# Geometric inequalities and their stabilities for modified quermassintegrals
in hyperbolic space

Chaoqun Gao  School of Mathematical Sciences, University of Science and
Technology of China, Hefei 230026, P.R. China<EMAIL_ADDRESS>and
Rong Zhou<EMAIL_ADDRESS>
###### Abstract.

In this paper, we first consider the curve case of Hu-Li-Wei’s flow for
shifted principal curvatures of h-convex hypersurfaces in $\mathbb{H}^{n+1}$
proposed in [10]. We prove that if the initial closed curve is smooth and
strictly h-convex, then the solution exists for all time and preserves strict
h-convexity along the flow. Moreover, the evolving curve converges smoothly
and exponentially to a geodesic circle centered at the origin. The key
ingredient in our proof is the Heintze-Karcher type inequality for h-convex
curves proved recently in [14].

As an application, we then provide a new proof of geometric inequalities
involving weighted curvature integrals and modified quermassintegrals for
h-convex curves in $\mathbb{H}^{2}$. We finally discuss the stability of these
inequalities as well as Alexandrov-Fenchel type inequalities for modified
quermassintegrals for strictly h-convex domains in $\mathbb{H}^{n+1}$.

###### Key words and phrases:

modified quermassintegrals, weighted curvature integrals, horospherically
convex

###### 2020 Mathematics Subject Classification:

53C42; 53E10

##  1\. Introduction

In this paper, we discuss some locally constrained curvature flows and their
applications to proving inequalities involving weighted curvature integrals
and modified quermassintegrals for smooth h-convex domains in hyperbolic space
$\mathbb{H}^{n+1}$.

We consider hyperbolic space as a warped product space
$\mathbb{H}^{n+1}=[0,\infty)\times\mathbb{S}^{n}$ equipped with metric
$\overline{g}=\mathrm{d}r^{2}+\phi^{2}(r)g_{\mathbb{S}^{n}}$, where
$\phi(r)=\sinh r,\ r\in[0,\infty)$. Let $\Phi(r)=\cosh r-1$. It’s well known
that $V=\overline{\nabla}\Phi=\phi(r)\partial_{r}$ is a conformal killing
field.

A smooth bounded domain $\Omega$ in $\mathbb{H}^{n+1}$ is said to be h-convex
(resp. strictly h-convex) if the principal curvatures of the boundary
$\partial\Omega$ satisfy $\kappa_{i}\geqslant 1$ (resp. $\kappa_{i}>1$) for
all $i=1,2,\cdots,n$. Let $M$ be a smooth closed hypersurface, denote $\nu$ as
its unit outward normal vector, and its support function is defined by
$u=\langle V,\nu\rangle$. We say that $M$ is star-shaped if its support
function $u$ is positive everywhere on $M$.

###  1.1. Curvature flows

Locally constrained curvature flows have been studied widely in recent years.
Brendle, Guan and Li [3] (see also [9]) introduced the flow
$X:M^{n}\times[0,T)\to\mathbb{N}^{n+1}(K)$

$\partial_{t}X(x,t)=\left(\dfrac{\phi^{\prime}(r)E_{k-1}(\kappa)}{E_{k}(\kappa)}-u\right)\nu(x,t),\
\ k=1,2,\cdots,n$ (1.1)

for a family of smooth closed hypersurfaces in $n+1$ dimensional space forms
with constant sectional curvature $K=1,-1$ and discuss some convergence
results under several assumptions when $n\geqslant 2$. Here $E_{k}(\kappa)$
denotes the $k$th normalized elementary symmetric function of principal
curvatures $\kappa$ of $M_{t}=X(M,t)$.

For $n\geqslant 2$, Hu, Li and Wei [10] used the tensor maximum principle to
verify that the h-convexity preserves along the flow (1.1) and thus proved the
long time existence and exponential convergence to a geodesic sphere if the
initial hypersurface is h-convex in hyperbolic space $\mathbb{H}^{n+1}$.

Kwong, Wei, Wheeler and Wheeler [13] considered the curve case of (1.1) in
space forms and proved that if the initial curve is strictly convex (i.e. the
curvature of curve $\kappa>0$), then the solution curve exists for all time
and preserves strict convexity. Moreover, the evolving curve smoothly and
exponentially converges to a geodesic circle centered at the origin.

In [10], Hu, Li and Wei introduced a new locally constrained curvature flow
$X:M^{n}\times[0,T)\to\mathbb{H}^{n+1}$,

$\partial_{t}X(x,t)=\left((\phi^{\prime}(r)-u)\dfrac{E_{k-1}(\tilde{\kappa})}{E_{k}(\tilde{\kappa})}-u\right)\nu(x,t),\
\ k=1,2,\cdots,n,$ (1.2)

where $\tilde{\kappa}_{i}=\kappa_{i}-1$ are shifted principal curvatures of
$M_{t}=X(M,t)$, $E_{k}(\tilde{\kappa})$ is the normalized $k$th elementary
symmetric function of $\tilde{\kappa}$. They proved the following long time
existence and convergence results of the flow (1.2) for strictly h-convex
hypersurfaces in higher dimensions ($n\geqslant 2$).

######  Theorem 1.1 ([10, Theorem 1.6]).

Let $X_{0}:M^{n}\to\mathbb{H}^{n+1}\ (n\geqslant 2)$ be a smooth embedding
such that $M_{0}=X_{0}(M)$ is a smooth, strictly h-convex hypersurface in
$\mathbb{H}^{n+1}$. Then the flow (1.2) has a smooth solution for all time
$t\in[0,\infty)$, and $M_{t}=X_{t}(M)$ is strictly h-convex for each $t>0$ and
converges smoothly and exponentially to a geodesic sphere centered at the
origin.

The convergence results for $n=1$ has not been obtained yet. In the first part
of this paper, we aim to make up this gap. When $n=1$, the flow (1.2) becomes

$\partial_{t}X(x,t)=\left(\dfrac{\phi^{\prime}(r)-u}{\kappa-1}-u\right)\nu(x,t).$
(1.3)

Let’s consider a family of immersed closed strictly h-convex curves
$X:\mathbb{S}^{1}\times[0,T)\to\mathbb{H}^{2}$ satisfying the above flow
equation and we will prove the long time existence and convergence results for
the flow (1.3).

######  Theorem 1.2.

Let $X_{0}:\mathbb{S}^{1}\to\mathbb{H}^{2}$ be a smooth strictly h-convex
closed curve containing the origin. There exists a unique solution
$X:\mathbb{S}^{1}\times[0,\infty)\to\mathbb{H}^{2}$ to the flow (1.3) such
that $X(\cdot,0)=X_{0}$. For each $t\in[0,\infty)$,
$\gamma_{t}=X(\mathbb{S}^{1},t)$ is strictly h-convex and converges smoothly
and exponentially to a geodesic circle
$\gamma_{\infty}=X_{\infty}(\mathbb{S}^{1})$ centered at the origin.

We remark that for a strictly h-convex initial curve, the short time existence
of the flow (1.3) is guaranteed by standard theory of parabolic equations. The
long time existence follows by the standard procedure after a priori
estimates. To show the convergence, the key ingredient is the Heintze-Karcher
type inequality for h-convex curves proved in [14, Proposition 8.2], from
which we know that the area of domains enclosed by h-convex curves is
monotonically non-decreasing along the flow (1.3). It also helps us to find
the limit curve be a geodesic circle and to prove smooth convergence. In the
proof of exponential convergence, inspired by [13, Section 4.3] and [21,
Section 9], we consider the linearization of the flow (3.13) at infinity and
used the spherical harmonics to expand functions in $L^{2}(\mathbb{S}^{1})$ to
obtain asymptotic behavior around the infinite time.

###  1.2. Applications to geometric inequalities

In the second part of this paper, we will use the flow (1.3) to provide a new
proof of some geometric inequalities involving weighted curvature integrals
and modified quermassintegrals for h-convex curves in $\mathbb{H}^{2}$. (See
Section 2.2 for precise definitions.)

Many scholars studied the proof of Alexandrov-Fenchel type inequalities in
$\mathbb{H}^{n+1}$ by using curvature flow methods. In 2014, Wang and Xia [20]
proved the quermassintegral inequalities for a h-convex domain $\Omega$ in
$\mathbb{H}^{n+1}\ (n\geqslant 2)$,

$W_{k}(\Omega)\geqslant f_{k}\circ f_{\ell}^{-1}(W_{\ell}(\Omega)),\ \
0\leqslant\ell<k\leqslant n$ (1.4)

with equality holds if and only if $\Omega$ is a geodesic ball. Here
$f_{k}:[0,\infty)\to\mathbb{R}^{+}$ is a monotone function defined by
$f_{k}(r)=W_{k}(\overline{B}_{r})$, the $k$th quermassintegral for the
geodesic ball of radius $r$, and $f_{\ell}^{-1}$ is the inverse function of
$f_{\ell}$. In 2021, Andrews, Chen and Wei [1] first introduced modified
quermassintegrals for h-convex domains and proved inequalities for a bounded,
smooth strictly h-convex domain $\Omega\subset\mathbb{H}^{n+1}\ (n\geqslant
2)$,

$\tilde{W}_{k}(\Omega)\geqslant\tilde{f}_{k}\circ\tilde{f}_{\ell}^{-1}(\tilde{W}_{\ell}(\Omega)),\
\ 0\leqslant\ell<k\leqslant n.$ (1.5)

Equality holds in (1.5) if and only if $\Omega$ is a geodesic ball. Here
$\tilde{f}_{k}:[0,\infty)\to\mathbb{R}^{+}$ is a monotone function defined by
$\tilde{f}_{k}(r)=\tilde{W}_{k}(\overline{B}_{r})$, the $k$th modified
quermassintegral for the geodesic ball of radius $r$, and
$\tilde{f}_{\ell}^{-1}$ is the inverse function of $\tilde{f}_{\ell}$.

Later, Hu-Li-Wei [10] applied the flow (1.1) and (1.2) for h-convex
hypersurfaces in $\mathbb{H}^{n+1}$ to provide a new proof of inequalities
(1.4) and (1.5) respectively. They also used the flow (1.2) to derive
inequalities involving weighted curvature integrals and modified
quermassintegrals for a strictly h-convex domain $\Omega$ in
$\mathbb{H}^{n+1}\ (n\geqslant 2)$ (see [10, Theorem 1.8]): For each
$k=1,2,\cdots,n$, there holds

$\int_{\partial\Omega}\left(\phi^{\prime}(r)-u\right)E_{k}(\tilde{\kappa})\mathrm{d}\mu\geqslant\tilde{g}_{k}\circ\tilde{f}_{k}^{-1}(\tilde{W}_{k}(\Omega)),$
(1.6)

where
$\tilde{g}_{k}(r)=\int_{\partial\overline{B}_{r}}\left(\phi^{\prime}(r)-u\right)E_{k}(\tilde{\kappa})\mathrm{d}\mu$,
$\tilde{f}_{k}(r)=\tilde{W}_{k}(\overline{B}_{r})$. Equality holds in (1.6) if
and only if $\Omega$ is a geodesic ball centered at the origin.

Recently, Kwong and Wei [12] investigated a family of three-term geometric
inequalities involving weighted curvature integrals and quermassintegrals for
hypersurfaces in space forms. As a special case of [12, Theorem 5.4], they
proved that for a smooth bounded h-convex domain $\Omega$ in
$\mathbb{H}^{n+1}\ (n\geqslant 2)$, there holds

$\int_{\partial\Omega}\Phi
E_{k}(\kappa)\mathrm{d}\mu+kW_{k-1}(\Omega)\geqslant(\xi_{k}+kf_{k-1})\circ
f_{\ell}^{-1}(W_{\ell}(\Omega))$ (1.7)

for all $k=1,2,\cdots,n$ and $l=0,1,\cdots,k$. Equality holds in (1.7) if and
only if $\Omega$ is a geodesic ball centered at the origin.
$\xi_{k}(r)=\int_{\partial\overline{B}_{r}}\Phi E_{k}(\kappa)\mathrm{d}\mu$,
$f_{k}(r)=W_{k}(\overline{B}_{r})$.

Li and Xu proposed a conjecture of horospherical $p$-Minkowski inequalities in
[14, Conjecture 10.5]. Especially, they proved some inequalities which are
special cases of the conjecture from a horospherically convex geometric point
of view. They used the flow (1.2) and the convergence result (Theorem 1.1) to
prove the following three-term inequality involving weighted curvature
integrals and modified quermassintegrals for strictly h-convex hypersurfaces.
We state their result in our setting.

######  Theorem 1.3 ([14, Theorem 10.9]).

Let $M$ be a smooth strictly h-convex hypersurface enclosing a domain $\Omega$
in $\mathbb{H}^{n+1}\ (n\geqslant 2)$. For any $k=1,2,\cdots,n$, there holds

$\int_{M}(\Phi-u)E_{k}(\tilde{\kappa})\mathrm{d}\mu+(n-2k)\tilde{W}_{k}(\Omega)\geqslant\tilde{h}_{k}\circ\tilde{f}_{k}^{-1}(\tilde{W}_{k}(\Omega)),$
(1.8)

where
$\tilde{h}_{k}(r)=\int_{\partial\overline{B}_{r}}(\Phi-u)E_{k}(\tilde{\kappa})\mathrm{d}\mu+(n-2k)\tilde{W}_{k}(\overline{B}_{r})$,
$\tilde{f}_{k}(r)=\tilde{W}_{k}(\overline{B}_{r})$, $\overline{B}_{r}$ is a
geodesic ball centered at the origin, $\tilde{f}_{k}^{-1}$ is the inverse
function of $\tilde{f}_{k}(r)$. Equality holds if and only if $M$ is a
geodesic sphere centered at the origin.

Note that when $n=1$, we have $\tilde{W}_{1}(\Omega)=L[\gamma]-A[\gamma]$ by
(2.10), where $L[\gamma]$ denotes the length of the curve $\gamma$,
$A[\gamma]$ denotes the area of the domain $\Omega$ enclosed by $\gamma$. It
has been proved in [14, Theorem 10.13] that (1.8) holds for a smooth strictly
h-convex curve in $\mathbb{H}^{2}$ by using horospherical Gauss map.

Motivated by the work [12, 14], we provide a new proof of inequality (1.8) for
h-convex curves in $\mathbb{H}^{2}$ by using the flow (1.3) and the
convergence result (Theorem 1.2).

######  Theorem 1.4.

Let $\gamma$ be a smooth h-convex closed curve enclosing a domain $\Omega$ in
$\mathbb{H}^{2}$. There holds

$\int_{\gamma}(\Phi-u)(\kappa-1)\mathrm{d}s+(L[\gamma]-A[\gamma])\geqslant\dfrac{1}{2\pi}(L[\gamma]-A[\gamma])^{2}$
(1.9)

with equality if and only if $\gamma$ is a geodesic circle centered at the
origin. Here $\Phi(r)=\cosh r-1$, $u=\langle V,\nu\rangle$ is the support
function of $\gamma$, $L[\gamma]$ denotes the length of $\gamma$, $A[\gamma]$
denotes the area of the domain $\Omega$ enclosed by $\gamma$.

We first use the flow (1.3) to prove inequality (1.9) for strictly h-convex
curves in $\mathbb{H}^{2}$. Then inequality (1.9) for h-convex curves follows
by an approximation argument following [7] and [13].

###  1.3. Stability results

We finally consider the stability of inequalities (1.9) for curves, (1.5) and
(1.8) for hypersurfaces characterized by the Hausdorff distance between the
hypersurface and an appropriate geodesic sphere. Recall that

$\mathrm{dist}(K,L)=\inf\\{\delta>0:K\subset B_{\delta}(L)\ \mbox{and}\
L\subset B_{\delta}(K)\\}$ (1.10)

is the Hausdorff distance between two compact sets $K$ and $L$.

For $n=1$, we will use Theorem 1.2 to prove the stability of inequality (1.9)
for strictly h-convex curves.

