\sigma_{\mathbb{S}^{n}}dt<\infty.$
(5.21)

By Theorem 4.1, there exists a subsequence $\\{t_{i}\\}$, such that
$M_{t_{i}}$ converge smoothly to $M_{\infty}$ and

$\displaystyle\displaystyle\int_{\mathbb{S}^{n}}\left(\psi^{-1}(r)f^{-1}(\theta)\hat{K}^{-1}-\varphi(\hat{u})\right)^{2}\mathrm{d}\sigma_{\mathbb{S}^{n}}\rightarrow
0$

as $t_{i}\rightarrow\infty$, since $\hat{K}$ is uniformly bounded from below
by (5.13) and Lemma 4.1. Thus $\hat{M}_{\infty}$ satisfies

$\displaystyle\psi(r)f(\theta)\hat{K}=\varphi(\hat{u})^{-1}.$ (5.22)

Combining with (5.1), (5.2), (5.3) etc., we obtain $M_{\infty}$ satisfies
(1.2).

Now we show the uniqueness of the solution of (1.2) when $\alpha\geq n+1$.
First we assume that there exists two solutions $\rho_{1}$ and $\rho_{2}$. Let
$\gamma(\rho)=\log(1-\frac{2}{e^{\rho}+1})$ and $G=\gamma_{1}-\gamma_{2}$,
then $\overline{\nabla}\gamma=\frac{\overline{\nabla}\rho}{\phi}$. Assume $G$
attains its maximal point at $\theta_{0}$ and $G(\theta_{0})>0$, which implies
$\rho_{1}(\theta_{0})>\rho_{2}(\theta_{0})$. At $\theta_{0}$, we have

$\displaystyle\overline{\nabla}\gamma_{1}=\overline{\nabla}\gamma_{2}$ (5.23)

and $\overline{\nabla}^{2}_{ij}G\leq 0$, i.e.,

$\displaystyle\overline{\nabla}^{2}_{ij}\gamma_{1}\leq\overline{\nabla}^{2}_{ij}\gamma_{2}.$
(5.24)

Substituting $\gamma$ into (2.14) and (2.16), we have

$\displaystyle
g_{ij}=\phi^{2}\left(\delta^{ij}+\gamma_{i}\gamma_{j}\right),\quad
g^{kj}=\frac{1}{\phi^{2}}\left(\delta^{kj}-\frac{\gamma_{k}\gamma_{j}}{1+|\overline{\nabla}\gamma|^{2}}\right)$
(5.25)

and

$\displaystyle
h_{ik}=\frac{\phi}{\sqrt{1+|\overline{\nabla}\gamma|^{2}}}\left(-\gamma_{ik}+\phi^{\prime}\gamma_{i}\gamma_{k}+\phi^{\prime}\delta_{ik}\right).$
(5.26)

Plugging (5.25) and (5.26) in (1.2) we have

$\displaystyle\sigma_{n}(h_{i}{}^{j})=\frac{f}{\phi(\rho)^{\alpha-1}\sqrt{1+|\overline{\nabla}\gamma|^{2}}},$
(5.27)

where

$\displaystyle
h_{i}{}^{j}=\frac{1}{\phi\sqrt{1+|\overline{\nabla}\gamma|^{2}}}\left(-\gamma_{ik}+\phi^{\prime}\gamma_{i}\gamma_{k}+\phi^{\prime}\delta_{ik}\right)\left(\delta^{kj}-\frac{\gamma_{k}\gamma_{j}}{1+|\overline{\nabla}\gamma|^{2}}\right).$
(5.28)

Then

$\displaystyle\sigma_{n}\left(\frac{1}{\sqrt{1+|\overline{\nabla}\gamma|^{2}}}\left(-\gamma_{ik}+\phi^{\prime}\gamma_{i}\gamma_{k}+\phi^{\prime}\delta_{ik}\right)\left(\delta^{kj}-\frac{\gamma_{k}\gamma_{j}}{1+|\overline{\nabla}\gamma|^{2}}\right)\right)=\frac{f}{\phi(\rho)^{\alpha-n-1}\sqrt{1+|\overline{\nabla}\gamma|^{2}}}.$
(5.29)

Using (5.23), we have at $\theta$

$\displaystyle\phi(\rho_{1})^{\alpha-n-1}\sigma_{n}\left(-(\gamma_{1}){}_{ik}+\phi(\rho_{1})^{\prime}(\gamma_{1}){}_{i}(\gamma_{1}){}_{k}+\phi(\rho_{1})^{\prime}\delta_{ik}\right)$
$\displaystyle=$
$\displaystyle\phi(\rho_{2})^{\alpha-n-1}\sigma_{n}\left(-(\gamma_{2}){}_{ik}+\phi(\rho_{2})^{\prime}(\gamma_{2}){}_{i}(\gamma_{2}){}_{k}+\phi(\rho_{2})^{\prime}\delta_{ik}\right).$
(5.30)

Since $\rho_{1}(\theta)>\rho_{2}(\theta)$ and both of the solutions are
uniformly convex, by (5.23) and (5.24)

$\displaystyle-(\gamma_{1}){}_{ik}+\phi^{\prime}(\rho_{1})(\gamma_{1}){}_{i}(\gamma_{1}){}_{k}+\phi^{\prime}(\rho_{1})\delta_{ik}>-(\gamma_{2}){}_{ik}+\phi(\rho_{2})^{\prime}(\gamma_{2}){}_{i}(\gamma_{2}){}_{k}+\phi(\rho_{2})^{\prime}\delta_{ik}>0.$
(5.31)

When $\alpha\geq n+1$, we have

$\displaystyle\phi(\rho_{1})^{\alpha-n-1}\sigma_{n}\left(-(\gamma_{1}){}_{ik}+\phi(\rho_{1})^{\prime}(\gamma_{1}){}_{i}(\gamma_{1}){}_{k}+\phi(\rho_{1})^{\prime}\delta_{ik}\right)$
$\displaystyle>$
$\displaystyle\phi(\rho_{2})^{\alpha-n-1}\sigma_{n}\left(-(\gamma_{2}){}_{ik}+\phi(\rho_{2})^{\prime}(\gamma_{2}){}_{i}(\gamma_{2}){}_{k}+\phi(\rho_{2})^{\prime}\delta_{ik}\right)$

which is contrary to (5). So $G\leq 0$, which implies
$\gamma_{1}\equiv\gamma_{2}$ and $\rho_{1}\equiv\rho_{2}$. Now we complete the
proof of Theorem 1.1.