######  Theorem 1.5.

Let $\gamma$ be a smooth strictly h-convex closed curve in $\mathbb{H}^{2}$
enclosing a domain $\Omega$. Regard
$\gamma=\\{(\rho(\theta),\theta):\theta\in\mathbb{S}^{1}\\}$ as a graph on
$\mathbb{S}^{1}$. There exists a constant
$C(\rho,\max\limits_{\gamma}|\rho_{\theta}|,\min\limits_{\gamma}\kappa)$ and a
geodesic circle $\gamma_{\mathbb{H}}$, such that

$\mathrm{dist}(\gamma,\gamma_{\mathbb{H}})\leqslant
Cf\left[\int_{\gamma}(\Phi-u)(\kappa-1)\mathrm{d}s+(L[\gamma]-A[\gamma])-\dfrac{1}{2\pi}(L[\gamma]-A[\gamma])^{2}\right],$
(1.11)

where $f(s)=s^{\frac{1}{2}}+s^{\frac{1}{6}}\ (s\geqslant 0)$, and
$\mathrm{dist}(\gamma,\gamma_{\mathbb{H}})$ denotes the Hausdorff distance
between $\gamma$ and $\gamma_{\mathbb{H}}$.

The proof relies on the Poincaré inequality, which is used to characterize the
distance between the curve and a geodesic circle.

In [17], Sahjwani and Scheuer used flow (1.1) to prove the stability of
quermassintegral inequalities (1.4) when $l=k-1$ for h-convex domains in
$\mathbb{H}^{n+1}\ (n\geqslant 2)$. Their proof relied on the following nearly
umbilical theorem.

######  Theorem 1.6 ([16, Theorem 1.2]).

Let $n\geqslant 2$, $p\in(1,\infty)$. There exists $\varepsilon(n,p)>0$, if
$\Sigma$ is a closed convex hypersurface in $\mathbb{R}^{n+1}$ satisfying

$|\Sigma|=|\mathbb{S}^{n}|,\ \
\int_{\Sigma}|\mathring{A}|^{p}\mathrm{d}\mu\leqslant\varepsilon,$ (1.12)

then there exist a smooth parametrization
$\varphi:\mathbb{S}^{n}\to\Sigma\subset\mathbb{R}^{n+1}$ and a point
$\mathcal{P}\in\mathbb{R}^{n+1}$ such that the following estimate holds:

$\|\varphi-\mathrm{id}-\mathcal{P}\|_{W^{2,p}(\mathbb{S}^{n})}\leqslant
C(n,p)\|\mathring{A}\|_{L^{p}(\Sigma)},$ (1.13)

where $\mathrm{id}:\mathbb{S}^{n}\to\mathbb{S}^{n}$ is the identity map,
$\mathring{A}$ is the trace free Weingarten matrix of $\Sigma$.

For $n\geqslant 2$, inspired by the work [17], we will use the flow (1.2) with
the convergence result (Theorem 1.1) and the nearly umbilical result (Theorem
1.6) to discuss the stability of inequalities (1.5) and (1.8) for strictly
h-convex hypersurfaces.

######  Theorem 1.7.

Let $M$ be a smooth strictly h-convex closed hypersurface in
$\mathbb{H}^{n+1}\ (n\geqslant 2)$ enclosing a domain $\Omega$. For any
$1\leqslant k\leqslant n-1$, there exist a constant
$C=C(\rho_{-}(\Omega),\mathrm{dist}(\tilde{\kappa},\partial\Gamma_{k}^{+}),M,n,k)$
and a geodesic sphere $S_{\mathbb{H}}$, such that

$\mathrm{dist}(\partial\Omega,S_{\mathbb{H}})\leqslant
Cf\left[\tilde{W}_{k+1}(\Omega)-\tilde{f}_{k+1}\circ\tilde{f}_{k}^{-1}\left(\tilde{W}_{k}(\Omega)\right)\right]$
(1.14)

and

$\mathrm{dist}(\partial\Omega,S_{\mathbb{H}})\leqslant
Cf\left[\int_{M}(\Phi-u)E_{k}(\tilde{\kappa})\mathrm{d}\mu+(n-2k)\tilde{W}_{k}(\Omega)-\tilde{h}_{k}\circ\tilde{f}_{k}^{-1}\left(\tilde{W}_{k}(\Omega)\right)\right]$
(1.15)

hold, where $f(s)=s^{\frac{1}{2}}+s^{\frac{1}{4}}\ (s\geqslant 0)$,
$\mathrm{dist}(\partial\Omega,S_{\mathbb{H}})$ is the Hausdorff distance
between $\partial\Omega$ and $S_{\mathbb{H}}$.

The paper is organized as follows. In Section 2, we will first gather some
useful properties of elementary symmetric functions, quermassintegrals and
modified quermassintegrals. Then we summarize important formulas for
hypersurfaces in hyperbolic space and general evolution equations of curvature
flows. In Section 3, we will prove Theorem 1.2, the long time existence and
convergence results for the curve flow (1.3). In Section 4, we will prove
geometric inequalities Theorem 1.4 and include the complete proof of Theorem
1.3. In Section 5, we will prove stability results Theorems 1.5 and 1.7.

######  Acknowledgements.

The authors would like to thank Professor Yong Wei for his helpful discussions
and support. The authors also appreciate Dr. Botong Xu for his advice on
improving this work. This work is supported by National Key Research and
Development Program of China 2021YFA1001800.

##  2\. Preliminaries

In this section, we will make some preparations for our proof, including
properties of elementary symmetric functions, definitions for
quermassintegrals and modified quermassintegrals, some formulas for
hypersurfaces in hyperbolic space and general evolution equations for
curvature flows.

###  2.1. Elementary symmetric functions

Here we review some properties of elementary symmetric functions. For integer
$k=1,2,\cdots,n$, for any point
$\kappa=(\kappa_{1},\kappa_{2},\cdots,\kappa_{n})\in\mathbb{R}^{n}$, the
normalized $k$th elementary symmetric polynomial $E_{k}(\kappa)$ is defined by

$E_{k}(\kappa)={n\choose k}^{-1}\sum\limits_{1\leqslant
i_{1}<i_{2}<\cdots<i_{k}\leqslant
n}\kappa_{i_{1}}\kappa_{i_{2}}\cdots\kappa_{i_{k}}$ (2.1)

and has the convention that $E_{0}=1$ and $E_{k}(\kappa)=0$ for $k>n$. The
definition can be extended to symmetric matrices. Let $A=(a_{ij})$ be a
$n\times n$ symmetric matrix, denote $\kappa=\kappa(A)$ as the eigenvalue
vector of $A$. Define $E_{k}(A)=E_{k}(\kappa(A))$, and we have

$E_{k}(A)=\dfrac{(n-m)!}{n!}\delta_{i_{1}i_{2}\cdots i_{k}}^{j_{1}j_{2}\cdots
j_{k}}a_{i_{1}j_{1}}a_{i_{2}j_{2}}\cdots a_{i_{k}j_{k}},\ \ k=1,2,\cdots,n.$
(2.2)

We include some useful properties here but omit their proofs. For more
details, we may refer to [6, Section 2].

######  Lemma 2.1.

Denote $\dot{E}^{ij}_{k}=\dfrac{\partial E_{k}(A)}{\partial a_{ij}}$, then we
have the following identities.

$\displaystyle\sum\limits_{i,j}\dot{E}^{ij}_{k}\delta^{j}_{i}=kE_{k-1}(A),$
$\displaystyle\sum\limits_{i,j}\dot{E}^{ij}_{k}a_{ij}=kE_{k}(A),$
$\displaystyle\sum\limits_{i,j}\dot{E}^{ij}_{k}(A^{2})_{ij}=nE_{1}(A)E_{k}(A)-(n-k)E_{k+1}(A).$
(2.3)

For each $k=1,2,\cdots,n$, let

$\Gamma_{k}^{+}=\\{x\in\mathbb{R}^{n}:E_{i}(x)>0,\ i=1,2,\cdots,k\\}$

be the Garding cone.

######  Lemma 2.2 (Newton-MacLaurin inequality).

If $\kappa\in\Gamma_{k}^{+}$, we have the following inequality

$E_{k+1}(\kappa)E_{l-1}(\kappa)\leqslant E_{l}(\kappa)E_{k}(\kappa),\ \
1\leqslant l\leqslant k.$ (2.4)

Equality holds if and only if $\kappa_{1}=\kappa_{2}=\cdots=\kappa_{n}$.

The following divergence free property will be used frequently in integration
by parts.

######  Lemma 2.3.

If $A=(a_{ij})$ is a Codazzi tensor on Riemannian manifold $(M^{n},g)$, let
$\\{e_{1},e_{2},\cdots,e_{n}\\}$ be a local frame, then

$\sum\limits_{j}\nabla_{j}\dot{E}_{k}^{ij}(A)=0.$ (2.5)

###  2.2. Quermassintegrals and modified quermassintegrals

Recall that for a convex domain $\Omega$ in hyperbolic space
$\mathbb{H}^{n+1}$, the quermassintegrals of $\Omega$ are defined as follows
(see [19, Definition 2.1]):

$W_{k}(\Omega)=\dfrac{\omega_{k-1}\cdots\omega_{0}}{\omega_{n-1}\cdots\omega_{n-k}}\int_{\mathcal{L}_{k}}\chi(L_{k}\cap\Omega)\mathrm{d}L_{k},\
k=1,2,\cdots,n,$ (2.6)

where $\omega_{k}=|\mathbb{S}^{k}|$ denotes the area of $k$-dimensional
sphere, $\mathcal{L}_{k}$ is the space of $k$-dimensional totally geodesic
subspaces $L_{k}$ in $\mathbb{H}^{n+1}$. The function $\chi$ is defined to be
1 if $L_{k}\cap\Omega\neq\varnothing$ and to be 0 otherwise. Furthermore, we
set

$W_{0}(\Omega)=|\Omega|,\ \
W_{n+1}(\Omega)=|\mathbb{B}^{n+1}|=\dfrac{\omega_{n}}{n+1}.$ (2.7)

If the boundary of $\Omega$ is smooth, the quermassintegrals and the curvature
integrals of a smooth domain $\Omega$ in $\mathbb{H}^{n+1}$ are related by the
following equations:

$\displaystyle W_{1}(\Omega)=\dfrac{|\partial\Omega|}{n},$
$\displaystyle\int_{\partial\Omega}E_{k}(\kappa)\mathrm{d}\mu=(n-k)W_{k+1}(\Omega)+kW_{k-1}(\Omega),\
k=1,2,\cdots,n-1,$ (2.8)
$\displaystyle\int_{\partial\Omega}E_{n}(\kappa)\mathrm{d}\mu=\omega_{n}+nW_{n-1}(\Omega).$
(2.9)

Recently, Andrews-Chen-Wei [1] introduced modified quermassintegrals for
h-convex domains in $\mathbb{H}^{n+1}$:

$\tilde{W}_{k}(\Omega):=\sum\limits_{i=0}^{k}(-1)^{k-i}{k\choose
i}W_{i}(\Omega),\ \ k=0,1,\cdots,n.$ (2.10)

We know that $\tilde{W}_{k}(\Omega)$ is a linear combination of the
quermassintegrals of $\Omega$. It can be verified that (see [14, Lemma 10.1])

$\displaystyle\int_{\partial\Omega}E_{k}(\tilde{\kappa})\mathrm{d}\mu$
$\displaystyle=$
$\displaystyle(n-k)\tilde{W}_{k+1}(\Omega)+(n-2k)\tilde{W}_{k}(\Omega),\
k=1,2,\cdots,n-1,$ (2.11)
$\displaystyle\int_{\partial\Omega}E_{n}(\tilde{\kappa})\mathrm{d}\mu$
$\displaystyle=$ $\displaystyle\omega_{n}-n\tilde{W}_{n}(\Omega),$ (2.12)

where $\tilde{\kappa}_{i}=\kappa_{i}-1$ are the shifted principal curvatures
of the hypersurface $\partial\Omega$. These new quermassintegrals satisfy the
following nice variational formula (see [1, Lemma 2.4])

$\dfrac{\partial}{\partial
t}\tilde{W}_{k}(\Omega_{t})=\int_{\partial\Omega_{t}}\eta
E_{k}(\tilde{\kappa})\mathrm{d}\mu_{t}$ (2.13)

along any variation in the direction of outward normal with speed function
$\eta$.

###  2.3. Hypersurfaces in hyperbolic space

In this paper, we consider $\mathbb{H}^{n+1}=\mathbb{S}^{n}\times[0,\infty)$
as a warped product manifold equipped with metric

$\overline{g}=\mathrm{d}r^{2}+\phi^{2}(r)g_{\mathbb{S}^{n}},$ (2.14)

where $\phi(r)=\sinh r$, $r\in[0,\infty)$, $g_{\mathbb{S}^{n}}$ denotes the
standard metric on $\mathbb{S}^{n}$. Let

$\Phi(r)=\int_{0}^{r}\phi(s)\mathrm{d}s=\cosh r-1.$ (2.15)

It’s well known that $V=\overline{\nabla}\Phi=\phi(r)\partial_{r}$ is a
conformal Killing field on $\mathbb{H}^{n+1}$, that is

$\overline{\nabla}V=\phi^{\prime}(r)\overline{g}.$ (2.16)

Let $n=1$. Let $\gamma=X(\theta)$ be a smooth closed curve with its position
vector $X:\mathbb{S}^{1}\to\mathbb{H}^{2}$. Denote
$v=|X_{\theta}|=\sqrt{\phi^{2}(r)+r_{\theta}^{2}}$, and define the arc-length
parameter $s$ by

$\dfrac{\partial}{\partial s}=\dfrac{1}{v}\dfrac{\partial}{\partial\theta},$

then $T=\dfrac{\partial X}{\partial s}=\dfrac{1}{v}X_{\theta}$ is the unit
tangent vector. Denote $\nu$ as the unit outward normal vector and $\kappa$ as
the curvature of $\gamma$, we have the Frenet formula

$\overline{\nabla}_{T}T=-\kappa\nu,\ \ \overline{\nabla}_{T}\nu=\kappa T.$
(2.17)

It has been verified in [13, Proposition 3.2] that the following formulas hold
for smooth curves, which can be regarded as the special case for higher
dimensions.

######  Lemma 2.4.

Let $\gamma$ be a smooth closed curve in $\mathbb{H}^{2}$ with $s$ be its arc-
length parameter. Let $u=\langle V,\nu\rangle$ be its support function,
$\kappa$ be its curvature, then

$\displaystyle\dfrac{\partial\phi^{\prime}}{\partial s}=\langle V,T\rangle,\ \
\dfrac{\partial^{2}\phi^{\prime}}{\partial s^{2}}=\phi^{\prime}-\kappa u,$
(2.18) $\displaystyle\dfrac{\partial u}{\partial s}=\kappa\langle V,T\rangle,\
\,\dfrac{\partial^{2}u}{\partial s^{2}}=\dfrac{\partial\kappa}{\partial
s}\langle V,T\rangle+\kappa(\phi^{\prime}-\kappa u).$ (2.19)

Moreover,

$\int_{\gamma}\left(\phi^{\prime}(r)-\kappa u\right)\mathrm{d}s=0.$ (2.20)

Let $n\geqslant 2$. Let $M$ be a smooth closed hypersurface in
$\mathbb{H}^{n+1}$ with induced metric $g$ and unit outward normal vector
$\nu$. Let $\\{x^{1},x^{2},\cdots,x^{n}\\}$ be a local coordinate system on
$M$, we denote $g_{ij}=g(\partial_{i},\partial_{j})$ and its second
fundamental form
$h_{ij}=\langle\overline{\nabla}_{\partial_{i}}\nu,\partial_{j}\rangle$. The
Weingarten matrix is denoted by $A=(h^{i}_{j})$, where
$h^{i}_{j}=g^{ik}h_{kj}$, and $(g^{ij})$ is the inverse matrix of $(g_{ij})$.
The principal curvatures $\kappa=(\kappa_{1},\kappa_{2},\cdots,\kappa_{n})$
are eigenvalues of the Weingarten matrix $A$. The connection on $M$ is denoted
by $\nabla$.

The following formulas hold for smooth hypersurfaces in $\mathbb{H}^{n+1}$,
see [8, Lemmas 2.2 and 2.6].

######  Lemma 2.5.

Let $(M,g)$ be a smooth hypersurface in $\mathbb{H}^{n+1}$ with local frame
$\\{\partial_{1},\cdots,\partial_{n}\\}$, then $\Phi|_{M}$ satisfies

$\nabla_{i}\Phi=\langle V,e_{i}\rangle,\ \
\nabla_{j}\nabla_{i}\Phi=\phi^{\prime}g_{ij}-uh_{ij},$ (2.21)

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

$\nabla_{i}u=\langle V,e_{l}\rangle h_{i}^{l},\ \
\nabla_{j}\nabla_{i}u=\langle V,\nabla
h_{ij}\rangle+\phi^{\prime}h_{ij}-u(h^{2})_{ij},$ (2.22)

where $(h^{2})_{ij}=h_{ik}h^{k}_{j}$.

We also have the following Minkowski formulas.

######  Lemma 2.6 ([8, Proposition 2.5]).

Let $M$ be a smooth closed hypersurface in $\mathbb{H}^{n+1}$. Then

$\int_{M}\phi^{\prime}E_{k}(\kappa)\mathrm{d}\mu=\int_{M}uE_{k+1}(\kappa)\mathrm{d}\mu,\
\ k=0,1,\cdots,n-1,$ (2.23)

where $E_{k}(\kappa)$ is the $k$th mean curvature of $M$.

For shifted $k$th mean curvature, the Minkowski type formulas also hold.

######  Lemma 2.7 ([10, Lemma 2.6]).

Let $M$ be a smooth closed hypersurface in $\mathbb{H}^{n+1}$. Let
$\tilde{\kappa}_{i}=\kappa_{i}-1$ be the shifted principal curvatures of $M$,
then

$\int_{M}(\phi^{\prime}-u)E_{k}(\tilde{\kappa})\mathrm{d}\mu=\int_{M}uE_{k+1}(\tilde{\kappa})\mathrm{d}\mu,\
\ k=0,1,\cdots,n-1.$ (2.24)

###  2.4. General evolution equations

Let $X:M^{n}\times[0,T)\to\mathbb{H}^{n+1}$ be a family of immersed
hypersurfaces satisfying the outward normal variation with speed $\eta$

$\partial_{t}X(x,t)=\eta(x,t)\nu(x,t).$ (2.25)

Denote $M_{t}=X_{t}(M)$ and let
$\\{\partial_{1},\partial_{2},\cdots,\partial_{n}\\}$ be local frame on
$M_{t}$. Then the induced metric $g_{ij}$, the unit outward normal vector
$\nu$, the second fundamental form $h_{ij}$, the weingarten tensor
$h^{i}_{j}$, the induced area element $\mathrm{d}\mu_{t}$ evolve as follows
(see [8, Lemma 3.1]):

$\displaystyle\dfrac{\partial}{\partial t}g_{ij}$ $\displaystyle=$
$\displaystyle 2\eta h_{ij},$ (2.26) $\displaystyle\dfrac{\partial}{\partial
t}\nu$ $\displaystyle=$ $\displaystyle-\nabla\eta,$ (2.27)
$\displaystyle\dfrac{\partial}{\partial t}h_{ij}$ $\displaystyle=$
$\displaystyle-\nabla_{j}\nabla_{i}\eta+\eta(h^{2})_{ij}+\eta g_{ij},$ (2.28)
$\displaystyle\dfrac{\partial}{\partial t}h^{i}_{j}$ $\displaystyle=$
$\displaystyle-\nabla_{j}\nabla^{i}\eta-\eta(h^{2})^{i}_{j}+\eta\delta^{i}_{j},$
(2.29) $\displaystyle\dfrac{\partial}{\partial t}\mathrm{d}\mu_{t}$
$\displaystyle=$ $\displaystyle\eta H\mathrm{d}\mu_{t},$ (2.30)

where $H=nE_{1}(\kappa)$ is the mean curvature of $M_{t}$, $\nabla$ denotes
the Levi-Civita connection with respect to the induced metric $g_{ij}$ on
$M_{t}$.

Moreover, along the flow (2.25), $\Phi(r)=\cosh r-1$ and the support function
$u$ evolve as follows:

$\displaystyle\dfrac{\partial}{\partial t}\Phi$ $\displaystyle=$
$\displaystyle\left\langle\overline{\nabla}\Phi,\partial_{t}X\right\rangle=\langle
V,\eta\nu\rangle=\eta u,$ (2.31) $\displaystyle\dfrac{\partial}{\partial t}u$
$\displaystyle=$
$\displaystyle\langle\overline{\nabla}_{\partial_{t}X}V,\nu\rangle+\langle
V,\partial_{t}\nu\rangle=\eta\phi^{\prime}(r)-\langle V,\nabla\eta\rangle.$
(2.32)

Let $\Omega_{t}$ be the domain enclosed by $M_{t}$, then the volume of
$\Omega_{t}$ denoted by $\mathrm{Vol}(\Omega_{t})$ evolves by

$\dfrac{\partial}{\partial
t}\mathrm{Vol}(\Omega_{t})=\int_{M_{t}}\eta\mathrm{d}\mu_{t}.$ (2.33)

Now we would like to derive the evolution equation for the shifted $k$th mean
curvature $E_{k}(\tilde{\kappa})$ (see also [10, Section 9.1]). Set
$S^{i}_{j}=h^{i}_{j}-\delta^{i}_{j}$, then the shifted weingarten tensor
satisfies

$\displaystyle\dfrac{\partial}{\partial t}S^{i}_{j}$ $\displaystyle=$
$\displaystyle-\nabla_{j}\nabla^{i}\eta-\eta\left((S^{2})^{i}_{j}+2S^{i}_{j}\right).$
(2.34)

Denote $\dot{E}_{k}^{ij}(\tilde{\kappa})=\dfrac{\partial
E_{k}(\tilde{\kappa})}{\partial S^{i}_{j}}$, then by Lemma 2.1, we have

$\displaystyle\dot{E}_{k}^{ij}(\tilde{\kappa})\delta^{i}_{j}$ $\displaystyle=$
$\displaystyle kE_{k-1}(\tilde{\kappa}),$ (2.35)
$\displaystyle\dot{E}_{k}^{ij}(\tilde{\kappa})S^{i}_{j}$ $\displaystyle=$
$\displaystyle kE_{k}(\tilde{\kappa}),$ (2.36)
$\displaystyle\dot{E}_{k}^{ij}(\tilde{\kappa})\left(S^{2}\right)^{i}_{j}$
$\displaystyle=$ $\displaystyle
nE_{1}(\tilde{\kappa})E_{k}(\tilde{\kappa})-(n-k)E_{k+1}(\tilde{\kappa}).$
(2.37)

From Lemma 2.5, we get

$\displaystyle\nabla_{j}\nabla^{i}\Phi$ $\displaystyle=$
$\displaystyle(\phi^{\prime}-u)\delta^{i}_{j}-uS^{i}_{j},$ (2.38)
$\displaystyle\nabla_{j}\nabla^{i}u$ $\displaystyle=$ $\displaystyle\langle
V,\nabla
S^{i}_{j}\rangle-u\left((S^{2})^{i}_{j}+2S^{i}_{j}+\delta^{i}_{j}\right)+\phi^{\prime}(S^{i}_{j}+\delta^{i}_{j}),$
(2.39) $\displaystyle\nabla_{j}\nabla^{i}(\Phi-u)$ $\displaystyle=$
$\displaystyle-\phi^{\prime}S^{i}_{j}-\langle V,\nabla
S^{i}_{j}\rangle+u\left((S^{2})^{i}_{j}+S^{i}_{j}\right).$ (2.40)

Combining the above identities, we obtain

$\dot{E}_{k}^{ij}(\tilde{\kappa})\nabla_{j}\nabla^{i}\Phi=k\left((\phi^{\prime}-u)E_{k-1}(\tilde{\kappa})-uE_{k}(\tilde{\kappa})\right),$
(2.41)

and the shifted $k$th mean curvature evolves as follows:

$\displaystyle\dfrac{\partial}{\partial t}E_{k}(\tilde{\kappa})$
$\displaystyle=$
$\displaystyle-\dot{E}_{k}^{ij}(\tilde{\kappa})\nabla_{j}\nabla^{i}\eta-\left(nE_{1}(\tilde{\kappa})E_{k}(\tilde{\kappa})-(n-k)E_{k+1}(\tilde{\kappa})\right)\eta-2kE_{k}(\tilde{\kappa})\eta.$
(2.42)

##  3\. Flow for strictly h-convex curves in hyperbolic space

In this section, we will prove Theorem 1.2, the long time existence and
convergence results for the flow (1.3).

Recall that Heintze-Karcher type inequality for h-convex curves is obtained in
[14, Proposition 8.2]. Let $\gamma$ be a smooth strictly h-convex curve in
$\mathbb{H}^{2}$ enclosing a domain $\Omega$, then

$\int_{\partial\Omega}\dfrac{\phi^{\prime}(r)-u}{\kappa-1}-u\mathrm{d}s\geqslant
0.$ (3.1)

Equality holds in (3.1) if and only if $\Omega$ is a geodesic ball. This is an
important ingredient in our proof.

We will first verify that the difference between the length of curve and the
area of the domain enclosed by the curve $L[\gamma_{t}]-A[\gamma_{t}]$
preserves along the flow (1.3).

######  Lemma 3.1.

Along the flow (1.3), the smooth closed curve $\gamma_{t}$ satisfies

$\displaystyle\dfrac{\partial}{\partial
t}\left(L[\gamma_{t}]-A[\gamma_{t}]\right)=0,$ (3.2)

where $L[\gamma_{t}]$ is the length of $\gamma_{t}$, $A[\gamma_{t}]$ denotes
the area of the domain enclosed by $\gamma_{t}$.

###### Proof.

Set $\eta=\dfrac{\phi^{\prime}(r)-u}{\kappa-1}-u$. By the evolution equation
of length (2.30) and area (2.33), and Minkowski formula (2.20), we know

$\dfrac{\partial}{\partial
t}\left(L[\gamma_{t}]-A[\gamma_{t}]\right)=\int_{\gamma_{t}}\eta(\kappa-1)\mathrm{d}s=\int_{\gamma_{t}}\left(\phi^{\prime}(r)-\kappa
u\right)\mathrm{d}s=0,$ (3.3)

as desired. ∎

###  3.1. A priori estimates

We derive some a priori estimates for the flow (1.3).

######  Proposition 3.2 (Curvature Estimate).

Let $X_{0}$ be a strictly h-convex curve, then $\gamma_{t}$ is strictly
h-convex along the flow (1.3). Furthermore, we have the uniform curvature
estimate

$\min\kappa_{0}\leqslant\kappa(\cdot,t)\leqslant\max\kappa_{0},$ (3.4)

where $\kappa_{0}$ is the curvature of $X_{0}$.

###### Proof.

We calculate the evolution equation of curvature $\kappa(\cdot,t)$ for
$\gamma_{t}$. Let $s$ be the arc-length parameter. After a direct computation,
we get

$\displaystyle\dfrac{\partial\kappa}{\partial t}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\kappa-1}\left(-\dfrac{\partial^{2}\phi^{\prime}}{\partial
s^{2}}+\dfrac{\partial^{2}u}{\partial
s^{2}}\right)+\dfrac{2}{(\kappa-1)^{2}}\left(\dfrac{\partial\phi^{\prime}}{\partial
s}-\dfrac{\partial u}{\partial s}\right)\dfrac{\partial\kappa}{\partial
s}+\dfrac{\phi^{\prime}-u}{(\kappa-1)^{2}}\dfrac{\partial^{2}\kappa}{\partial
s^{2}}$ (3.5)
$\displaystyle-\dfrac{2(\phi^{\prime}-u)}{(\kappa-1)^{3}}\left(\dfrac{\partial\kappa}{\partial
s}\right)^{2}+\dfrac{\partial^{2}u}{\partial
s^{2}}-(\kappa^{2}-1)\left(\dfrac{\phi^{\prime}-u}{\kappa-1}-u\right).$

By using

$\displaystyle\dfrac{\partial\phi^{\prime}}{\partial s}=\langle V,T\rangle,\ \
\dfrac{\partial^{2}\phi^{\prime}}{\partial s^{2}}=\phi^{\prime}-\kappa u,\ \
\dfrac{\partial u}{\partial s}=\kappa\langle V,T\rangle,\ \
\dfrac{\partial^{2}u}{\partial s^{2}}=\dfrac{\partial\kappa}{\partial
s}\langle V,T\rangle+\kappa(\phi^{\prime}-\kappa u),$ (3.6)

we get

$\dfrac{\partial\kappa}{\partial
t}=\dfrac{\phi^{\prime}-u}{(\kappa-1)^{2}}\dfrac{\partial^{2}\kappa}{\partial
s^{2}}+\dfrac{\partial\kappa}{\partial s}\left[\dfrac{\partial}{\partial
s}\left(\dfrac{\phi^{\prime}-u}{(\kappa-1)^{2}}\right)+\langle
V,T\rangle\right].$ (3.7)

Note that $\phi^{\prime}-u>0$, by the maximum principle, we have

$\dfrac{\partial}{\partial
t}\max\limits_{x\in\mathbb{S}^{1}}\kappa(x,t)\leqslant 0,\
\dfrac{\partial}{\partial
t}\min\limits_{x\in\mathbb{S}^{1}}\kappa(x,t)\geqslant 0.$ (3.8)

Then (3.4) follows and $\gamma_{t}$ remains strictly h-convex. ∎

Note that h-convexity implies star-shapedness with respect to some point
inside the domain. In the following, we will regard the solution curve of the
flow (1.3) as the graph over $\mathbb{S}^{1}$,
$\gamma_{t}=\\{(\rho(\theta),\theta):\theta\in\mathbb{S}^{1}\\}$, where
$\rho:\mathbb{S}^{1}\to(0,\infty)$ is the radial distance. The unit outward
normal vector is

$\nu=\dfrac{1}{\sqrt{1+\rho_{\theta}^{2}\phi(\rho)^{-2}}}\left(\partial\rho-\rho_{\theta}\phi(\rho)^{-2}\partial_{\theta}\right),$
(3.9)

the curvature is

$\kappa=\dfrac{\phi(\rho)^{2}\phi^{\prime}(\rho)+2\rho_{\theta}^{2}\phi^{\prime}(\rho)-\rho_{\theta\theta}\phi(\rho)}{\left(\phi(\rho)^{2}+\rho_{\theta}^{2}\right)^{\frac{3}{2}}},$
(3.10)

the support function is

$u=\langle
V,\nu\rangle=\dfrac{\phi(\rho)}{\sqrt{1+\rho_{\theta}^{2}\phi(\rho)^{-2}}}.$
(3.11)

For a family of curves which can be regarded as the graph over
$\mathbb{S}^{1}$, we can write

$X(x,t)=(\rho(\psi(x,t),t),\psi(x,t)),$

where $\psi(\cdot,t):\mathbb{S}^{1}\to\mathbb{S}^{1}$ is a family of
diffeomorphism and $\rho(\cdot,t):\mathbb{S}^{1}\to(0,\infty)$ is the radial
distance. Then

$\dfrac{\partial X}{\partial
t}=\left(\dfrac{\partial\rho}{\partial\theta}\dfrac{\partial\psi}{\partial
t}+\dfrac{\partial\rho}{\partial
t}\right)\dfrac{\partial}{\partial\rho}+\dfrac{\partial\psi}{\partial
t}\dfrac{\partial}{\partial\theta}.$

By a direct computation,

$\left\langle\dfrac{\partial X}{\partial
t},\nu\right\rangle=\dfrac{1}{\sqrt{1+\rho_{\theta}^{2}\phi(\rho)^{-2}}}\dfrac{\partial\rho}{\partial
t}.$ (3.12)

Therefore, we express the flow (1.3) equivalently as the initial value
problem:

$\left\\{\begin{array}[]{ll}\dfrac{\partial\rho}{\partial
t}=\dfrac{\phi^{\prime}(\rho)\sqrt{1+\rho_{\theta}^{2}\phi(\rho)^{-2}}-\phi(\rho)}{\kappa-1}-\phi(\rho),&(\theta,t)\in\mathbb{S}^{1}\times[0,T),\\\
\rho(\theta,0)=\rho_{0}(\theta).\end{array}\right.$ (3.13)

In the following, we will use (3.13) to obtain some estimates.