##  6\. Proof of Theorem 1.2

In this section, we consider the flow (1.3) for $\alpha=n+1$ and the
corresponding even Alexandrov problem in $\mathbb{H}^{n+1}$. Throughout this
section, if not specified, we regard $c_{i}$, $C_{i}$ for $i\in\mathbb{N}$ as
some positive constants. By (2.17), when $\alpha=n+1$ the scalar equation of
the radial function $\rho$ becomes

$\left\\{\begin{aligned} \frac{\partial}{\partial
t}\rho(\theta,t)=&-\left(\phi(\rho)\right)^{n+1}f(\theta)wK+\phi(\theta,t),\quad\text{for
}(\theta,t)\in\mathbb{S}^{n}\times[0,+\infty),\\\
\rho(\cdot,0)=&\rho_{0}(\cdot).\end{aligned}\right.$ (6.1)

By the projection $\pi_{p}$ in Section 5 and the relation between $M_{t}$ and
$\pi_{p}(M_{t})$, (5.1)-(5.6) and (5.13), we obtain the scalar parabolic PDE
of the radial function $r$ of $\pi_{p}(M_{t})$ along the flow (1.3)

$\displaystyle\partial_{t}r=-r^{n+2}(1-r^{2})\hat{u}^{-1}(1-\hat{u}^{2})^{-\frac{n+1}{2}}f\hat{K}+r\sqrt{1-r^{2}}.$
(6.2)

Then we have the following results.

######  Lemma 6.1.

Let $\rho$ be a smooth, positive, uniformly convex and origin-symmetric
solution to (6.1) on $\mathbb{S}^{n}\times[0,T)$. If $f$ is a smooth positive
even function on $\mathbb{S}^{n}$ satisfying
$\int_{\mathbb{S}^{n}}f^{-1}\mathrm{d}\theta_{\mathbb{S}^{n}}>|S^{n}|$, then
there exists a positive constant $C$ depending on $n$, $\max f$, $\min f$ and
the initial hypersurface, such that

$\displaystyle\frac{1}{C}\leq\rho\leq C,\quad\forall t\in[0,T).$ (6.3)

###### Proof.

The upper bound of $\rho$ is obtained directly from the proof of Lemma 3.1,
i.e., there exists a positive constant $c_{1}$, such that $\rho\leq c_{1}$.
Combining with (5.3), we obtain that the radial function $r$ of
$\pi_{p}(M_{t})$ is away from $1$, i.e., there exists a positive constant
$c_{2}$, such that $\hat{u}\leq r\leq c_{2}<1$. Under the flow (6.2), we have
the following monotone non-decreasing function by Lemma 5.1

$\displaystyle
Q(t)=\displaystyle\int_{\mathbb{S}^{n}}\int_{a}^{r}s^{-1}(1-s^{2})^{-\frac{1}{2}}f^{-1}\mathrm{d}s\mathrm{d}\theta_{\mathbb{S}^{n}}-\displaystyle\int_{\mathbb{S}^{n}}\int_{a}^{\hat{u}}s^{-1}(1-s^{2})^{-\frac{n+1}{2}}\mathrm{d}s\mathrm{d}\sigma_{\mathbb{S}^{n}}.$
(6.4)

where we can choose $a=c_{2}$ without loss of generality. Then there exists a
positive constant $C_{1}$ depending on the initial hypersurface of (1.3), such
that

$\displaystyle-C_{1}\leq Q(t)\leq$ $\displaystyle-
f_{\max}^{-1}\int_{\mathbb{S}^{n}}\int_{r}^{c_{2}}s^{-1}\mathrm{d}s\mathrm{d}\theta_{\mathbb{S}^{n}}+(1-c_{2}^{2})^{-\frac{n+1}{2}}\int_{\mathbb{S}^{n}}\int_{\hat{u}}^{c_{2}}s^{-1}\mathrm{d}s\mathrm{d}\sigma_{\mathbb{S}^{n}}$
(6.5) $\displaystyle=$ $\displaystyle-f_{\max}^{-1}\int_{\mathbb{S}^{n}}(\log
c_{2}-\log
r)\mathrm{d}\theta_{\mathbb{S}^{n}}+(1-c_{2}^{2})^{-\frac{n+1}{2}}\int_{\mathbb{S}^{n}}(\log
c_{2}-\log\hat{u})\mathrm{d}\sigma_{\mathbb{S}^{n}}$ $\displaystyle\leq$
$\displaystyle C_{2}+f_{\max}^{-1}\int_{\mathbb{S}^{n}}\log r\
\mathrm{d}\theta_{\mathbb{S}^{n}}-(1-c_{2}^{2})^{-\frac{n+1}{2}}\int_{\mathbb{S}^{n}}\log\hat{u}\
\mathrm{d}\sigma_{\mathbb{S}^{n}}.$

Assume $r_{\min}(t)$ is attained at $\theta_{0}=(1,\vec{0})$ at any fixed time
$t$. Here we define by $\vec{0}$ an $n$-dimensional zero vector. We
parametrize any point $\theta\in\mathbb{S}^{n}$ as

$\displaystyle\theta=(\cos\theta_{1},\sin\theta_{1}\vec{x}),$ (6.6)

where $0\leq\theta_{1}\leq\pi$, and
$\vec{x}=(x_{2},\cdots,x_{n+1})\in\mathbb{S}^{n-1}$ is an $n$-dimensional unit
vector. We have $\langle\theta,\theta_{0}\rangle=\cos\theta_{1}$. Then
$r(\theta,t)\leq\frac{r_{\min}(t)}{|\cos\theta_{1}|}$ because the flow
hypersurface $M_{t}$ is strictly convex and origin-symmetric. Assume
$\hat{u}_{\max}(t)$ is attained at $v_{0}\in\mathbb{S}^{n}$. We have
$\hat{u}(v,t)\geq\hat{u}_{\max}(t)|\langle v,v_{0}\rangle|$ (referring to
[Sch13, p.44]). Take the direction of $v_{0}$ as $x$-axis and regard $v_{1}$
as the angle with the $x$-axis. We parametrize any point $v$ in
$\mathbb{S}^{n}$ as

$\displaystyle v=\left(\cos v_{1},\sin v_{1}\vec{y}\right),$ (6.7)

where $0\leq v_{1}\leq\pi$, and
$\vec{y}=(y_{2},\cdots,y_{n+1})\in\mathbb{S}^{n-1}$ is an $n$-dimensional unit
vector. We have $\langle v,v_{0}\rangle=\cos v_{1}$. By a direct calculation,
the area measure $\mathrm{d}\sigma_{\mathbb{S}^{n}}$ becomes

$\displaystyle\mathrm{d}\sigma_{\mathbb{S}^{n}}=(\sin
v_{1})^{n-1}\mathrm{d}v_{1}\mathrm{d}\sigma_{\mathbb{S}^{n-1}}.$ (6.8)

Similarly,

$\displaystyle\mathrm{d}\theta_{\mathbb{S}^{n}}=(\sin\theta_{1})^{n-1}\mathrm{d}\theta_{1}\mathrm{d}\theta_{\mathbb{S}^{n-1}}.$
(6.9)

Then

$\displaystyle\int\limits_{\mathbb{S}^{n}}\log\hat{u}\mathrm{d}\sigma_{\mathbb{S}^{n}}\geq$
$\displaystyle\int\limits_{\mathbb{S}^{n}}\log(\hat{u}_{\max}|\cos
v_{1}|)\mathrm{d}\sigma_{\mathbb{S}^{n}}$ (6.10) $\displaystyle=$
$\displaystyle
2\int_{0}^{\frac{\pi}{2}}\int_{\mathbb{S}^{n-1}}\left(\log\hat{u}_{\max}+\log\cos
v_{1}\right)(\sin
v_{1})^{n-1}\mathrm{d}v_{1}\mathrm{d}\sigma_{\mathbb{S}^{n-1}}$
$\displaystyle\geq$
$\displaystyle|S^{n}|\log\hat{u}_{\max}+2|S^{n-1}|\int_{0}^{\frac{\pi}{2}}\log\cos
v_{1}\mathrm{d}v_{1}\geq|S^{n}|\log\hat{u}_{\max}-C_{3}.$