######  Proposition 3.3 ($C^{0}$ Estimate).

Let $\rho(\cdot,t)\in\mathbb{S}^{1}\times[0,T)$ be the solution to (3.13),
then

$\min\limits_{\mathbb{S}^{1}}\rho(\cdot,0)\leqslant\rho(\cdot,t)\leqslant\max\limits_{\mathbb{S}^{1}}\rho(\cdot,0).$
(3.14)

###### Proof.

At the spatial extremal point of $\rho$, we have $\rho_{\theta}=0$. The
equation (3.13) then becomes

$\dfrac{\partial\rho}{\partial
t}=\dfrac{\phi^{\prime}(\rho)-\phi(\rho)}{\kappa-1}-\phi(\rho).$ (3.15)

At the maximum point of $\rho$, i.e.
$\rho(\theta,t)=\max\limits_{\mathbb{S}^{1}}\rho(\cdot,t)$, we have
$\kappa\geqslant\dfrac{\phi^{\prime}(\rho)}{\phi(\rho)}$. Thus

$\displaystyle\partial_{t}\rho_{\max}(t)\leqslant\dfrac{\phi^{\prime}-\phi}{\frac{\phi^{\prime}}{\phi}-1}-\phi=0.$
(3.16)

At the minimum point of $\rho$, i.e.
$\rho(\theta,t)=\min\limits_{\mathbb{S}^{1}}\rho(\cdot,t)$, we have
$\kappa\leqslant\dfrac{\phi^{\prime}(\rho)}{\phi(\rho)}$. Thus

$\displaystyle\partial_{t}\rho_{\min}(t)\geqslant\dfrac{\phi^{\prime}-\phi}{\frac{\phi^{\prime}}{\phi}-1}-\phi=0.$
(3.17)

The conclusion follows by the standard maximum principle. ∎

######  Proposition 3.4 ($C^{1}$ Estimate).

Along the flow (3.13), there exists a constant $C>0$ depending only on
$\rho_{0}$, such that

$\|\rho_{\theta}\|_{C^{0}(\mathbb{S}^{1}\times[0,T))}\leqslant C.$ (3.18)

###### Proof.

1\. Firstly, we prove that star-shapedness preserves along the flow (1.3). For
any $t\in[0,T)$, at the spatial minimum point $P$ of $u$, we have

$\dfrac{\partial u}{\partial s}=\kappa\langle V,T\rangle=0.$ (3.19)

Since $\gamma_{t}$ is strictly h-convex, that is $\kappa>1$, then $\langle
V,T\rangle=0$. This implies that $\dfrac{\partial}{\partial\rho}$ is parallel
to the unit outward normal vector $\nu$ at $P$. Hence

$u(P)=\langle V,\nu\rangle(P)=\phi(\rho)|_{P}.$ (3.20)

By Proposition 3.3, we have

$u\geqslant\min_{\gamma_{t}}\phi(\rho)>0.$ (3.21)

2\. Combining (3.11) and (3.21), we get

$\dfrac{|\rho_{\theta}|}{\phi(\rho)}\leqslant\sqrt{1+\rho_{\theta}^{2}\phi(\rho)^{-2}}\leqslant\dfrac{\max\limits_{\gamma_{t}}\phi(\rho)}{\min\limits_{\gamma_{t}}\phi(\rho)}\leqslant\dfrac{\max\limits_{\gamma_{0}}\phi}{\min\limits_{\gamma_{0}}\phi},$
(3.22)

where we have used that $\phi(\rho)=\sinh\rho$ is monotonically increasing in
the last inequality. Therefore,

$|\rho_{\theta}|\leqslant\dfrac{\max\limits_{\gamma_{0}}\phi^{2}}{\min\limits_{\gamma_{0}}\phi}:=C,$
(3.23)

where the constant $C$ depends only on $\rho_{0}$. ∎

###  3.2. Long time existence

By the above a priori estimates, we know that if $\rho$ is a positive
admissible solution to the flow (3.13) in $\mathbb{S}^{1}\times[0,T)$, then by
the expression (3.10) of $\kappa$, we have

$\|\rho\|_{C^{2}(\mathbb{S}^{1}\times[0,T))}\leqslant C,$ (3.24)

where $C$ depends only on $\rho_{0}$. By Krylov and Safanov’s theorem [11,
Section 5.5], we get $\|\rho\|_{C^{2,\alpha}}\leqslant C$, where $C$ depends
only on $\rho_{0}$. By Schauder’s estimate, we get the uniform $C^{k}$
estimate of $\rho$ ($k\geqslant 2$). Therefore, the flow (1.3) exists in
$t\in(0,\infty)$.

###  3.3. Smooth convergence

We first obtain subsequential convergence.

######  Lemma 3.5.

There exists a sequence $\\{t_{j}\\}$ such that when $t_{j}\to\infty$,
$Q(t_{j})$ monotonically decreasing converges to 0, where

$Q(t)=\int_{\gamma_{t}}\left(\dfrac{\phi^{\prime}(r)-u}{\kappa-1}-u\right)\mathrm{d}s,$
(3.25)

and $\gamma_{t_{j}}$ converges smoothly to a geodesic circle centered at the
origin.

###### Proof.

1\. Existence of $\gamma_{\infty}$. Note that $A^{\prime}(t)=Q(t)$, then

$\int_{0}^{t}Q(\tau)\mathrm{d}\tau=A(t)-A(0).$ (3.26)

By Heintze-Karcher type inequality (3.1), we know $Q(t)\geqslant 0$. By
Proposition 3.3, $A(t)\leqslant
2\pi\Phi(\max\limits_{\mathbb{S}^{1}}\rho_{0})$. Thus $Q(t)$ is a nonnegative
$L^{1}(0,\infty)$ function. Then by Arzelà-Ascoli theorem and uniform
estimates, there exists $\\{t_{j}\\}$ such that $\gamma_{t_{j}}$ converges
smoothly to $\gamma_{\infty}$, and the following holds:

$\int_{\gamma_{\infty}}\left(\dfrac{\phi^{\prime}-u}{\kappa-1}-u\right)\mathrm{d}s=0.$
(3.27)

By the rigidity of Heintze-Karcher type inequality proved in [14, Proposition
8.2], we know that $\gamma_{\infty}$ is a geodesic circle.

2\. We prove that $\gamma_{\infty}$ is a geodesic circle centered at the
origin. Let $\rho_{\infty}$ be the radius of $\gamma_{\infty}$, and its
curvature is $\dfrac{\phi^{\prime}(\rho_{\infty})}{\phi(\rho_{\infty})}$. We
argue by contradiction. Suppose $\gamma_{\infty}$ is not centered at the
origin, then at the spatial point where $\rho$ attains its maximum of
$\theta$,

$\dfrac{\partial\rho}{\partial
t}=\dfrac{\phi^{\prime}(\rho)-\phi(\rho)}{\dfrac{\phi^{\prime}(\rho_{\infty})}{\phi(\rho_{\infty})}-1}-\phi(\rho)=\dfrac{\phi(\rho_{\infty}-\rho)}{\phi^{\prime}(\rho_{\infty})-\phi(\rho_{\infty})}<0,$
(3.28)

which contradicts that $\gamma_{\infty}$ is the stationary solution to the
flow (1.3). ∎

Then we get smooth convergence.

######  Proposition 3.6.

There exists $\rho_{\infty}>0$ such that along the flow (3.13), when
$t\to\infty$, there holds
$\|\rho(\cdot,t)-\rho_{\infty}\|_{C^{k}(\mathbb{S}^{1})}\to 0$, $\forall\
k\in\mathbb{N}$.

###### Proof.

We have proved that there exist $t_{j}\to\infty$ and $\rho_{\infty}$ such that
$\|\rho(\cdot,t_{j})-\rho_{\infty}\|_{C^{k}(\mathbb{S}^{1})}\to 0,\ \forall\
k\in\mathbb{N}$. Since
$\rho_{\max}(t)=\max\limits_{\mathbb{S}^{1}}\rho(\theta,t)$ is monotonically
non-increasing, and
$\rho_{\min}(t)=\min\limits_{\mathbb{S}^{1}}\rho(\theta,t)$ is monotonically
non-decreasing, we have
$\|\rho(\cdot,t)-\rho_{\infty}\|_{C^{0}(\mathbb{S}^{1})}\to 0\ (t\to\infty)$.

We recall the special case of Gagliardo-Nirenberg interpolation inequality
[15]: For any positive integer $j$ and $k$, where $j<k$, and let
$\delta=\frac{j}{k}$, there exists a constant $C(j,k)$, such that for any
$f\in W^{k,2}(\mathbb{S}^{1})$,

$\left\|\dfrac{\partial^{j}}{\partial\theta^{j}}f\right\|_{L^{2}(\mathbb{S}^{1})}\leqslant\left\|\dfrac{\partial^{k}}{\partial\theta^{k}}f\right\|_{L^{2}(\mathbb{S}^{1})}^{\delta}\|f\|_{L^{2}(\mathbb{S}^{1})}^{1-\delta}.$
(3.29)

Choose $f=\rho(\theta,t)-\rho_{\infty}$, we have

$\left\|\dfrac{\partial^{j}}{\partial\theta^{j}}\rho(\theta,t)\right\|_{L^{2}(\mathbb{S}^{1})}\leqslant\left\|\dfrac{\partial^{k}}{\partial\theta^{k}}\rho(\theta,t)\right\|_{L^{2}(\mathbb{S}^{1})}^{\delta}\|\rho(\theta,t)-\rho_{\infty}\|_{L^{2}(\mathbb{S}^{1})}^{1-\delta}.$
(3.30)

For any positive integer $j$, and any $\varepsilon>0$, for a large enough time
$t$, by the boundedness of $\|\rho\|_{C^{k}(\mathbb{S}^{1})}$ and
$\|\rho(\cdot,t)-\rho_{\infty}\|_{C^{0}(\mathbb{S}^{1})}<\varepsilon$, we know
that
$\left\|\dfrac{\partial^{j}}{\partial\theta^{j}}\rho(\theta,t)\right\|_{L^{2}(\mathbb{S}^{1})}<C\varepsilon$.
Then Sobolev embedding theorem implies that
$\|\rho(\cdot,t)-\rho_{\infty}\|_{C^{j-1}(\mathbb{S}^{1})}<C\varepsilon$. This
shows that $\rho(\cdot,t)$ converges smoothly to $\rho_{\infty}$ when
$t\to\infty$. ∎

###  3.4. Exponential convergence

We have shown that when $t\to\infty$, $\gamma_{t}$ converges smoothly to a
geodesic circle centered at the origin with radius $\rho_{\infty}$. Therefore,
for each large enough time $t$, $\gamma_{t}$ lies in the neighborhood of
$\gamma_{\infty}$. We consider the linearization of flow (3.13) at
$\rho=\rho_{\infty}$ (See [13, Lemma 4.3]). Let

$\sigma(\theta,t)=\rho(\theta,t)-\rho_{\infty},$ (3.31)

then as $t\to\infty$, $\sigma(\theta,t)\to 0$. Denote $G[\rho(\theta,t)]$ as
the right hand side of (3.13), then

$\displaystyle\dfrac{\partial}{\partial t}\sigma(\theta,t)$ $\displaystyle=$
$\displaystyle\dfrac{\partial}{\partial
t}(\rho(\theta,t)-\rho_{\infty})=G[\rho(\theta,t)]$ (3.32) $\displaystyle=$
$\displaystyle
G[\rho_{\infty}]+DG|_{\rho_{\infty}}(\sigma(\theta,t))+O\left(\|\sigma(\theta,t)\|_{C^{2}(\mathbb{S}^{1})}^{2}\right)$
$\displaystyle=$
$\displaystyle\dfrac{1}{\phi^{\prime}(\rho_{\infty})-\phi(\rho_{\infty})}\dfrac{\partial^{2}}{\partial\theta^{2}}\sigma(\theta,t)+O\left(\|\sigma(\theta,t)\|_{C^{2}(\mathbb{S}^{1})}^{2}\right).$

Consider the eigenfunction of laplacian on $\mathbb{S}^{1}$ (see also [21,
Section 9]), i.e.

$\dfrac{\partial^{2}\phi_{k}(\theta)}{\partial\theta^{2}}=-\lambda_{k}\phi_{k}(\theta),$
(3.33)

where $0=\lambda_{0}<\lambda_{1}\leqslant\cdots\leqslant\lambda_{k}\to\infty$,
$\theta\in\mathbb{S}^{1}$, and $\\{\phi_{k}(\theta)\\}$ constitutes a set of
orthonormal basis in $L^{2}(\mathbb{S}^{1})$. Let

$\sigma(\theta,t)=\sum\limits_{k=0}^{\infty}\varphi_{k}(\theta,t),$ (3.34)

where $\varphi_{k}(\theta,t)=f_{k}(t)\phi_{k}(\theta)$, $f_{k}(t)\in
C^{\infty}(I)$, then

$\varphi_{0}=\dfrac{1}{2\pi}\int_{\mathbb{S}^{1}}\sigma(\theta,t)\mathrm{d}\theta.$
(3.35)

Therefore,

$\displaystyle\dfrac{\partial}{\partial
t}\int_{\mathbb{S}^{1}}\sigma(\theta,t)^{2}\mathrm{d}\theta$ $\displaystyle=$
$\displaystyle\int_{\mathbb{S}^{1}}2\sigma(\theta,t)\dfrac{\partial}{\partial
t}\sigma(\theta,t)\mathrm{d}\theta$ (3.36) $\displaystyle\leqslant$
$\displaystyle\dfrac{2}{\phi^{\prime}(\rho_{\infty})-\phi(\rho_{\infty})}\int_{\mathbb{S}^{1}}\sigma(\theta,t)\dfrac{\partial^{2}}{\partial\theta^{2}}\sigma(\theta,t)\mathrm{d}\theta+C\|\sigma(\theta,t)\|_{C^{2}(\mathbb{S}^{1})}^{2}\int_{\mathbb{S}^{1}}|\sigma(\theta,t)|\mathrm{d}\theta$
$\displaystyle=$
$\displaystyle\dfrac{2}{\phi^{\prime}(\rho_{\infty})-\phi(\rho_{\infty})}\int_{\mathbb{S}^{1}}-\sum\limits_{k\geqslant
1}\lambda_{k}\varphi_{k}^{2}(\theta,t)\mathrm{d}\theta+C\|\sigma(\theta,t)\|^{2}_{C^{2}(\mathbb{S}^{1})}\int_{\mathbb{S}^{1}}|\sigma(\theta,t)|\mathrm{d}\theta$
$\displaystyle=$
$\displaystyle-\dfrac{2}{\phi^{\prime}(\rho_{\infty})-\phi(\rho_{\infty})}\int_{\mathbb{S}^{1}}\sum\limits_{k\geqslant
1}\lambda_{1}\varphi_{k}^{2}(\theta,t)\mathrm{d}\theta$
$\displaystyle-\dfrac{2}{\phi^{\prime}(\rho_{\infty})-\phi(\rho_{\infty})}\int_{\mathbb{S}^{1}}\sum\limits_{k\geqslant
1}(\lambda_{k}-\lambda_{1})\varphi_{k}^{2}(\theta,t)\mathrm{d}\theta$
$\displaystyle+C\|\sigma(\theta,t)\|^{2}_{C^{2}(\mathbb{S}^{1})}\int_{\mathbb{S}^{1}}|\sigma(\theta,t)|\mathrm{d}\theta$
$\displaystyle\leqslant$
$\displaystyle-\dfrac{2\lambda_{1}}{\phi^{\prime}(\rho_{\infty})-\phi(\rho_{\infty})}\int_{\mathbb{S}^{1}}\sum\limits_{k\geqslant
0}\varphi_{k}^{2}(\theta,t)\mathrm{d}\theta+\dfrac{2\lambda_{1}}{\phi^{\prime}(\rho_{\infty})-\phi(\rho_{\infty})}\int_{\mathbb{S}^{1}}\varphi_{0}^{2}(\theta,t)\mathrm{d}\theta$
$\displaystyle+C\|\sigma(\theta,t)\|^{2}_{C^{2}(\mathbb{S}^{1})}\int_{\mathbb{S}^{1}}|\sigma(\theta,t)|\mathrm{d}\theta$
$\displaystyle=$
$\displaystyle-\dfrac{2\lambda_{1}}{\phi^{\prime}(\rho_{\infty})-\phi(\rho_{\infty})}\int_{\mathbb{S}^{1}}\sigma(\theta,t)^{2}\mathrm{d}\theta+\underbrace{\dfrac{\lambda_{1}}{\pi\left(\phi^{\prime}(\rho_{\infty})-\phi(\rho_{\infty})\right)}\left(\int_{\mathbb{S}^{1}}\sigma(\theta,t)\mathrm{d}\theta\right)^{2}}_{I}$
$\displaystyle+\underbrace{C\|\sigma(\theta,t)\|^{2}_{C^{2}(\mathbb{S}^{1})}\int_{\mathbb{S}^{1}}|\sigma(\theta,t)|\mathrm{d}\theta}_{II}.$

By Lemma 3.1, we know that $[L-A](\rho(\theta,t))=[L-A](\rho_{\infty})$. We
claim that: There exists a constant $C$ such that

$\left(\int_{\mathbb{S}^{1}}\sigma(\theta,t)\mathrm{d}\theta\right)^{2}\leqslant
C\left(\int_{\mathbb{S}^{1}}\|\sigma(\theta,t)\|^{2}_{C^{2}(\mathbb{S}^{1})}\mathrm{d}\theta\right)^{2}\leqslant
C\|\sigma(\theta,t)\|_{C^{2}(\mathbb{S}^{1})}^{4}.$ (3.37)

Indeed, we use the Taylor expansion of
$[L-A](\rho(\theta,t))=[L-A](\sigma(\theta,t)+\rho_{\infty})$ at
$\rho_{\infty}$ to obtain

$[L-A](\rho(\theta,t))=[L-A](\rho_{\infty})+\left(\phi^{\prime}(\rho_{\infty})-\phi(\rho_{\infty})\right)\int_{\mathbb{S}^{1}}\sigma(\theta,t)\mathrm{d}\theta+\int_{\mathbb{S}^{1}}O\left(\|\sigma(\theta,t)\|_{C^{2}(\mathbb{S}^{1})}^{2}\right)\mathrm{d}\theta.$
(3.38)

This implies

$\left(\phi^{\prime}(\rho_{\infty})-\phi(\rho_{\infty})\right)\int_{\mathbb{S}^{1}}\sigma(\theta,t)\mathrm{d}\theta+\int_{\mathbb{S}^{1}}O\left(\|\sigma(\theta,t)\|_{C^{2}(\mathbb{S}^{1})}^{2}\right)\mathrm{d}\theta=0,$
(3.39)

and thus there exists a constant $C$ such that

$\left(\int_{\mathbb{S}^{1}}\sigma(\theta,t)\mathrm{d}\theta\right)^{2}\leqslant
C\left(\int_{\mathbb{S}^{1}}\|\sigma(\theta,t)\|^{2}_{C^{2}(\mathbb{S}^{1})}\mathrm{d}\theta\right)^{2}\leqslant
C\|\sigma(\theta,t)\|_{C^{2}(\mathbb{S}^{1})}^{4}.$

In the following, we will choose $f=\sigma(\theta,t)$ in Gagliardo-Nirenberg
interpolation inequality (3.29) to estimate
$\|\sigma(\theta,t)\|_{C^{2}(\mathbb{S}^{1})}^{2}$. For any positive integer
$j,\ k$ satisfying $j<k$, let $\delta=\frac{j}{k}$. There exists a constant
$C(j,k)$ such that

$\left\|\dfrac{\partial^{j}}{\partial\theta^{j}}\sigma(\theta,t)\right\|_{L^{2}(\mathbb{S}^{1})}\leqslant\left\|\dfrac{\partial^{k}}{\partial\theta^{k}}\sigma(\theta,t)\right\|_{L^{2}(\mathbb{S}^{1})}^{\delta}\|\sigma(\theta,t)\|_{L^{2}(\mathbb{S}^{1})}^{1-\delta}.$
(3.40)

By Sobolev embedding theorem, we have

$\|\sigma(\theta,t)\|_{C^{2}(\mathbb{S}^{1})}\leqslant
C\|\sigma(\theta,t)\|_{W^{j,2}(\mathbb{S}^{1})},\ \ j\geqslant 3.$ (3.41)

Combining (3.40) and (3.41), taking $k$ large enough, we obtain that $\forall\
0<\varepsilon<1$, there exists a positive constant $C(\varepsilon)$ such that

$\|\sigma(\theta,t)\|_{C^{2}(\mathbb{S}^{1})}^{2}\leqslant
C(\varepsilon)\|\sigma(\theta,t)\|_{L^{2}(\mathbb{S}^{1})}^{1+\varepsilon}.$
(3.42)

Choose $\varepsilon=\dfrac{1}{2}$, we get

$\|\sigma(\theta,t)\|_{C^{2}(\mathbb{S}^{1})}^{2}\leqslant
C\|\sigma(\theta,t)\|_{L^{2}(\mathbb{S}^{1})}^{\frac{3}{2}}.$ (3.43)

Now from (3.37) and (3.43), we know

$\left(\int_{\mathbb{S}^{1}}\sigma(\theta,t)\mathrm{d}\theta\right)^{2}\leqslant
C\|\sigma(\theta,t)\|_{L^{2}(\mathbb{S}^{1})}^{3}=C\left(\int_{\mathbb{S}^{1}}\sigma(\theta,t)^{2}\mathrm{d}\theta\right)^{1+\frac{1}{2}}.$
(3.44)

By (3.43) and Hölder inequality, we obtain

$\displaystyle\|\sigma(\theta,t)\|_{C^{2}(\mathbb{S}^{1})}^{2}\int_{\mathbb{S}^{1}}|\sigma(\theta,t)|\mathrm{d}\theta\leqslant
C\|\sigma(\theta,t)\|_{L^{2}(\mathbb{S}^{1})}^{\frac{5}{2}}=C\left(\int_{\mathbb{S}^{1}}\sigma(\theta,t)^{2}\mathrm{d}\theta\right)^{1+\frac{1}{4}}.$
(3.45)

Substituting (3.44) and (3.45) into (3.36), we get

$\displaystyle\dfrac{\partial}{\partial
t}\int_{\mathbb{S}^{1}}\sigma(\theta,t)^{2}\mathrm{d}\theta$ $\displaystyle=$
$\displaystyle-\dfrac{2\lambda_{1}}{\phi^{\prime}(\rho_{\infty})-\phi(\rho_{\infty})}\int_{\mathbb{S}^{1}}\sigma(\theta,t)^{2}\mathrm{d}\theta+C\left(\int_{\mathbb{S}^{1}}\sigma(\theta,t)^{2}\mathrm{d}\theta\right)^{1+\frac{1}{2}}+C\left(\int_{\mathbb{S}^{1}}\sigma(\theta,t)^{2}\mathrm{d}\theta\right)^{1+\frac{1}{4}}$
$\displaystyle=$
$\displaystyle-\dfrac{2\lambda_{1}}{\phi^{\prime}(\rho_{\infty})-\phi(\rho_{\infty})}\int_{\mathbb{S}^{1}}\sigma(\theta,t)^{2}\mathrm{d}\theta\left(1-C^{\prime}\left(\int_{\mathbb{S}^{1}}\sigma(\theta,t)^{2}\mathrm{d}\theta\right)^{\frac{1}{2}}-C^{\prime\prime}\left(\int_{\mathbb{S}^{1}}\sigma(\theta,t)^{2}\mathrm{d}\theta\right)^{\frac{1}{4}}\right).$

Since $\int_{\mathbb{S}^{1}}\sigma(\theta,t)^{2}\mathrm{d}\theta\to 0\
(t\to\infty)$, for any $\delta>0$, there exists $t_{\delta}>0$ such that for
any $t>t_{\delta}$,

$\displaystyle\dfrac{\partial}{\partial
t}\int_{\mathbb{S}^{1}}\sigma(\theta,t)^{2}\mathrm{d}\theta\leqslant-\dfrac{2\lambda_{1}}{\phi^{\prime}(\rho_{\infty})-\phi(\rho_{\infty})}\int_{\mathbb{S}^{1}}\sigma(\theta,t)^{2}\mathrm{d}\theta\left(1-\delta\right).$
(3.46)

Choose $\delta=\frac{1}{2}$, we obtain

$\int_{\mathbb{S}^{1}}\sigma(\theta,t)^{2}\mathrm{d}\theta\leqslant
C\mathrm{e}^{-\frac{\lambda_{1}}{\phi^{\prime}(\rho_{\infty})-\phi(\rho_{\infty})}t}.$
(3.47)

By (3.40) and Sobolev embedding theorem:

$\|\sigma(\theta,t)\|_{C^{j-1}(\mathbb{S}^{1})}\leqslant
C\|\sigma(\theta,t)\|_{W^{j,2}(\mathbb{S}^{1})},\ \ j\geqslant 1,$ (3.48)

we obtain that

$\|\sigma(\theta,t)\|_{C^{k}(\mathbb{S}^{1})}\leqslant
C_{1}\mathrm{e}^{-C_{2}t},\ \ \forall\ k\geqslant 0,$ (3.49)

where $C_{1},\ C_{2}$ depends only on $\rho_{0},\ k$.

Above all, we complete the proof of Theorem 1.2.

##  4\. Geometric inequalities

In this section, we will prove geometric inequalities involving weighted
curvature integrals and modified quermassintegrals for h-convex domains.

###  4.1. Inequalities for curves

We will prove Theorem 1.4 in this subsection.

######  Lemma 4.1.

Along the flow (1.3), if the initial curve is strictly h-convex, then there
holds

$\dfrac{\partial}{\partial
t}\int_{\gamma_{t}}\left(\Phi-u\right)(\kappa-1)\mathrm{d}s\leqslant 0$ (4.1)

with equality if and only if $\gamma_{t}$ is a geodesic circle centered at the
origin.

###### Proof.

Along the flow $\partial_{t}X=\eta\nu$, we calculate that

$\displaystyle\dfrac{\partial}{\partial
t}\int_{\gamma_{t}}(\Phi-u)(\kappa-1)\mathrm{d}s$ $\displaystyle=$
$\displaystyle\int_{\gamma_{t}}\left(\partial_{t}\Phi-\partial_{t}u\right)(\kappa-1)\mathrm{d}s+\int_{\gamma_{t}}(\Phi-u)\partial_{t}\kappa\mathrm{d}s$
(4.2)
$\displaystyle+\int_{\gamma_{t}}(\Phi-u)(\kappa-1)\partial_{t}\mathrm{d}s.$

Note that

$\displaystyle\partial_{t}\Phi$ $\displaystyle=$
$\displaystyle\langle\overline{\nabla}\Phi,\partial_{t}X\rangle=\langle
V,\eta\nu\rangle=\eta u,$ (4.3) $\displaystyle\partial_{t}u$ $\displaystyle=$
$\displaystyle\langle\overline{\nabla}_{\partial_{t}X}V,\nu\rangle+\langle
V,\partial_{t}\nu\rangle$ (4.4) $\displaystyle=$
$\displaystyle\eta\langle\overline{\nabla}_{\nu}V,\nu\rangle-\langle
V,\dfrac{\partial\eta}{\partial s}T\rangle$ $\displaystyle=$
$\displaystyle\eta\phi^{\prime}-\dfrac{\partial\eta}{\partial
s}\dfrac{\partial\Phi}{\partial s},$

we get the evolution equation

$\displaystyle\partial_{t}\int_{\gamma_{t}}(\Phi-u)(\kappa-1)\mathrm{d}s$
$\displaystyle=$
$\displaystyle\int_{\gamma_{t}}\left[(u-\phi^{\prime})\eta+\dfrac{\partial\eta}{\partial
s}\dfrac{\partial\Phi}{\partial s}\right](\kappa-1)\mathrm{d}s$ (4.5)
$\displaystyle-\int_{\gamma_{t}}(\Phi-u)\left[\dfrac{\partial^{2}\eta}{\partial
s^{2}}+(\kappa^{2}-1)\eta\right]\mathrm{d}s$
$\displaystyle+\int_{\gamma_{t}}(\Phi-u)(\kappa-1)\kappa\eta\mathrm{d}s.$

Using integration by parts, we have

$\displaystyle\int_{\gamma_{t}}(\Phi-u)\dfrac{\partial^{2}\eta}{\partial
s^{2}}\mathrm{d}s=-\int_{\gamma_{t}}\left(\dfrac{\partial\Phi}{\partial
s}-\dfrac{\partial u}{\partial s}\right)\dfrac{\partial\eta}{\partial
s}\mathrm{d}s=\int_{\gamma_{t}}(\kappa-1)\dfrac{\partial\Phi}{\partial
s}\dfrac{\partial\eta}{\partial s}\mathrm{d}s.$ (4.6)

We obtain that

$\displaystyle\partial_{t}\int_{\gamma_{t}}(\Phi-u)(\kappa-1)\mathrm{d}s$
$\displaystyle=$
$\displaystyle\int_{\gamma_{t}}(u-\phi^{\prime})(\kappa-1)\eta\mathrm{d}s-\int_{\gamma_{t}}(\Phi-u)(\kappa-1)\eta\mathrm{d}s$
(4.7) $\displaystyle=$
$\displaystyle\int_{\gamma_{t}}(2u-2\Phi-1)(\kappa-1)\eta\mathrm{d}s,$

where we used $\phi^{\prime}=\Phi+1$. Substituting

$\eta=\dfrac{\phi^{\prime}-u}{\kappa-1}-u=\dfrac{\phi^{\prime}-\kappa
u}{\kappa-1}$

into (4.7), using (2.18) and integration by parts, we obtain that

$\displaystyle\partial_{t}\int_{\gamma_{t}}(\Phi-u)(\kappa-1)\mathrm{d}s$
$\displaystyle=$
$\displaystyle\int_{\gamma_{t}}(2u-2\Phi-1)(\phi^{\prime}-\kappa
u)\mathrm{d}s$ (4.8) $\displaystyle=$
$\displaystyle\int_{\gamma_{t}}(2u-2\Phi-1)\dfrac{\partial^{2}\Phi}{\partial
s^{2}}\mathrm{d}s$ $\displaystyle=$
$\displaystyle-2\int_{\gamma_{t}}\left(\dfrac{\partial u}{\partial
s}-\dfrac{\partial\Phi}{\partial s}\right)\dfrac{\partial\Phi}{\partial
s}\mathrm{d}s$ $\displaystyle=$
$\displaystyle-2\int_{\gamma_{t}}(\kappa-1)\left|\dfrac{\partial\Phi}{\partial
s}\right|^{2}\mathrm{d}s\leqslant 0,$

where we used that the curvature of $\gamma_{t}$ is $\kappa>1$ by Proposition
3.2. Equality holds in (4.8) if and only if $\dfrac{\partial\Phi}{\partial
s}\equiv 0$ holds along $\gamma_{t}$, which is equivalent to that $r\equiv
r_{0}$. Thus $\gamma_{t}$ is a geodesic circle centered at the origin. ∎

###### Proof of Theorem 1.4.

We divide the proof into two steps.

Step 1. Suppose $\gamma$ is a smooth strictly h-convex closed curve in
$\mathbb{H}^{2}$. Let $\gamma$ be the initial value of the flow (1.3) and we
obtain the solution $\\{\gamma_{t}\\}_{t\in[0,\infty)}$ by Theorem 1.2.
Moreover, when $t\to\infty$, $\gamma_{t}$ converges smoothly to a geodesic
circle $\gamma_{\infty}$ centered at the origin with radius $\rho_{\infty}$.
By (3.2) and (4.1), we have

$\displaystyle\int_{\gamma}(\Phi-u)(\kappa-1)\mathrm{d}s+(L[\gamma]-A[\gamma])$
$\displaystyle\geqslant$
$\displaystyle\int_{\gamma_{\infty}}(\Phi-u)(\kappa-1)\mathrm{d}s+(L[\gamma_{\infty}]-A[\gamma_{\infty}])$
$\displaystyle=$ $\displaystyle
2\pi\left(\Phi(\rho_{\infty})-\phi(\rho_{\infty})\right)\left(\phi^{\prime}(\rho_{\infty})-\phi(\rho_{\infty})\right)$
$\displaystyle+2\pi\left(\phi(\rho_{\infty})-\Phi(\rho_{\infty})\right)$
$\displaystyle(\mbox{by}\ \phi^{\prime}=\Phi+1)$ $\displaystyle=$
$\displaystyle 2\pi\left(\phi(\rho_{\infty})-\Phi(\rho_{\infty})\right)^{2}$
(4.9) $\displaystyle=$
$\displaystyle\dfrac{1}{2\pi}\left(L[\gamma_{\infty}]-A[\gamma_{\infty}]\right)^{2}$
$\displaystyle=$
$\displaystyle\dfrac{1}{2\pi}\left(L[\gamma]-A[\gamma]\right)^{2}.$

By Lemma 4.1, equality holds in (4.9) if and only if $\gamma$ is a geodesic
circle centered at the origin.