The second term in the last inequality is convergent, since $\log(\cos
v_{1})\geq\log\left(\frac{\pi}{4}-\frac{v_{1}}{2}\right)$ at
$v_{1}\in[0,\frac{\pi}{2}]$. Besides, we have

$\displaystyle\int\limits_{\mathbb{S}^{n}}\log
r\mathrm{d}\theta_{\mathbb{S}^{n}}\leq$
$\displaystyle\int\limits_{\mathbb{S}^{n}}\log\frac{r_{\min}}{|\cos\theta_{1}|}\mathrm{d}\theta_{\mathbb{S}^{n}}$
(6.11) $\displaystyle=$ $\displaystyle
2\int_{0}^{\frac{\pi}{2}}\int_{\mathbb{S}^{n-1}}\left(\log
r_{\min}-\log\cos\theta_{1}\right)(\sin\theta_{1})^{n-1}\mathrm{d}\theta_{1}\mathrm{d}\theta_{\mathbb{S}^{n-1}}$
$\displaystyle\leq$ $\displaystyle|S^{n}|\log
r_{\min}-2|S^{n-1}|\int_{0}^{\frac{\pi}{2}}\log\cos\theta_{1}\mathrm{d}\theta_{1}\leq|S^{n}|\log
r_{\min}+C_{3},$

where the second term in the last inequality is convergent for the same reason
as (6.10). Plugging (6.10) and (6.11) in (6.5), we obtain

$\displaystyle-C_{1}\leq C_{4}+f_{\max}^{-1}|S^{n}|\log
r_{\min}-(1-c_{2}^{2})^{-\frac{n+1}{2}}|S^{n}|\log\hat{u}_{\max}.$ (6.12)

Hence, there exists positive constants $c_{3}$ and $c_{4}$, such that

$\displaystyle r_{\min}\geq c_{3}r_{\max}^{c_{4}},$ (6.13)

where $c_{4}=f_{\max}(1-c_{2}^{2})^{-\frac{n+1}{2}}$.
Dividing the both sides of (6.2) by $r\sqrt{1-r^{2}}f$, we have

$\displaystyle\frac{\partial_{t}r}{r\sqrt{1-r^{2}}f}=-r^{n+1}\sqrt{1-r^{2}}\hat{u}^{-1}(1-\hat{u}^{2})^{-\frac{n+1}{2}}\hat{K}+f^{-1}(\theta).$
(6.14)

Integrating (6.14) over $\mathbb{S}^{n}$, we get

$\displaystyle\int_{\mathbb{S}^{n}}\frac{\partial_{t}r}{r\sqrt{1-r^{2}}f}\mathrm{d}\theta_{\mathbb{S}^{n}}=$
$\displaystyle-\int_{\mathbb{S}^{n}}r^{n+1}\sqrt{1-r^{2}}\hat{u}^{-1}(1-\hat{u}^{2})^{-\frac{n+1}{2}}\hat{K}\mathrm{d}\theta_{\mathbb{S}^{n}}+\int_{\mathbb{S}^{n}}f^{-1}(\theta)\mathrm{d}\theta_{\mathbb{S}^{n}}$
(6.15) $\displaystyle=$
$\displaystyle-\int_{\mathbb{S}^{n}}\sqrt{1-r^{2}}(1-\hat{u}^{2})^{-\frac{n+1}{2}}\mathrm{d}\sigma_{\mathbb{S}^{n}}+\int_{\mathbb{S}^{n}}f^{-1}(\theta)\mathrm{d}\theta_{\mathbb{S}^{n}},$

where we use
$r^{n+1}d\theta_{\mathbb{S}^{n}}=\frac{\hat{u}}{\hat{K}}d\sigma_{\mathbb{S}^{n}}$
in the last equality. Note that the left hand side of (6.15) is

$\displaystyle\int_{\mathbb{S}^{n}}\frac{\partial_{t}r}{r\sqrt{1-r^{2}}f}\mathrm{d}\theta_{\mathbb{S}^{n}}=\partial_{t}\displaystyle\int_{\mathbb{S}^{n}}\int_{c_{2}}^{r}s^{-1}(1-s^{2})^{-\frac{1}{2}}f^{-1}\mathrm{d}s\mathrm{d}\theta_{\mathbb{S}^{n}}.$
(6.16)

Besides, we have

$\displaystyle\int_{\mathbb{S}^{n}}\int_{c_{2}}^{r}s^{-1}(1-s^{2})^{-\frac{1}{2}}f^{-1}\mathrm{d}s\mathrm{d}\theta_{\mathbb{S}^{n}}$
(6.17) $\displaystyle\leq$ $\displaystyle-
f_{\max}^{-1}\int_{\mathbb{S}^{n}}\int_{r}^{c_{2}}s^{-1}\mathrm{d}s\mathrm{d}\theta_{\mathbb{S}^{n}}$
$\displaystyle=$ $\displaystyle-|S^{n}|f_{\max}^{-1}\log
c_{2}+f_{\max}^{-1}\int_{\mathbb{S}^{n}}\log
r\mathrm{d}\theta_{\mathbb{S}^{n}}$ $\displaystyle\leq$ $\displaystyle
C_{5}+|S^{n}|f_{\max}^{-1}\log r_{\max}$

and

$\displaystyle\int_{\mathbb{S}^{n}}\int_{c_{2}}^{r}s^{-1}(1-s^{2})^{-\frac{1}{2}}f^{-1}\mathrm{d}s\mathrm{d}\theta_{\mathbb{S}^{n}}$
(6.18) $\displaystyle\geq$
$\displaystyle-\left(1-c_{2}^{2}\right)^{-\frac{1}{2}}f_{\min}^{-1}\int_{\mathbb{S}^{n}}\int_{r}^{c_{2}}s^{-1}\mathrm{d}s\mathrm{d}\theta_{\mathbb{S}^{n}}$
$\displaystyle=$
$\displaystyle-\left(1-c_{2}^{2}\right)^{-\frac{1}{2}}|S^{n}|f_{\min}^{-1}\log
c_{2}+\left(1-c_{2}^{2}\right)^{-\frac{1}{2}}f_{\min}^{-1}\int_{\mathbb{S}^{n}}\log
r\mathrm{d}\theta_{\mathbb{S}^{n}}$ $\displaystyle\geq$
$\displaystyle\left(1-c_{2}^{2}\right)^{-\frac{1}{2}}|S^{n}|f_{\min}^{-1}\log
r_{\min}.$

Since
$r(\theta(v),t)=\sqrt{\hat{u}(v)^{2}+|\nabla\hat{u}(v)|^{2}}\geq\hat{u}(v,t)$
on $\mathbb{S}^{n}$, we have