Step 2. Now we consider $\gamma$ as a smooth h-convex closed curve in
$\mathbb{H}^{2}$. We evolve $\gamma$ along curve shortening flow

$\partial_{t}X=-\kappa\nu,$ (4.10)

and obtain solution curve $\gamma_{t}$ with curvature $\kappa(x,t)$ satisfying

$\dfrac{\partial}{\partial t}(\kappa-1)=\dfrac{\partial^{2}}{\partial
s^{2}}(\kappa-1)+(\kappa-1)\kappa(\kappa+1).$ (4.11)

Note that for a compact h-convex curve, there exists at least one point on
$\gamma$ with $\kappa>1$. By strong maximum principle, $\gamma_{t}\ (\forall\
t>0)$ is strictly h-convex. Note that for $\gamma_{t}\ (t>0)$, inequality
(1.9) holds by Step 1. Now let $t\to 0$, we obtain that inequality (1.9) holds
for the h-convex curve $\gamma$.

If equality in (1.9) holds for a h-convex curve $\gamma$, we are going to show
that $\gamma$ is strictly h-convex. Let

$\tilde{\gamma}=\\{p\in\gamma:\kappa(p)>1\\}.$ (4.12)

Since there exists at least one point on $\gamma$ satisfying $\kappa>1$ for a
compact h-convex curve $\gamma$, we know that $\tilde{\gamma}$ is a nonempty
open set in $\gamma$. We next show that $\tilde{\gamma}$ is also closed in
$\gamma$. Following [7], we choose a smooth function $\varphi\in
C_{c}^{\infty}(\tilde{\gamma})$ with compact support in $\tilde{\gamma}$, and
consider a normal variation of $\gamma$

$\partial_{t}X=-\varphi\nu,$ (4.13)

where $t\in(-\varepsilon,\varepsilon)$, and $\varepsilon>0$ is small enough
such that $\gamma_{t}=X(\mathbb{S}^{1},t)$ is h-convex. Let

$Q(t)=\int_{\gamma_{t}}(\Phi-u)(\kappa-1)\mathrm{d}s+(L[\gamma_{t}]-A[\gamma_{t}])-\dfrac{1}{2\pi}(L[\gamma_{t}]-A[\gamma_{t}])^{2},$
(4.14)

we have

$Q(t)\geqslant 0,\ t\in(-\varepsilon,\varepsilon),\ \ Q(0)=0,$ (4.15)

which implies that $Q(t)\ (t\in(-\varepsilon,\varepsilon))$ attains its
minimum at $t=0$. Thus

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\left.\dfrac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}Q(t)$ (4.16)
$\displaystyle=$
$\displaystyle-\int_{\gamma}(2u-2\Phi-1)(\kappa-1)\varphi\mathrm{d}s-\int_{\gamma}(\kappa-1)\varphi\mathrm{d}s+\dfrac{L[\gamma]-A[\gamma]}{\pi}\int_{\gamma}(\kappa-1)\varphi\mathrm{d}s$
$\displaystyle=$
$\displaystyle\int_{\gamma}\left(\dfrac{L[\gamma]-A[\gamma]}{\pi}-(2u-2\Phi)\right)(\kappa-1)\varphi\mathrm{d}s,\
\ \forall\ \varphi\in C_{c}^{\infty}(\tilde{\gamma}).$

In particular, the curve $\tilde{\gamma}$ satisfies

$2u-2\Phi\equiv\dfrac{L[\gamma]-A[\gamma]}{\pi},$ (4.17)

and hence

$\tilde{\gamma}=\left\\{p\in\gamma:(u-\Phi)(p)=\dfrac{L[\gamma]-A[\gamma]}{2\pi}\right\\}.$
(4.18)

This implies that $\tilde{\gamma}$ is closed in $\gamma$. By the connectness
of $\gamma$, $\gamma\equiv\tilde{\gamma}$, thus $\gamma$ is strictly h-convex.
By the equality case in Step 1, we conclude that $\gamma$ is a geodesic circle
centered at the origin. ∎

###  4.2. Inequalities for hypersurfaces

We include the complete proof of Theorem 1.3 in this subsection though it has
been proved in [14]. Some details in this proof will be used in proving
stability results later. Let’s first derive the general evolution equation.

######  Lemma 4.2.

Along the flow $\partial_{t}X=\eta\nu$, hypersurface
$M_{t}=\partial\Omega_{t}\subset\mathbb{H}^{n+1}$ $(n\geqslant 2)$ satisfies

$\displaystyle\dfrac{\partial}{\partial
t}\left[\int_{M_{t}}(\Phi-u)E_{k}(\tilde{\kappa})\mathrm{d}\mu_{t}+(n-2k)\tilde{W}_{k}(\Omega_{t})\right]$
$\displaystyle=$ $\displaystyle(n-k)\int_{M_{t}}\Phi
E_{k+1}(\tilde{\kappa})\eta\mathrm{d}\mu_{t}$ (4.19)
$\displaystyle-(k+1)\int_{M_{t}}(\phi^{\prime}-u)E_{k}(\tilde{\kappa})\eta\mathrm{d}\mu_{t}.$

###### Proof.

Denote $S^{i}_{j}=h^{i}_{j}-\delta^{i}_{j}$, we will compute

$\displaystyle\dfrac{\partial}{\partial
t}\left(\int_{M_{t}}(\Phi-u)E_{k}(\tilde{\kappa})\mathrm{d}\mu_{t}+(n-2k)\tilde{W}_{k}(\Omega_{t})\right)$
(4.20) $\displaystyle=$
$\displaystyle\int_{M_{t}}(\partial_{t}\Phi-\partial_{t}u)E_{k}(\tilde{\kappa})\mathrm{d}\mu_{t}+\int_{M_{t}}(\Phi-u)\partial_{t}E_{k}(\tilde{\kappa})\mathrm{d}\mu_{t}$
$\displaystyle+\int_{M_{t}}(\Phi-u)E_{k}(\tilde{\kappa})\eta
H\mathrm{d}\mu_{t}+(n-2k)\int_{M_{t}}\eta
E_{k}(\tilde{\kappa})\mathrm{d}\mu_{t}.$

Substituting (2.13), (2.31), (2.32) and (2.42), and combining
$nE_{1}(\tilde{\kappa})=H-n$ and $\phi^{\prime}=\Phi+1$, we obtain

$\displaystyle\dfrac{\partial}{\partial
t}\left(\int_{M_{t}}(\Phi-u)E_{k}(\tilde{\kappa})\mathrm{d}\mu_{t}+(n-2k)\tilde{W}_{k}(\Omega_{t})\right)$
(4.21) $\displaystyle=$
$\displaystyle\int_{M_{t}}\left[(u-\phi^{\prime})\eta+\langle
V,\nabla\eta\rangle\right]E_{k}(\tilde{\kappa})\mathrm{d}\mu_{t}-\int_{M_{t}}(\Phi-u)\dot{E}_{k}^{ij}(\tilde{\kappa})\nabla_{j}\nabla^{i}\eta\mathrm{d}\mu_{t}$
$\displaystyle+(n-2k)\int_{M_{t}}(\phi^{\prime}-u)E_{k}(\tilde{\kappa})\eta\mathrm{d}\mu_{t}+(n-k)\int_{M_{t}}(\Phi-u)E_{k+1}(\tilde{\kappa})\eta\mathrm{d}\mu_{t}.$

By divergence free property of $E_{k}(\tilde{\kappa})$ (Lemma 2.3) and using
integration by parts, we have

$\displaystyle-\int_{M_{t}}(\Phi-u)\dot{E}_{k}^{ij}(\tilde{\kappa})\nabla_{j}\nabla^{i}\eta\mathrm{d}\mu_{t}$
(4.22) $\displaystyle=$
$\displaystyle-\int_{M_{t}}\dot{E}_{k}^{ij}(\tilde{\kappa})\nabla_{j}\nabla^{i}(\Phi-u)\eta\mathrm{d}\mu_{t}$
$\displaystyle=$
$\displaystyle-\int_{M_{t}}\dot{E}_{k}^{ij}(\tilde{\kappa})\left[-\phi^{\prime}S^{i}_{j}-\langle
V,\nabla
S^{i}_{j}\rangle+u\left((S^{2})^{i}_{j}+S^{i}_{j}\right)\right]\eta\mathrm{d}\mu_{t}$
$\displaystyle=$
$\displaystyle\int_{M_{t}}\left[\phi^{\prime}kE_{k}(\tilde{\kappa})+\langle
V,\nabla E_{k}(\tilde{\kappa})\rangle\right]\eta\mathrm{d}\mu_{t}$
$\displaystyle-\int_{M_{t}}u\left(nE_{1}(\tilde{\kappa})E_{k}(\tilde{\kappa})-(n-k)E_{k+1}(\tilde{\kappa})+kE_{k}(\tilde{\kappa})\right)\eta\mathrm{d}\mu_{t}.$

Note that by using integration by parts and
$\Delta\Phi=n\phi^{\prime}-Hu=n\phi^{\prime}-(nE_{1}(\tilde{\kappa})+n)u$, we
obtain

$\displaystyle\int_{M_{t}}\langle V,\nabla\eta\rangle
E_{k}(\tilde{\kappa})\mathrm{d}\mu_{t}+\int_{M_{t}}\langle V,\nabla
E_{k}(\tilde{\kappa})\rangle\eta\mathrm{d}\mu_{t}$ (4.23) $\displaystyle=$
$\displaystyle\int_{M_{t}}\langle V,\nabla(\eta
E_{k}(\tilde{\kappa}))\rangle\mathrm{d}\mu_{t}$ $\displaystyle=$
$\displaystyle-\int_{M_{t}}\Delta\Phi
E_{k}(\tilde{\kappa})\eta\mathrm{d}\mu_{t}$ $\displaystyle=$
$\displaystyle-\int_{M_{t}}\left[n\phi^{\prime}-\left(nE_{1}(\tilde{\kappa})+n\right)u\right]E_{k}(\tilde{\kappa})\eta\mathrm{d}\mu_{t}.$

Combining (4.21), (4.22) and (4.23), and after a direct computation we will
obtain (4.19). ∎

###### Proof of Theorem 1.3.

For $k=1,2,\cdots,n$, we evolve $M$ along the flow (1.2) and get a family of
strictly h-convex hypersurfaces $\\{M_{t}\\}_{t\in[0,\infty)}$ by Theorem 1.1.
Moreover, as $t\to\infty$, $M_{t}$ converges smoothly to a geodesic sphere
$M_{\infty}=\partial\Omega_{\infty}$ centered at the origin with radius
$\rho_{\infty}$.

Substituting
$\eta=\dfrac{(\phi^{\prime}-u)E_{k-1}(\tilde{\kappa})}{E_{k}(\tilde{\kappa})}-u$
in (4.19), by Newton-MacLaurin inequality (2.4) and (2.41), we have

$\displaystyle\dfrac{\partial}{\partial
t}\left[\int_{M_{t}}(\Phi-u)E_{k}(\tilde{\kappa})\mathrm{d}\mu_{t}+(n-2k)\tilde{W}_{k}(\Omega_{t})\right]$
(4.24) $\displaystyle\leqslant$
$\displaystyle(n-k)\int_{M_{t}}\Phi\left((\phi^{\prime}-u)E_{k}(\tilde{\kappa})-uE_{k+1}(\tilde{\kappa})\right)\mathrm{d}\mu_{t}$
$\displaystyle-(k+1)\int_{M_{t}}(\phi^{\prime}-u)\left((\phi^{\prime}-u)E_{k-1}(\tilde{\kappa})-uE_{k}(\tilde{\kappa})\right)\mathrm{d}\mu_{t}$
$\displaystyle=$
$\displaystyle\dfrac{n-k}{k+1}\int_{M_{t}}\Phi\dot{E}_{k+1}^{ij}(\tilde{\kappa})\nabla_{j}\nabla^{i}\phi^{\prime}\mathrm{d}\mu_{t}-\dfrac{k+1}{k}\int_{M_{t}}(\phi^{\prime}-u)\dot{E}^{ij}_{k}(\tilde{\kappa})\nabla_{j}\nabla^{i}\phi^{\prime}\mathrm{d}\mu_{t}$
$\displaystyle=$
$\displaystyle-\dfrac{n-k}{k+1}\int_{M_{t}}\dot{E}^{ij}_{k+1}(\tilde{\kappa})\nabla_{j}\Phi\nabla^{i}\phi^{\prime}\mathrm{d}\mu_{t}+\dfrac{k+1}{k}\int_{M_{t}}\dot{E}_{k}^{ij}(\tilde{\kappa})\nabla_{j}(\phi^{\prime}-u)\nabla^{i}\phi^{\prime}\mathrm{d}\mu_{t}$
$\displaystyle=$
$\displaystyle-\dfrac{n-k}{k+1}\int_{M_{t}}\dot{E}^{ij}_{k+1}(\tilde{\kappa})\nabla_{j}\phi^{\prime}\nabla^{i}\phi^{\prime}\mathrm{d}\mu_{t}-\dfrac{k+1}{k}\int_{M_{t}}\dot{E}_{k}^{ij}(\tilde{\kappa})S_{j}^{l}\nabla_{l}\phi^{\prime}\nabla^{i}\phi^{\prime}\mathrm{d}\mu_{t}.$

Note that $\dot{E}_{k}^{ij}(\tilde{\kappa})$,
$\dot{E}_{k+1}^{ij}(\tilde{\kappa})$, $S_{j}^{l}$ are all symmetric positive
definite, we conclude that

$\dfrac{\partial}{\partial
t}\left[\int_{M_{t}}(\Phi-u)E_{k}(\tilde{\kappa})\mathrm{d}\mu_{t}+(n-2k)\tilde{W}_{k}(\Omega_{t})\right]\leqslant
0$ (4.25)

with equality holds if and only if $\nabla\phi^{\prime}\equiv 0$, which
implies that $M_{t}$ is a geodesic sphere centered at the origin.

By Minkowski identity (2.24), along the flow (1.2), we have

$\dfrac{\partial}{\partial
t}\tilde{W}_{k}(\Omega_{t})=\int_{M_{t}}\left[(\phi^{\prime}-u)E_{k-1}(\tilde{\kappa})-uE_{k}(\tilde{\kappa})\right]\mathrm{d}\mu_{t}=0.$
(4.26)

Therefore,

$\displaystyle\int_{M}(\Phi-u)E_{k}(\tilde{\kappa})\mathrm{d}\mu+(n-2k)\tilde{W}_{k}(\Omega)$
$\displaystyle\geqslant$
$\displaystyle\int_{M_{\infty}}(\Phi-u)E_{k}(\tilde{\kappa})\mathrm{d}\mu_{\infty}+(n-2k)\tilde{W}_{k}(\Omega_{\infty})$
(4.27) $\displaystyle=$
$\displaystyle\tilde{h}_{k}\circ\tilde{f}_{k}^{-1}\left(\tilde{W}_{k}(\Omega_{\infty})\right)$
$\displaystyle=$
$\displaystyle\tilde{h}_{k}\circ\tilde{f}_{k}^{-1}\left(\tilde{W}_{k}(\Omega)\right).$

Here
$\tilde{h}_{k}(r)=\int_{\partial\overline{B}_{r}}(\Phi-u)E_{k}(\tilde{\kappa})\mathrm{d}\mu+(n-2k)\tilde{W}_{k}(\overline{B}_{r})$,
$\tilde{f}_{k}(r)=\tilde{W}_{k}(\overline{B}_{r})$, $\overline{B}_{r}$ is a
geodesic ball centered at the origin with radius $r$. Equality holds in (4.27)
if and only if $M_{t}$ is a geodesic sphere centered at the origin. Let $t\to
0$, we conclude that $M$ is a geodesic sphere centered at the origin. ∎

##  5\. Stability results

In this section, we will prove our stability results Theorems 1.5 and 1.7.

###  5.1. Stability of inequalities for curves

We will discuss the stability of inequality (1.9).

###### Proof of Theorem 1.5.

Let

$\varepsilon:=\int_{\gamma}(\Phi-u)(\kappa-1)\mathrm{d}s+(L[\gamma]-A[\gamma])-\dfrac{1}{2\pi}(L[\gamma]-A[\gamma])^{2}.$
(5.1)

We evolve $\gamma$ along flow (1.3) and obtain the solution curve
$\\{\gamma_{t}\\}_{t\in[0,\infty)}$ by Theorem 1.2. Moreover, $\gamma_{t}$ is
strictly h-convex and smoothly converges to a geodesic circle
$\gamma_{\infty}$ centered at the origin as $t\to\infty$. Let

$Q(t):=\int_{\gamma_{t}}(\Phi-u)(\kappa-1)\mathrm{d}s+(L[\gamma_{t}]-A[\gamma_{t}]).$
(5.2)

On the one hand, by Lemma 3.1 and (5.1), we have

$\displaystyle\int_{0}^{\infty}\partial_{t}Q(t)\mathrm{d}t$ $\displaystyle=$
$\displaystyle Q(\infty)-Q(0)$ (5.3) $\displaystyle=$
$\displaystyle\dfrac{1}{2\pi}\left(L[\gamma_{\infty}]-A[\gamma_{\infty}]\right)^{2}-Q(0)$
$\displaystyle=$
$\displaystyle\dfrac{1}{2\pi}\left(L[\gamma]-A[\gamma]\right)^{2}-Q(0)$
$\displaystyle=$ $\displaystyle-\varepsilon.$

On the other hand, by (4.8),

$\displaystyle\int_{0}^{\infty}\partial_{t}Q(t)\mathrm{d}t=-\int_{0}^{\infty}\int_{\gamma_{t}}2(\kappa-1)\left|\dfrac{\partial\Phi}{\partial
s}\right|^{2}\mathrm{d}s\mathrm{d}t.$ (5.4)

Combining (5.3) and (5.4), we obtain

$\int_{0}^{\infty}\int_{\gamma_{t}}2(\kappa-1)\left|\dfrac{\partial\Phi}{\partial
s}\right|^{2}\mathrm{d}s\mathrm{d}t=\varepsilon.$ (5.5)

Since $\gamma_{t}$ is strictly h-convex, there exists a constant
$C=C(\min\limits_{\gamma}\kappa)>0$ such that $2(\kappa-1)\geqslant C$. Thus

$\int_{0}^{\infty}\int_{\gamma_{t}}\left|\dfrac{\partial\Phi}{\partial
s}\right|^{2}\mathrm{d}s\mathrm{d}t\leqslant
C(\min\limits_{\gamma}\kappa)\varepsilon.$ (5.6)

Since $\gamma_{t}$ is star-shaped, we regard it as a graph over
$\mathbb{S}^{1}$ and denote
$\gamma_{t}=\\{(\rho(\theta),\theta):\theta\in\mathbb{S}^{1}\\}$. Since by
Propositions 3.2 and 3.3, along the flow (1.3), we have

$\left|\dfrac{\phi^{\prime}-u}{\kappa-1}-u\right|\leqslant
C(\rho_{0},\min\limits_{\gamma}\kappa).$ (5.7)

We can characterise the Hausdorff distance from $\gamma$ to $\gamma_{t}$ by

$\displaystyle\mathrm{dist}(\gamma,\gamma_{t})\leqslant\int_{0}^{t}\left|\partial_{\tau}X(x,\tau)\right|\mathrm{d}\tau=\int_{0}^{t}\left|\dfrac{\phi^{\prime}-u}{\kappa-1}-u\right|(x,\tau)\mathrm{d}\tau\leqslant
C(\rho_{0},\min\limits_{\gamma}\kappa)t.$ (5.8)

Since

$\int_{0}^{\sqrt{\varepsilon}}\int_{\gamma_{t}}\left|\dfrac{\partial\Phi}{\partial
s}\right|^{2}\mathrm{d}s\mathrm{d}t\leqslant
C(\min\limits_{\gamma}\kappa)\varepsilon,$ (5.9)

there exists $t_{\varepsilon}\in(0,\sqrt{\varepsilon}]$ such that

$\int_{\gamma_{t_{\varepsilon}}}\left|\dfrac{\partial\Phi}{\partial
s}\right|^{2}\mathrm{d}s\leqslant
C(\min\limits_{\gamma}\kappa)\sqrt{\varepsilon}.$ (5.10)

Meanwhile,

$\mathrm{dist}(\gamma,\gamma_{t_{\varepsilon}})\leqslant
C(\rho_{0},\min\limits_{\gamma}\kappa)\sqrt{\varepsilon}.$ (5.11)

We next show that there exists a geodesic circle $\gamma_{\mathbb{H}}$ such
that $\mathrm{dist}(\gamma,\gamma_{\mathbb{H}})\leqslant
C\varepsilon^{\frac{1}{6}}$. Consider the strictly h-convex curve
$\gamma_{t_{\varepsilon}}=\\{(\rho(\theta),\theta):\theta\in\mathbb{S}^{1}\\}$
as a graph on $\mathbb{S}^{1}$. Note that the arc-length element is
$\mathrm{d}s=\sqrt{\phi^{2}(\rho)+\rho_{\theta}^{2}}\mathrm{d}\theta$, from
(5.10) we have,

$\displaystyle C(\min\limits_{\gamma}\kappa)\sqrt{\varepsilon}$
$\displaystyle\geqslant$
$\displaystyle\int_{\gamma_{t_{\varepsilon}}}\left|\dfrac{\partial\Phi}{\partial
s}\right|^{2}\mathrm{d}s=\int_{\mathbb{S}^{1}}\dfrac{\phi^{2}(\rho)\rho_{\theta}^{2}}{\sqrt{\phi^{2}(\rho)+\rho_{\theta}^{2}}}\mathrm{d}\theta$
(5.12) $\displaystyle\geqslant$ $\displaystyle
C(\max\limits_{\gamma}|\rho_{\theta}|,\rho_{0})\int_{\mathbb{S}^{1}}\rho_{\theta}^{2}\mathrm{d}\theta$
$\displaystyle\geqslant$ $\displaystyle
C(\max\limits_{\gamma}|\rho_{\theta}|,\rho_{0})\int_{\mathbb{S}^{1}}\left|\rho-a\right|^{2}\mathrm{d}\theta,$

where we used Poincaré inequality in (5.12), and
$a=\dfrac{1}{2\pi}\displaystyle\int_{\mathbb{S}^{1}}\rho(\theta)\mathrm{d}\theta$.
This implies that

$\int_{\mathbb{S}^{1}}|\rho-a|^{2}\mathrm{d}\theta\leqslant
C_{0}(\rho_{0},\max\limits_{\gamma}|\rho_{\theta}|,\min\limits_{\gamma}\kappa)\sqrt{\varepsilon}.$
(5.13)

We claim that: There exists a constant
$C_{1}(\rho_{0},\max\limits_{\gamma}|\rho_{\theta}|,\min\limits_{\gamma}\kappa)$
such that

$\max\left\\{\max\limits_{\mathbb{S}^{1}}\rho-a,a-\min\limits_{\mathbb{S}^{1}}\rho\right\\}\leqslant
C_{1}\varepsilon^{\frac{1}{6}}=:L.$ (5.14)

Suppose it is not true, without loss of generality, we assume that

$\max\limits_{\mathbb{S}^{1}}\rho-a=\rho(\theta_{0})-a>L.$ (5.15)

Then for $l:=\dfrac{L}{2\max\limits_{\mathbb{S}^{1}}|\rho_{\theta}|}>0$, there
holds

$|\rho(\xi)-a|>\dfrac{L}{2},\ \forall\
\xi\in\overline{B}_{l}(\theta_{0})\subset\mathbb{S}^{1}.$ (5.16)

Thus,

$C_{0}\sqrt{\varepsilon}\geqslant\int_{\mathbb{S}^{1}}|\rho-a|^{2}\mathrm{d}\theta\geqslant\int_{\overline{B}_{l}(\theta_{0})}|\rho-a|^{2}\mathrm{d}\theta\geqslant
CL^{3}=CC_{1}^{3}\sqrt{\varepsilon},$ (5.17)

which makes a contradiction if we choose
$C_{1}=\left(\frac{2C_{0}}{C}\right)^{\frac{1}{3}}$. Therefore, there exists a
geodesic circle $\gamma_{\mathbb{H}}$ with radius $a$ such that

$\mathrm{dist}(\gamma_{t_{\varepsilon}},\gamma_{\mathbb{H}})\leqslant
C\|\rho-a\|_{L^{\infty}(\mathbb{S}^{1})}\leqslant
C(\rho_{0},\max\limits_{\gamma}|\rho_{\theta}|,\min\limits_{\gamma}\kappa)\varepsilon^{\frac{1}{6}}.$
(5.18)

Combining (5.11) with (5.18), we conclude that

$\displaystyle\mathrm{dist}(\gamma,\gamma_{\mathbb{H}})$
$\displaystyle\leqslant$
$\displaystyle\mathrm{dist}(\gamma,\gamma_{t_{\varepsilon}})+\mathrm{dist}(\gamma_{t_{\varepsilon}},\gamma_{\mathbb{H}})$
(5.19) $\displaystyle\leqslant$ $\displaystyle
C(\rho_{0},\max\limits_{\gamma}|\rho_{\theta}|,\min\limits_{\gamma}\kappa)\left(\varepsilon^{\frac{1}{2}}+\varepsilon^{\frac{1}{6}}\right),$

which is (1.11). ∎

###  5.2. Stability of inequalities for hypersurfaces

We will apply the nearly umbilical result (Theorem 1.6) to discuss the
stability of inequalities (1.5) and (1.8).

###### Proof of Theorem 1.7.

For (1.15), we set

$\varepsilon:=\int_{M}(\Phi-u)E_{k}(\tilde{\kappa})\mathrm{d}\mu+(n-2k)\tilde{W}_{k}(\Omega)-\tilde{h}_{k}\circ\tilde{f}_{k}^{-1}\left(\tilde{W}_{k}(\Omega)\right).$
(5.20)

Set the center of the interior ball of $\Omega$ with radius $\rho_{-}(\Omega)$
denoted by $o$ as the origin. We evolve $M$ along the flow

$\partial_{t}X=\left(\dfrac{(\phi^{\prime}-u)E_{k-1}(\tilde{\kappa})}{E_{k}(\tilde{\kappa})}-u\right)\nu$
(5.21)

and obtain a family of strictly h-convex hypersurfaces
$\\{M_{t}\\}_{t\in[0,\infty)}$. Denote $\Omega_{t}$ as the domain enclosed by
$M_{t}$. By Theorem 1.1, $M_{t}$ is strictly h-convex and $M_{t}$ converges
smoothly to a geodesic sphere $M_{\infty}=\partial\Omega_{\infty}$ centered at
the origin as $t\to\infty$. From [2, Theorem 1], we know

$\rho_{-}(\Omega)\leqslant r(x,t)\leqslant\rho_{-}(\Omega)+\ln 2,\ \forall\
(x,t)\in M_{t}.$ (5.22)

Thus there exists a constant $C=C(\rho_{-}(\Omega))>0$ such that
$\Phi(r)\geqslant C>0$. Denote

$Q(t)=\int_{M_{t}}(\Phi-u)E_{k}(\tilde{\kappa})\mathrm{d}\mu_{t}+(n-2k)\tilde{W}_{k}(\Omega_{t}),$
(5.23)

then by (4.19) we have

$\displaystyle\dfrac{\partial}{\partial t}Q(t)$ $\displaystyle=$
$\displaystyle(n-k)\int_{M_{t}}(\Phi-C)E_{k+1}(\tilde{\kappa})\left(\dfrac{(\phi^{\prime}-u)E_{k-1}(\tilde{\kappa})}{E_{k}(\tilde{\kappa})}-u\right)\mathrm{d}\mu_{t}$
(5.26)
$\displaystyle+(n-k)C\int_{M_{t}}E_{k+1}(\tilde{\kappa})\left(\dfrac{(\phi^{\prime}-u)E_{k-1}(\tilde{\kappa})}{E_{k}(\tilde{\kappa})}-u\right)\mathrm{d}\mu_{t}$
$\displaystyle-(k+1)\int_{M_{t}}(\phi^{\prime}-u)E_{k}(\tilde{\kappa})\left(\dfrac{(\phi^{\prime}-u)E_{k-1}(\tilde{\kappa})}{E_{k}(\tilde{\kappa})}-u\right)\mathrm{d}\mu_{t}$
$\displaystyle\leqslant$
$\displaystyle(n-k)C\int_{M_{t}}E_{k+1}(\tilde{\kappa})\left(\dfrac{(\phi^{\prime}-u)E_{k-1}(\tilde{\kappa})}{E_{k}(\tilde{\kappa})}-u\right)\mathrm{d}\mu_{t}$
(5.27) $\displaystyle=$
$\displaystyle(n-k)C\int_{M_{t}}(\phi^{\prime}-u)\left[\dfrac{E_{k+1}(\tilde{\kappa})E_{k-1}(\tilde{\kappa})}{E_{k}(\tilde{\kappa})}-E_{k}(\tilde{\kappa})\right]\mathrm{d}\mu_{t}.$
(5.28)

Here $(\ref{equ-Phic1})+(\ref{equ-Phic2})\leqslant 0$ for the same reason as
(4.24) and (4.25). We also used Minkowski type fomulas (2.24) in (5.28).
Integrating (5.28) on $[0,\infty)$, and combining the fact

$\phi^{\prime}-u\geqslant\cosh r-\sinh
r\geqslant\dfrac{1}{2}\mathrm{e}^{-\rho_{-}(\Omega)}:=C(\rho_{-}(\Omega)),$
(5.29)

we have

$\int_{0}^{\infty}\dfrac{\partial}{\partial
t}Q(t)\mathrm{d}t\leqslant(n-k)C\int_{0}^{\infty}\int_{M_{t}}\dfrac{E_{k+1}(\tilde{\kappa})E_{k-1}(\tilde{\kappa})}{E_{k}(\tilde{\kappa})}-E_{k}(\tilde{\kappa})\mathrm{d}\mu_{t}\mathrm{d}t.$
(5.30)

Besides,

$\displaystyle\int_{0}^{\infty}\dfrac{\partial}{\partial t}Q(t)\mathrm{d}t$
$\displaystyle=$ $\displaystyle Q(\infty)-Q(0)$ (5.31) $\displaystyle=$
$\displaystyle\tilde{h}_{k}\circ\tilde{f}_{k}^{-1}\left(\tilde{W}_{k}(\Omega_{\infty})\right)-Q(0)$
$\displaystyle=$
$\displaystyle\tilde{h}_{k}\circ\tilde{f}_{k}^{-1}\left(\tilde{W}_{k}(\Omega)\right)-Q(0)$
$\displaystyle=$ $\displaystyle-\varepsilon,$ (5.32)

where in (5.31) we used that $\tilde{W}_{k}(\Omega_{t})$ preserves along the
flow (1.2), and in (5.32) we used (5.20). Combining (5.30) and (5.32), we get

$\int_{0}^{\infty}\int_{M_{t}}E_{k}(\tilde{\kappa})-\dfrac{E_{k+1}(\tilde{\kappa})E_{k-1}(\tilde{\kappa})}{E_{k}(\tilde{\kappa})}\mathrm{d}\mu_{t}\mathrm{d}t\leqslant\dfrac{C(\rho_{-}(\Omega))}{n-k}\varepsilon.$
(5.33)

For (1.14), we also set

$\varepsilon:=\tilde{W}_{k+1}(\Omega)-\tilde{f}_{k+1}\circ\tilde{f}_{k}^{-1}(\tilde{W}_{k}(\Omega)).$
(5.34)

Note that along the flow (1.2), we have

$\displaystyle\dfrac{\partial}{\partial t}\tilde{W}_{k+1}(\Omega_{t})$
$\displaystyle=$
$\displaystyle\int_{M_{t}}\left[(\phi^{\prime}-u)\dfrac{E_{k-1}(\tilde{\kappa})}{E_{k}(\tilde{\kappa})}-u\right]E_{k+1}(\tilde{\kappa})\mathrm{d}\mu_{t}$
(5.35) $\displaystyle=$
$\displaystyle\int_{M_{t}}(\phi^{\prime}-u)\left[\dfrac{E_{k-1}(\tilde{\kappa})E_{k+1}(\tilde{\kappa})}{E_{k}(\tilde{\kappa})}-E_{k}(\tilde{\kappa})\right]\mathrm{d}\mu_{t}.$

Then we obtain that

$\displaystyle\int_{0}^{\infty}\int_{M_{t}}(\phi^{\prime}-u)\left[\dfrac{E_{k-1}(\tilde{\kappa})E_{k+1}(\tilde{\kappa})}{E_{k}(\tilde{\kappa})}-E_{k}(\tilde{\kappa})\right]\mathrm{d}\mu_{t}\mathrm{d}t$
$\displaystyle=$ $\displaystyle-\varepsilon,$ (5.36)

and (5.33) also holds by using (5.29).