$\displaystyle\int_{\mathbb{S}^{n}}\sqrt{1-r^{2}}(1-\hat{u}^{2})^{-\frac{n+1}{2}}\mathrm{d}\sigma_{\mathbb{S}^{n}}\leq\int_{\mathbb{S}^{n}}(1-\hat{u}^{2})^{-\frac{n}{2}}\mathrm{d}\sigma_{\mathbb{S}^{n}}\leq(1-\hat{u}_{\max}^{2})^{-\frac{n}{2}}|S^{n}|.$
(6.19)

When $\hat{u}_{\max}\rightarrow 0$,
$(1-\hat{u}_{\max}^{2})^{-\frac{n}{2}}|S^{n}|\rightarrow|S^{n}|$. By (6.19)
and the condition of $f$, there exists a positive constant $c_{5}$, such that
when $u_{\max}\leq c_{5}$,
$\int_{\mathbb{S}^{n}}\sqrt{1-r^{2}}(1-\hat{u}^{2})^{-\frac{n+1}{2}}\mathrm{d}\sigma_{\mathbb{S}^{n}}\leq\frac{1}{2}(1+\int_{\mathbb{S}^{n}}f^{-1}(\theta)\mathrm{d}\theta_{\mathbb{S}^{n}})<\int_{\mathbb{S}^{n}}f^{-1}(\theta)\mathrm{d}\theta_{\mathbb{S}^{n}}$.
This together with (6.15) and (6.16) implies

$\displaystyle\partial_{t}\int_{\mathbb{S}^{n}}\int_{c_{2}}^{r}s^{-1}(1-s^{2})^{-\frac{1}{2}}f^{-1}\mathrm{d}s\mathrm{d}\theta_{\mathbb{S}^{n}}=-\int_{\mathbb{S}^{n}}\sqrt{1-r^{2}}(1-\hat{u}^{2})^{-\frac{n+1}{2}}\mathrm{d}\sigma_{\mathbb{S}^{n}}+\int_{\mathbb{S}^{n}}f^{-1}(\theta)\mathrm{d}\theta_{\mathbb{S}^{n}}>0.$
(6.20)

When $\hat{u}_{\max}=c_{5}$, inserting (6.13) into (6.18), we have

$\displaystyle\int_{\mathbb{S}^{n}}\int_{c_{2}}^{r}s^{-1}(1-s^{2})^{-\frac{1}{2}}f^{-1}\mathrm{d}s\mathrm{d}\theta_{\mathbb{S}^{n}}\geq$
$\displaystyle\left(1-c_{2}^{2}\right)^{-\frac{1}{2}}|S^{n}|f_{\min}^{-1}\log
c_{3}+c_{4}\left(1-c_{2}^{2}\right)^{-\frac{1}{2}}|S^{n}|f_{\min}^{-1}\log
r_{\max}$ (6.21) $\displaystyle\geq$ $\displaystyle-
C_{6}+(1-c_{2}^{2})^{-\frac{n+2}{2}}|S^{n}|f_{\max}f_{\min}^{-1}\log c_{5},$

where we use
$\max\limits_{\hat{M}_{t}}\hat{u}=\max\limits_{\hat{M}_{t}}r_{\max}$.

If the maximal radial function of the initial hypersurface satisfies
$\hat{u}_{\max}(0)>c_{5}$, then once $\hat{u}_{\max}(t)\leq c_{5}$, by (6.17),
(6.20) and (6.21) we obtain

$\displaystyle\log r_{\max}\geq-
C_{7}+(1-c_{2}^{2})^{-\frac{n+1}{2}}f_{\max}^{2}f_{\min}^{-1}\log c_{5},$
(6.22)

which implies that there exists a positive constant
$c_{6}=e^{-C_{7}}c_{5}^{(1-c_{2}^{2})^{-\frac{n+1}{2}}f_{\max}^{2}f_{\min}^{-1}}<c_{5}$,
such that $r_{\max}\geq c_{6}$, untill $\hat{u}_{\max}>c_{5}$ again.

If the initial hypersurface of (6.2) satisfies $\hat{u}_{\max}(0)\leq c_{5}$,
then from (6.20), we note that
$\int_{\mathbb{S}^{n}}\int_{a}^{r}s^{-1}(1-s^{2})^{-\frac{1}{2}}f^{-1}\mathrm{d}s\mathrm{d}\theta_{\mathbb{S}^{n}}$
will monotone increasing till $\hat{u}_{\max}(t)>c_{5}$. This together with
(6.17) implies that

$\displaystyle|S^{n}|f_{\max}^{-1}\log
r_{\max}+C_{5}\geq\int_{\mathbb{S}^{n}}\int_{c_{2}}^{r}s^{-1}(1-s^{2})^{-\frac{1}{2}}f^{-1}\mathrm{d}s\mid_{t=0}.$
(6.23)

Hence there exists a positive constant $c_{7}$, such that $r_{\max}\geq
c_{7}$. Clearly, we can deduce from (6.17) that
$c_{7}\leq\hat{u}_{\max}(0)\leq c_{5}$. Since $\hat{u}_{\max}=r_{\max}$ in
$\hat{M}_{t}$, we obtain $r_{\max}(t)\geq\min\\{c_{6},c_{7}\\}$. All those
constants only depend on $n$, $f$ and the initial hypersurface $M_{0}$. By
(6.13), there exists a positive constant $c_{8}$, such that $r_{\min}\geq
c_{8}$. Thus the radial function $r$ of $\hat{M}_{t}$ satisfying $0<c_{8}\leq
r\leq c_{2}<1$. By (5.3), we obtain the uniform bounds of $\rho$ and complete
the proof. ∎

Proof of Theorem 1.2 Similar to the proof of Theorem 1.1, by Lemma 6.1, Lemma
3.2 and Lemma 4.3, we establish the a priori estimates and obtain the long
time existence of (1.3) for the case $\alpha=n+1$. Using the same argument in
Section 5, we show that the flow (1.3) converges smoothly to the unique smooth
even solution of (1.2). Hence we complete the proof of Theorem 1.2.∎

######  Remark 6.1.

Note that the conditions of $\alpha$ and $f$ in Theorem 1.1 and Theorem 1.2
are necessary to the convergence and asymptotic results of (1.3). When
$\alpha<n+1$, or When $\alpha=n+1$ and $f\geq 1$ at the same time, consider a
geodesic sphere with its radial function $\rho\equiv c$. Note that
$\sinh\rho\rightarrow 0$ and $\cosh\rho\rightarrow 1$ as $\rho\rightarrow 0$.
When $\alpha<n+1$,

$\displaystyle\sinh\rho^{\alpha-n-1}\cosh\rho^{n}\rightarrow\infty.$

When $\alpha=n+1$,

$\displaystyle\sinh\rho^{\alpha-n-1}\cosh\rho^{n}\rightarrow 1^{+}.$

Hence there exists a constant $\rho_{0}>0$, such that for any
$\rho\in(0,\rho_{0})$, $-\sinh\rho^{\alpha-n-1}\cosh\rho^{n}f+1<0$ when
$\alpha<n+1$, or when $\alpha=n+1$ and $f\geq 1$ at the same time. Assume the
initial hypersurface of the flow (1.3) is a geodesic sphere with its radial
function $\rho\in(0,\rho_{0})$. Then combining (2.17) ($\tilde{\eta}(t)\equiv
1$) and (3.1), we obtain

$\displaystyle\frac{\partial}{\partial t}\rho(\theta,t)=$
$\displaystyle-\phi(\rho)^{\alpha}f(\theta)w\sigma_{k}+\phi(\theta,t)$
$\displaystyle=$
$\displaystyle\phi(\rho)\left(-\phi(\rho)^{\alpha-n-1}\phi^{\prime}(\rho)^{n}+1)\right)$
$\displaystyle<$ $\displaystyle 0.$

So the initial geodesic sphere keeps shrinking along (1.3).

However, for the cases $\alpha>n+1$, the same initial geodesic sphere of (1.3)
will expand untill it converges smoothly to the solution of (1.2). For the
case $\alpha=n+1$, convergence results of the flow (1.3) need the assumption
of $f$ in addition.

##  7\. Proof of Theorem 1.3

In this part, we consider the flow (1.6) for the cases $2<\alpha\leq n+1$.
First, we parametrize $M_{t}$ as a graph of the radial function
$\rho(\theta,t):\mathbb{S}^{n}\times[0,T)\to\mathbb{R}$. By (2.17), the scalar
parabolic PDE of the radial function turns to

$\frac{\partial}{\partial
t}\rho(\theta,t)=-\phi(\rho)^{\alpha}f(\theta)K(x,t)w+\frac{\displaystyle\int_{\mathbb{S}^{n}}\frac{K}{u}\phi^{n+1}\mathrm{d}\theta_{\mathbb{S}^{n}}}{\displaystyle\int_{\mathbb{S}^{n}}\phi^{n+1-\alpha}f^{-1}\mathrm{d}\theta_{\mathbb{S}^{n}}}\phi.$
(7.1)

Throughout this section, if not specified, we will regard $C_{i}$ and
$C_{I}^{\prime}$ for $i\in\mathbb{N}$ as some positive constants. Observe that
along (1.6), the hypersurfaces have the following properties.

######  Lemma 7.1.

Denote $\Omega(\rho)=\displaystyle\int_{b}^{\rho(\theta,t)}(\sinh
s)^{n-\alpha}\mathrm{d}s$. Then along the flow (1.6),

$\displaystyle\displaystyle\int_{\mathbb{S}^{n}}\frac{\Omega(\rho)}{f}\mathrm{d}\theta_{\mathbb{S}^{n}}=const,\quad\text{for
}t\geq 0.$ (7.2)

Here $\rho(\dot{,}t)$ is the radial function of $M_{t}$.

###### Proof.

By (7.1),

$\displaystyle\partial_{t}\int_{\mathbb{S}^{n}}\frac{\Omega(\rho)}{f}\mathrm{d}\theta_{\mathbb{S}^{n}}=$
$\displaystyle\int_{\mathbb{S}^{n}}\phi(\rho)^{n-\alpha}f^{-1}\partial_{t}\rho\mathrm{d}\theta_{\mathbb{S}^{n}}=0$
(7.3)

where we use $w=\frac{\phi}{u}$ in the last equality. ∎

######  Remark 7.1.

Without loss of generality, we can choose
$b=\frac{1}{2}\min\limits_{M_{0}}\rho$.

From Lemma 7.1, we have the following proposition.

######  Proposition 7.1.

Along (7.1), the maximal radial functions $\rho_{\max}(t)$ of $M_{t}$ have
uniform lower bounds.

###### Proof.

Observe that when $\alpha<n+1$, since $\cosh s>0$ and it is monotone
increasing, we have

$\displaystyle\phi^{n+1-\alpha}(\rho_{\max})|\mathbb{S}^{n}|-\phi^{n+1-\alpha}(b)|\mathbb{S}^{n}|$
(7.4) $\displaystyle\geq$
$\displaystyle\int_{\mathbb{S}^{n}}\left(\phi^{n+1-\alpha}(\rho)-\phi^{n+1-\alpha}(b)\right)\mathrm{d}\theta_{\mathbb{S}^{n}}$
$\displaystyle=$
$\displaystyle(n+1-\alpha)\int_{\mathbb{S}^{n}}\int_{b}^{\rho}(\sinh
s)^{n-\alpha}\cosh s\ \mathrm{d}s\mathrm{d}\theta_{\mathbb{S}^{n}}$
$\displaystyle\geq$
$\displaystyle(n+1-\alpha)\int_{\mathbb{S}^{n}}\int_{b}^{\rho}(\sinh
s)^{n-\alpha}\cosh b\ \mathrm{d}s\mathrm{d}\theta_{\mathbb{S}^{n}}\geq c>0.$

Here we use Lemma 7.1 in the last inequality. The positive constant $c$
depends on the initial hypersurface, $\min f$ and $a$. Similarly, when
$\alpha=n+1$, there exists a positive constant $c^{\prime}$, such that

$\displaystyle\log\phi(\rho_{\max})|\mathbb{S}^{n}|-\log\phi(b)|\mathbb{S}^{n}|\geq$
$\displaystyle\int_{\mathbb{S}^{n}}\left(\log\phi(\rho)-\log\phi(b)\right)\mathrm{d}\theta_{\mathbb{S}^{n}}$
(7.5) $\displaystyle=$
$\displaystyle\int_{\mathbb{S}^{n}}\int_{b}^{\rho}(\sinh s)^{-1}\cosh s\
\mathrm{d}s\mathrm{d}\theta_{\mathbb{S}^{n}}$ $\displaystyle\geq$
$\displaystyle\int_{\mathbb{S}^{n}}\int_{b}^{\rho}(\sinh s)^{-1}\cosh b\
\mathrm{d}s\mathrm{d}\theta_{\mathbb{S}^{n}}\geq c^{\prime}>0.$

If $\rho_{\max}\rightarrow 0$, we have
$\phi(\rho_{\max})^{n+1-\alpha}\rightarrow 0$ and
$\log\phi(\rho_{\max})\rightarrow-\infty$, which is contrary to (7.4) and
(7.5). Hence, we obtain the uniform lower bound of $\phi_{\max}$ and complete
the proof.

∎

Using the projection $\pi_{p}$ in Section 5 and the relation between $M_{t}$
and $\pi_{p}(M_{t})$, (5.1)-(5.6) and (5.13), we obtain the scalar equation of
the support function $\hat{u}$ of $\pi_{p}(M_{t})$,

$\displaystyle\partial_{t}\hat{u}=-r^{\alpha}(1-r^{2})^{\frac{n+3-\alpha}{2}}(1-\hat{u}^{2})^{-\frac{n+1}{2}}f\hat{K}+\eta(t)\hat{u}\sqrt{1-r^{2}},$
(7.6)

where
$\eta(t)=\frac{\displaystyle\int\sqrt{1-r^{2}}(1-\hat{u}^{2})^{-\frac{n+1}{2}}\mathrm{d}\sigma_{\mathbb{S}^{n}}}{\displaystyle\int\frac{r^{n+1-\alpha}}{(1-r^{2})^{\frac{n+1-\alpha}{2}}}f^{-1}\mathrm{d}\theta_{\mathbb{S}^{n}}}$.

Under the flow (7.6), we have the following monotone function

$\displaystyle\mathcal{J}(\hat{u})=\int_{\mathbb{S}^{n}}\Psi(\hat{u})\mathrm{d}\sigma_{\mathbb{S}^{n}}$
(7.7)

where
$\Psi(\hat{u})=\int_{a}^{\hat{u}(x,t)}\frac{1}{s}(1-s^{2})^{-\frac{n+1}{2}}\mathrm{d}s$.
Without loss of generality, we can choose
$a=\frac{1}{2}\min\limits_{\hat{M_{0}}}\hat{u}$.

######  Lemma 7.2.

Along (7.6), $\mathcal{J}(\hat{u})$ is non-increasing and the equality holds
if and only if $\pi_{p}(M_{t})$ satisfies the following equation

$\displaystyle
r^{\alpha}(1-r^{2})^{\frac{n+3-\alpha}{2}}(1-\hat{u}^{2})^{-\frac{n+1}{2}}f\hat{K}=c\hat{u}\sqrt{1-r^{2}}$
(7.8)

for some positive constant $c$.

###### Proof.

By (7.6),

$\displaystyle\partial_{t}\mathcal{J}(\hat{u})=$
$\displaystyle\int_{\mathbb{S}^{n}}\frac{1}{\hat{u}}(1-\hat{u}^{2})^{-\frac{n+1}{2}}\partial_{t}u\mathrm{d}\sigma_{\mathbb{S}^{n}}$
(7.9) $\displaystyle=$
$\displaystyle\frac{\left(\displaystyle\int\sqrt{1-r^{2}}(1-\hat{u}^{2})^{-\frac{n+1}{2}}\mathrm{d}\sigma_{\mathbb{S}^{n}}\right)^{2}}{\displaystyle\int\frac{r^{-\alpha}\hat{u}}{(1-r^{2})^{\frac{n+1-\alpha}{2}}\hat{K}}f^{-1}\mathrm{d}\sigma_{\mathbb{S}^{n}}}-\int
r^{\alpha}(1-r^{2})^{\frac{n+3-\alpha}{2}}(1-\hat{u}^{2})^{-(n+1)}f\frac{\hat{K}}{\hat{u}}\mathrm{d}\sigma_{\mathbb{S}^{n}}$
$\displaystyle\leq$ $\displaystyle 0.$

Here we use H$\ddot{\text{o}}$lder inequality in the last inequality and the
equality holds if and only if (7.8) holds.

∎

Next, we will show the radial function $\rho(\cdot,t)$ is uniformly bounded
along the flow (1.6) under the following assumption.

######  Proposition 7.2.

Let $\rho$ be a smooth, positive, uniformly convex and origin-symmetric
solution to (7.1) on $\mathbb{S}^{n}\times[0,T)$. If $2<\alpha\leq n+1$ and
$f$ is a positive even function on $\mathbb{S}^{n}$, then there exists a
positive constant $C$ depending on $\alpha$, $n$, $f$ and the initial
hypersurface, such that

$\displaystyle\frac{1}{C}\leq\rho\leq C,\quad\forall t\in[0,T).$ (7.10)

###### Proof.

Firstly, we prove $\rho$ has a uniform upper bound. Motivated by the proof in
Lemma 3.1, we will show that
$\frac{\displaystyle\int_{\mathbb{S}^{n}}{\frac{K}{u}\phi^{n+1}\mathrm{d}\theta_{\mathbb{S}^{n}}}}{\displaystyle\int_{\mathbb{S}^{n}}\phi^{n+1-\alpha}f^{-1}\mathrm{d}\theta_{\mathbb{S}^{n}}}$
is bounded from above by multiple $\phi_{\max}$. By Lemma 7.2, there exists a
positive constant $C_{1}>0$, such that

$\displaystyle C_{1}\geq$ $\displaystyle\mathcal{J}(\hat{u})$ (7.11)
$\displaystyle=$
$\displaystyle\int_{\mathbb{S}^{n}}\int_{a}^{\hat{u}}\frac{1}{s}(1-s^{2})^{-\frac{n+1}{2}}\mathrm{d}s\mathrm{d}\sigma_{\mathbb{S}^{n}}$
$\displaystyle\geq$ $\displaystyle
2^{\frac{n+1}{2}}\int\limits_{\mathbb{S}^{n}\bigcap\\{x|\hat{u}>a\\}}\int_{a}^{\hat{u}}(1-s)^{-\frac{n+1}{2}}\mathrm{d}s\mathrm{d}\sigma_{\mathbb{S}^{n}}-(1-a^{2})^{-\frac{n+1}{2}}\int\limits_{\mathbb{S}^{n}\bigcap\\{x|\hat{u}\leq
a\\}}\int_{\hat{u}}^{a}\frac{1}{s}\mathrm{d}s\mathrm{d}\sigma_{\mathbb{S}^{n}}$
$\displaystyle\geq$ $\displaystyle
C_{2}\int\limits_{\mathbb{S}^{n}\bigcap\\{x|\hat{u}>a\\}}\left((1-\hat{u})^{-\frac{n-1}{2}}-(1-a)^{\frac{n-1}{2}}\right)\mathrm{d}\sigma_{\mathbb{S}^{n}}-C_{3}\int\limits_{\mathbb{S}^{n}\bigcap\\{x|\hat{u}\leq
a\\}}\left(\log a-\log\hat{u}\right)\mathrm{d}\sigma_{\mathbb{S}^{n}}$
$\displaystyle\geq$ $\displaystyle-
C_{4}+C_{2}\int_{\mathbb{S}^{n}}(1-\hat{u})^{-\frac{n-1}{2}}\mathrm{d}\sigma_{\mathbb{S}^{n}}-C_{2}\int\limits_{\mathbb{S}^{n}\bigcap\\{x|\hat{u}\leq
a\\}}(1-\hat{u})^{-\frac{n-1}{2}}\mathrm{d}\sigma_{\mathbb{S}^{n}}+C_{3}\int\limits_{\mathbb{S}^{n}}\log\hat{u}\
\mathrm{d}\sigma_{\mathbb{S}^{n}}$ $\displaystyle\geq$ $\displaystyle-
C_{4}+C_{2}\int_{\mathbb{S}^{n}}(1-\hat{u})^{-\frac{n-1}{2}}\mathrm{d}\sigma_{\mathbb{S}^{n}}+C_{3}\int\limits_{\mathbb{S}^{n}}\log\hat{u}\
\mathrm{d}\sigma_{\mathbb{S}^{n}}.$

Similar to the proof in (6.5), we have when $n=1$

$\displaystyle-
C_{5}^{\prime}\int_{\mathbb{S}^{1}}\log(1-\hat{u})\mathrm{d}\sigma_{\mathbb{S}^{n}}+C_{6}^{\prime}\int_{\mathbb{S}^{1}}\log\hat{u}\mathrm{d}\sigma_{\mathbb{S}^{1}}\leq
C_{7}^{\prime},$ (7.12)

and when $n\geq 2$

$\displaystyle
C_{5}\int\limits_{\mathbb{S}^{n}}(1-\hat{u})^{-\frac{n-1}{2}}\mathrm{d}\sigma_{\mathbb{S}^{n}}+C_{6}\int\limits_{\mathbb{S}^{n}}\log\hat{u}\mathrm{d}\sigma_{\mathbb{S}^{n}}\leq
C_{7}.$ (7.13)

Assume $\hat{u}_{\max}(t)$ is attained at $v_{0}\in\mathbb{S}^{n}$ at any
fixed time $t$. Because the flow hypersurfaces are uniformly convex and
origin-symmetric, we have $\hat{u}(v)\geq\hat{u}_{\max}|\langle
v,v_{0}\rangle|$. Then

$\displaystyle\int_{\mathbb{S}^{n}}\log\hat{u}\mathrm{d}\sigma_{\mathbb{S}^{n}}\geq\int_{\mathbb{S}^{n}}(\log\hat{u}_{\max}+\log|\langle
v,v_{0}\rangle|)\mathrm{d}\sigma_{\mathbb{S}^{n}}.$ (7.14)

The second integral in (7.14) is convergent, for the same reason as (6.10).
Thus by (7.12) and (7.13), there exists positive constants $C_{5}$ and
$C_{5}^{\prime}$ such that, when $n=1$

$\displaystyle\int_{\mathbb{S}^{1}}\log(1-\hat{u}^{2})\mathrm{d}\sigma_{\mathbb{S}^{1}}\geq\int_{\mathbb{S}^{1}}\log(1-\hat{u})\mathrm{d}\sigma_{\mathbb{S}^{1}}\geq-
C_{8}^{\prime},$ (7.15)

and when $n\geq 2$

$\displaystyle
C_{8}\geq\int_{\mathbb{S}^{n}}(1-\hat{u})^{-\frac{n-1}{2}}\mathrm{d}\sigma_{\mathbb{S}^{n}}\geq\int_{\mathbb{S}^{n}}(1-\hat{u}^{2})^{-\frac{n-1}{2}}\mathrm{d}\sigma_{\mathbb{S}^{n}}.$
(7.16)

(7.4) and (7.5) implies that there exists a constant $c_{1}>0$, such that
$\rho_{max}\geq c_{1}$, i.e., by (5.3) there exists $c_{2}>0$, such that
$r\geq c_{2}$. Recall the relation between $M_{t}$ and $\pi_{p}(M_{t})$,
(5.1)-(5.6) and (5.13). Note that
$r(\theta,t)=\sqrt{\hat{u}^{2}+|\nabla\hat{u}|^{2}}\geq\hat{u}(v,t)$ and
$\max\limits_{\hat{M}_{t}}r=\max\limits_{\hat{M}_{t}}\hat{u}$. At a fixed time
$t$, we have

$\displaystyle\int_{\mathbb{S}^{n}}\frac{K}{u}\phi^{n+1}\mathrm{d}\theta_{\mathbb{S}^{n}}=$
$\displaystyle\int_{\mathbb{S}^{n}}\sqrt{1-r^{2}}(1-\hat{u}^{2})^{-\frac{n+1}{2}}\mathrm{d}\sigma_{\mathbb{S}^{n}}$
(7.17) $\displaystyle\leq$
$\displaystyle\frac{1}{\sqrt{1-\hat{u}_{\max}^{2}}}\int_{\mathbb{S}^{n}}(1-\hat{u}^{2})^{-\frac{n-1}{2}}\mathrm{d}\sigma_{\mathbb{S}^{n}}$
$\displaystyle\leq$
$\displaystyle\frac{C_{8}}{c_{2}}\frac{r_{\max}}{\sqrt{1-r_{\max}^{2}}}=C_{9}\phi_{\max},$

where we use (7.16) in the second inequality and (5.3) in the last equality.
Moreover, the uniform lower bound of
$\displaystyle\int_{\mathbb{S}^{n}}\phi^{n+1-\alpha}f^{-1}\mathrm{d}\theta_{\mathbb{S}^{n}}$
is obtained directly by (7.4). At the maximal point of $\rho$, inserting
(7.17) into (7.1), similar to the proof in Lemma 3.1, we have

$\displaystyle\partial_{t}\rho_{\max}\leq$
$\displaystyle-\phi_{\max}^{\alpha-n}\phi_{\max}^{\prime
n}f+\frac{\displaystyle\int_{\mathbb{S}^{n}}\frac{K}{u}\phi^{n+1}\mathrm{d}\theta_{\mathbb{S}^{n}}}{\displaystyle\int_{\mathbb{S}^{n}}\phi^{n+1-\alpha}f^{-1}\mathrm{d}\theta_{\mathbb{S}^{n}}}\phi_{\max}$
(7.18) $\displaystyle\leq$
$\displaystyle\phi_{\max}^{2}(-\phi_{\max}^{\alpha-2}f+C_{7}).$

Hence when $\alpha>2$, $\phi=\sinh\rho$ has an uniform upper bound. Thus by
(5.3) there exists a constant $c_{3}>0$, such that $r\leq c_{3}<1$.

In order to obtain the lower bound of $\rho$, we parametrize any point
$\theta$ in $\mathbb{S}^{n}$ as in (6.6). Assume $r_{\min}$ is attained at
$\theta_{0}=(1,\vec{0})$. We have
$\langle\theta,\theta_{0}\rangle=\cos\theta_{1}$. Since $r\leq c_{3}<1$, there
exists a positive constant $c_{4}$, such that
$\phi=\frac{r}{\sqrt{1-r^{2}}}\leq c_{4}r$. Denote by
$\delta=\sqrt{r_{\min}}$. For $2<\alpha<n+1$, we have

$\displaystyle\int_{\mathbb{S}^{n}}\phi^{n+1-\alpha}(\rho)\mathrm{d}\theta_{\mathbb{S}^{n}}\leq$
$\displaystyle
c_{4}^{n+1-\alpha}\int\limits_{S_{1}=\mathbb{S}^{n}\bigcap\\{r\leq\delta\\}}r^{n+1-\alpha}\mathrm{d}\theta_{\mathbb{S}^{n}}+c_{4}^{n+1-\alpha}\int\limits_{S_{2}=\mathbb{S}^{n}\bigcap\\{r>\delta\\}}r^{n+1-\alpha}\mathrm{d}\theta_{\mathbb{S}^{n}}$
(7.19) $\displaystyle\leq$ $\displaystyle
C_{8}\delta^{n+1-\alpha}|S^{n}|+C_{8}c_{3}^{n+1-\alpha}|S_{2}|.$

We also have $r(\theta)\leq\frac{r_{\min}(\theta_{0})}{|\cos\theta_{1}|}$
because the flow hypersurface $M_{t}$ are strictly convex and origin
symmetric. Then $S_{2}=\\{\theta|\cos\theta_{1}<\sqrt{r_{\min}}\\}$. Suppose
$r_{\min}\rightarrow 0$. Clearly, $\delta=\sqrt{r_{\min}}\rightarrow 0$ and
$|S_{2}|\rightarrow 0$ at the same time. By (7.19), we have
$\int_{\mathbb{S}^{n}}\phi^{n+1-\alpha}(\rho)\mathrm{d}\theta_{\mathbb{S}^{n}}\rightarrow
0$, which is contrary to (7.4). For $\alpha=n+1$, we have

$\displaystyle\int_{\mathbb{S}^{n}}\log\phi(\rho)\mathrm{d}\theta_{\mathbb{S}^{n}}\leq$
$\displaystyle\int_{\mathbb{S}^{n}}\log
r(\theta)\mathrm{d}\theta_{\mathbb{S}^{n}}+|S^{n}|\log c_{4}$ (7.20)
$\displaystyle\leq$ $\displaystyle|S^{n}|\log
r_{\min}(\theta_{0})-\int_{\mathbb{S}^{n}}\log|\cos\theta_{1}|\mathrm{d}\theta_{\mathbb{S}^{n}}+|S^{n}|\log
c_{4}.$

For the same reason as (6.11), the second term in (7.20) is convergent. So
$\int_{\mathbb{S}^{n}}\log\phi(\rho)\mathrm{d}\theta_{\mathbb{S}^{n}}\rightarrow-\infty$
as $r_{\min}\rightarrow 0$, which is contrary to (7.5). Hence we obtain the
uniform lower bounds of $r$ as well as $\rho$ by (5.3) and complete the proof.
∎

######  Remark 7.2.

Similar to the Euclidean space, we hope (7.11) implies the uniform upper bound
of the support function $\hat{u}$ away from $1$. Unfortunately, we find an
example here with unbounded $u$, i.e, $\hat{u}$ is very close to $1$
satisfying (7.11). Suppose $\hat{E}(e_{1},e_{2},\cdots,e_{2})$ is a rotation
symmetric ellipsoid in $\mathbb{R}^{n+1}$ with its centroid in the origin
represented by
$\frac{x_{1}^{2}}{e_{1}^{2}}+\frac{x_{2}^{2}}{e_{2}^{2}}+\cdots+\frac{x_{n+1}^{2}}{e_{2}^{2}}=1$.
Here we assume $0<e_{2}<e_{1}<1$. Then $\hat{E}$ can be parametrized by
$x=(e_{1}\cos\theta,e_{2}\sin\theta\vec{x})$ where
$\vec{x}\in\mathbb{S}^{n-1}$ is an $n$-dimensional unit vector.
Correspondingly, the unit outward norm vector is

$\displaystyle\frac{(e_{2}\cos\theta,e_{1}\sin\theta\vec{x})}{\sqrt{e_{2}^{2}\cos^{2}\theta+e_{1}^{2}\sin^{2}\theta}}.$
(7.21)

The support function becomes

$\displaystyle\hat{u}(\theta)=\frac{e_{1}e_{2}}{\sqrt{e_{2}^{2}\cos^{2}\theta+e_{1}^{2}\sin^{2}\theta}}.$
(7.22)

If we parametrize $\mathbb{S}^{n}$ as in (6.7), by comparing with (7.21) we
have

$\displaystyle\tan v_{1}=\frac{e_{1}}{e_{2}}\tan\theta.$ (7.23)

By (7.22) and (7.23), the support function can be parametrized by $v_{1}$ as

$\displaystyle\hat{u}(v_{1})=$
$\displaystyle\frac{e_{1}e_{2}\sqrt{\tan\theta^{2}+1}}{\sqrt{e_{2}^{2}+e_{1}^{2}\tan\theta^{2}}}=\frac{e_{1}\sqrt{\tan
v_{1}^{2}\frac{e_{2}^{2}}{e_{1}^{2}}+1}}{\sqrt{1+\tan v_{1}^{2}}}$ (7.24)
$\displaystyle=$ $\displaystyle\sqrt{\tan v_{1}^{2}e_{2}^{2}+e_{1}^{2}}\cos
v_{1}=\sqrt{e_{1}^{2}-(e_{1}^{2}-e_{2}^{2})\sin v_{1}^{2}}.$

Substitute (6.7) into (7.15) and (7.16) respectively. When $n=1$, we obtain

$\displaystyle\int_{\mathbb{S}^{1}}\log(1-\hat{u}^{2})\mathrm{d}\sigma_{\mathbb{S}^{1}}=$
$\displaystyle\int_{0}^{2\pi}\log(1-e_{1}^{2}+(e_{1}^{2}-e_{2}^{2})(\sin
v_{1})^{2})dv_{1}$ (7.25) $\displaystyle>$ $\displaystyle
2\int_{0}^{\pi}\left(\log(e_{1}^{2}-e_{2}^{2})+2\log\sin
v_{1}\right)\mathrm{d}v_{1}\geq-C(e_{1},e_{2}).$

When $n\geq 2$

$\displaystyle\int_{\mathbb{S}^{n}}(1-\hat{u}^{2})^{-\frac{n-1}{2}}\mathrm{d}\sigma_{\mathbb{S}^{n}}$
(7.26) $\displaystyle=$
$\displaystyle\int_{0}^{\pi}\int_{\mathbb{S}^{n-1}}(1-e_{1}^{2}+(e_{1}^{2}-e_{2}^{2})\sin
v_{1}^{2})^{-\frac{n-1}{2}}(\sin
v_{1})^{n-1}\mathrm{d}v_{1}\mathrm{d}\sigma_{\mathbb{S}^{n-1}}$
$\displaystyle<$
$\displaystyle\int_{0}^{\pi}\int_{\mathbb{S}^{n-1}}\left((e_{1}^{2}-e_{2}^{2})\sin
v_{1}^{2}\right)^{-\frac{n-1}{2}}(\sin
v_{1})^{n-1}\mathrm{d}v_{1}\mathrm{d}\sigma_{\mathbb{S}^{n-1}}$
$\displaystyle\leq$ $\displaystyle C(e_{1},e_{2},n).$

Now we construct a sequence of $\hat{E}(e_{1}^{(n)},e_{2}^{(n)})$ satisfying
$e_{1}^{(n)}\rightarrow 1$ and $e_{2}^{(n)}\leq c<1$. Note that by (5.6),
$\max u_{n}\rightarrow\infty$ as $\max\hat{u}_{n}=e_{1}\rightarrow 1$, which
implies that the preimages of $\hat{E}(e_{1}^{(n)},e_{2}^{(n)})$ in Hyperbolic
space tend to $\infty$. However, by inserting (7.25), (7.26) into (7.12)and
(7.13), we show that $\mathcal{J}(\hat{u})_{\hat{E}(e_{1}^{(n)},e_{2}^{(n)})}$
still can be bounded from above by some positive constant $C$ independent of
how far $e_{1}$ is away from $1$.

Proof of Theorem 1.3 Combining Lemma 3.2, we established the $C^{0}$, $C^{1}$
estimates of the flow (1.6) under the evenness assumption. Employing (5.3) and
(5.6), we obtain the uniform upper and lower bounds of $\eta(t)$ in (7.6).
Then the $C^{2}$ estimate follows from Lemma 4.3. Thus we obtain the long time
existence and regularity for the flow (1.6). Furthermore, Lemma 7.2 and the
similar argument in Section 5 imply the asymptotic behaviour of the flow (1.6)
and we complete the proof of the Theorem 1.3. ∎

## References