Denote $F=\dfrac{E_{k}(\tilde{\kappa})}{E_{k-1}(\tilde{\kappa})}$. It has been
verified in [10, Lemma 7.1, Lemma 7.2 and Proposition 7.4] that

$0<C_{1}\leqslant F\leqslant C_{2},\ \tilde{\kappa_{i}}\leqslant C_{3},\
i=1,2,\cdots,n,$ (5.37)

where the constants $C_{1},\ C_{2},\ C_{3}$ depend only on $M$. Thus

$\left|\dfrac{\phi^{\prime}-u}{F}-u\right|\leqslant C(\rho_{-}(\Omega),M),$
(5.38)

and the Hausdorff distance between $\partial\Omega$ and $M_{t}$ can be
estimated by

$\displaystyle\mathrm{dist}(\partial\Omega,M_{t})$ $\displaystyle\leqslant$
$\displaystyle\mathrm{dist}((X(x,0),X(x,t))$ (5.39) $\displaystyle\leqslant$
$\displaystyle\int_{0}^{t}\left|\partial_{\tau}X(x,\tau)\right|\mathrm{d}\tau=\int_{0}^{t}\left|\dfrac{\phi^{\prime}-u}{F}-u\right|(x,\tau)\mathrm{d}\tau$
$\displaystyle\leqslant$ $\displaystyle C(\rho_{-}(\Omega),M)t.$

Note that $S^{i}_{j}=h^{i}_{j}-\delta^{i}_{j}$. After a direct computation, we
have

$\mathring{S}^{i}_{j}=S^{i}_{j}-\dfrac{H-n}{n}\delta^{i}_{j}=h^{i}_{j}-\dfrac{H}{n}\delta^{i}_{j}=\mathring{h}^{i}_{j}.$
(5.40)

Thus $|\mathring{S}|^{2}=|\mathring{A}|^{2}$, and the following inequality
holds

$C(n)E_{k+1,n1}^{2}(\tilde{\kappa})|\mathring{A}|^{2}\leqslant
E_{k}(\tilde{\kappa})^{2}-E_{k-1}(\tilde{\kappa})E_{k+1}(\tilde{\kappa}),$
(5.41)

where
$\tilde{\kappa}_{1}\leqslant\tilde{\kappa}_{2}\leqslant\cdots\leqslant\tilde{\kappa}_{n}$,
$\tilde{\kappa}\in\Gamma_{k}^{+}$,
$E^{2}_{k+1,n1}(\tilde{\kappa})=\dfrac{\partial^{2}E_{k+1}(\tilde{\kappa})}{\partial\tilde{\kappa_{1}}\partial\tilde{\kappa_{n}}}>0$.
We refer to [4, Lemma 3.4] and [18, Lemma 4.2] for more details. Now

$\displaystyle\int_{0}^{\sqrt{\varepsilon}}\int_{M_{t}}|\mathring{A}|^{2}\mathrm{d}\mu_{t}\mathrm{d}t$
(5.42) $\displaystyle=$
$\displaystyle\int_{0}^{\sqrt{\varepsilon}}\int_{M_{t}}\dfrac{E_{k+1,n1}^{2}(\tilde{\kappa})|\mathring{A}|^{2}}{E_{k}(\tilde{\kappa})}\dfrac{E_{k}(\tilde{\kappa})}{E^{2}_{k+1,n1}(\tilde{\kappa})}\mathrm{d}\mu_{t}\mathrm{d}t$
$\displaystyle\leqslant$
$\displaystyle\dfrac{\max\limits_{t\in[0,\sqrt{\varepsilon}]}E_{k}(\tilde{\kappa})}{\min\limits_{t\in[0,\sqrt{\varepsilon}]}E_{k+1,n1}^{2}(\tilde{\kappa})}\int_{0}^{\sqrt{\varepsilon}}\int_{M_{t}}\dfrac{E^{2}_{k+1,n1}(\tilde{\kappa})|\mathring{A}|^{2}}{E_{k}(\tilde{\kappa})}\mathrm{d}\mu_{t}\mathrm{d}t$
$\displaystyle\leqslant$ $\displaystyle
C(n)\dfrac{\max\limits_{t\in[0,\sqrt{\varepsilon}]}E_{k}(\tilde{\kappa})}{\min\limits_{t\in[0,\sqrt{\varepsilon}]}E_{k+1,n1}^{2}(\tilde{\kappa})}\int_{0}^{\sqrt{\varepsilon}}\int_{M_{t}}\dfrac{E_{k}(\tilde{\kappa})^{2}-E_{k-1}(\tilde{\kappa})E_{k+1}(\tilde{\kappa})}{E_{k}(\tilde{\kappa})}\mathrm{d}\mu_{t}\mathrm{d}t$
$\displaystyle\leqslant$ $\displaystyle
C(M,\mathrm{dist}(\tilde{\kappa},\partial\Gamma_{k}^{+}),\rho_{-}(\Omega),n,k)\varepsilon,$
(5.43)

where we used (5.33) in the last inequality. There exists
$t_{\varepsilon}\in(0,\sqrt{\varepsilon}]$ such that

$\int_{M_{t_{\varepsilon}}}|\mathring{A}|^{2}\mathrm{d}\mu_{t_{\varepsilon}}\leqslant
C(M,\mathrm{dist}(\tilde{\kappa},\partial\Gamma_{k}^{+}),\rho_{-}(\Omega),n,k)\sqrt{\varepsilon}.$
(5.44)

At the same time,

$\mathrm{dist}(\partial\Omega,M_{t_{\varepsilon}})\leqslant
C(\rho_{-}(\Omega),M,n,k)\sqrt{\varepsilon}.$ (5.45)

In the following, we will use Theorem 1.6 to prove that there exists a
geodesic sphere $S_{\mathbb{H}}$ such that
$\mathrm{dist}(M_{t_{\varepsilon}},S_{\mathbb{H}})\leqslant
C(M,\mathrm{dist}(\tilde{\kappa},\partial\Gamma_{k}^{+}),\rho_{-}(\Omega),n,k)\varepsilon^{\frac{1}{4}}$.
We now follow [17] and contain the complete proof here. We view hyperbolic
space $\mathbb{H}^{n+1}$ as a ball of radius 2 denoted by $B_{2}(o)$ in
Euclidean space $\mathbb{R}^{n+1}$, and set

$r=\log(2+\rho)-\log(2-\rho).$ (5.46)

We have $\mathrm{d}r=\dfrac{4}{4-\rho^{2}}\mathrm{d}\rho$, $\sinh
r=\dfrac{4\rho}{4-\rho^{2}}$. Let
$\mathrm{e}^{\phi(\rho)}=\dfrac{4}{4-\rho^{2}}$, then

$\overline{g}=\mathrm{d}r^{2}+\sinh^{2}rg_{\mathbb{S}^{n}}=\mathrm{e}^{2\phi(\rho)}\left(\mathrm{d}\rho^{2}+\rho^{2}g_{\mathbb{S}^{n}}\right)=\mathrm{e}^{2\phi(\rho)}\tilde{g}.$
(5.47)

Therefore, hypersurface $M_{t_{\varepsilon}}$ in $\mathbb{H}^{n+1}$ is
conformally as a hypersurface $\tilde{M}_{t_{\varepsilon}}$ in $B_{2}(o)$.
Since $M_{t_{\varepsilon}}$ is h-convex, then $\tilde{M}_{t_{\varepsilon}}$ is
convex. Indeed, denote $\tilde{h}^{i}_{j}$ as the Weingarten matrix of
$\tilde{M}_{t_{\varepsilon}}$, and let $\tilde{\nu}$ be its unit outward
normal vector, then there holds (see [5, Proposition 1.1.11])

$\mathrm{e}^{\phi(\rho)}h^{i}_{j}=\tilde{h}^{i}_{j}+\mathrm{d}\phi(\tilde{\nu})\delta^{i}_{j}.$
(5.48)

Note that $\mathrm{d}\phi=\dfrac{2\rho}{4-\rho^{2}}\mathrm{d}\rho$, we have

$\displaystyle\tilde{h}^{i}_{j}$ $\displaystyle=$
$\displaystyle\mathrm{e}^{\phi(\rho)}h^{i}_{j}-\mathrm{d}\phi(\tilde{\nu})\delta^{i}_{j}$
$\displaystyle\geqslant$
$\displaystyle\dfrac{4}{4-\rho^{2}}h^{i}_{j}-\dfrac{2\rho}{4-\rho^{2}}\delta^{i}_{j}$
$\displaystyle((h^{i}_{j}-\delta^{i}_{j})\geqslant 0)$
$\displaystyle\geqslant$
$\displaystyle\dfrac{4-2\rho}{4-\rho^{2}}\delta^{i}_{j}$ (5.49)
$\displaystyle=$ $\displaystyle\dfrac{2}{2+\rho}\delta^{i}_{j}>0.$

We rescale $\tilde{M}_{t_{\varepsilon}}$ to be $\hat{M}_{t_{\varepsilon}}$
such that $|\hat{M}_{t_{\varepsilon}}|=|\mathbb{S}^{n}|$. Indeed, it suffices
to set

$\hat{M}_{t_{\varepsilon}}=\left(\dfrac{|\mathbb{S}^{n}|}{|\tilde{M}_{t_{\varepsilon}}|}\right)^{\frac{1}{n}}\tilde{M}_{t_{\varepsilon}}:=\gamma(n,\rho_{-}(\Omega))\tilde{M}_{t_{\varepsilon}}.$
(5.50)

Then $\hat{M}_{t_{\varepsilon}}$ is a convex hypersurface in
$B_{2}(o)\subset\mathbb{R}^{n+1}$. Denote $\hat{g}$, $\mathring{\hat{A}}$ as
the metric and trace free Weingarten matrix of $\hat{M}_{t_{\varepsilon}}$,
then we have

$\displaystyle\mathring{\hat{A}}=\dfrac{1}{\gamma}\mathring{\tilde{A}},\ \
\mathrm{d}\mu_{\hat{g}}=\gamma^{n}\mathrm{d}\mu_{\tilde{g}},$ (5.51)
$\mathring{\tilde{A}}=\mathrm{e}^{\phi(\rho)}\mathring{A},\ \
\mathrm{d}\mu_{\tilde{g}}=\mathrm{e}^{-n\phi(\rho)}\mathrm{d}\mu_{g}.$ (5.52)

Therefore,

$\displaystyle\|\mathring{\hat{A}}\|_{L^{2}(\hat{M}_{t_{\varepsilon}})}=\gamma^{\frac{n-2}{2}}\|\mathring{\tilde{A}}\|_{L^{2}(\tilde{M}_{t_{\varepsilon}})},$
(5.53)
$\displaystyle\|\mathring{\tilde{A}}\|_{L^{2}(\tilde{M}_{t_{\varepsilon}})}=\left(\int_{M_{t_{\varepsilon}}}\mathrm{e}^{(2-n)\phi(\rho)}|\mathring{A}|^{2}\mathrm{d}\mu_{g}\right)^{\frac{1}{2}}\leqslant
C(n,\rho_{-}(\Omega))\|\mathring{A}\|_{L^{2}(M_{t_{\varepsilon}})}.$ (5.54)

Combining (5.44), (5.53) and (5.54), we obtain that

$\|\mathring{\hat{A}}\|_{L^{2}(\hat{M}_{t_{\varepsilon}})}\leqslant
C(M,\mathrm{dist}(\tilde{\kappa},\partial\Gamma_{k}^{+}),\rho_{-}(\Omega),n,k)\varepsilon^{\frac{1}{4}}.$
(5.55)

We now set $p=2$ in Theorem 1.6, then there exist a parametrization
$\varphi:\mathbb{S}^{n}\to\hat{M}_{t_{\varepsilon}}\subset B_{2}(o)$ and a
point $\mathcal{P}\in B_{2}(o)$ such that

$\displaystyle\|\varphi-\mathrm{id}-\mathcal{P}\|_{W^{2,2}(\mathbb{S}^{n})}\leqslant
C(n)\|\mathring{\hat{A}}\|_{L^{2}(\hat{M}_{t_{\varepsilon}})}\leqslant
C(M,\mathrm{dist}(\tilde{\kappa},\partial\Gamma_{k}^{+}),\rho_{-}(\Omega),n,k)\varepsilon^{\frac{1}{4}}.$
(5.56)

This implies that $\hat{M}_{t_{\varepsilon}}$ is Hausdorff-close to some unit
sphere, thus $\tilde{M}_{t_{\varepsilon}}$ is Hausdorff-close to a sphere with
radius $\gamma^{-1}$. Therefore, there exists a geodesic sphere
$S_{\mathbb{H}}$ such that

$\mathrm{dist}(M_{t_{\varepsilon}},S_{\mathbb{H}})\leqslant
C(M,\mathrm{dist}(\tilde{\kappa},\partial\Gamma_{k}^{+}),\rho_{-}(\Omega),n,k)\varepsilon^{\frac{1}{4}}.$
(5.57)

Combining (5.45) and (5.57), we obtain that

$\displaystyle\mathrm{dist}(\partial\Omega,S_{\mathbb{H}})$
$\displaystyle\leqslant$
$\displaystyle\mathrm{dist}(\partial\Omega,M_{t_{\varepsilon}})+\mathrm{dist}(M_{t_{\varepsilon}},S_{\mathbb{H}})$
$\displaystyle\leqslant$ $\displaystyle
C(M,\mathrm{dist}(\tilde{\kappa},\partial\Gamma_{k}^{+}),\rho_{-}(\Omega),n,k)\left(\varepsilon^{\frac{1}{2}}+\varepsilon^{\frac{1}{4}}\right).$

These are (1.14) and (1.15). ∎

## References

  * Andrews et al. [2021] Ben Andrews, Xuzhong Chen, and Yong Wei.  Volume preserving flow and Alexandrov-Fenchel type inequalities in hyperbolic space.  _Journal of the European Mathematical Society_ , 23(7):2467–2509, 2021.
  * Borisenko and Miquel [1999] Alexandr A. Borisenko and Vicente Miquel.  Total curvatures of convex hypersurfaces in hyperbolic space.  _Illinois Journal of Mathematics_ , 43:61–78, 1999.
  * Brendle et al. [2018] Simon Brendle, Pengfei Guan, and Junfang Li.  An inverse curvature type hypersurface flow in $\mathbb{H}^{n+1}$.  _preprint_ , 2018.
  * Chen et al. [2022] Chuanqiang Chen, Pengfei Guan, Junfang Li, and Julian Scheuer.  A fully-nonlinear flow and quermassintegral inequalities in the sphere.  _Pure and Applied Mathematics Quarterly_ , 18(2):437–461, 2022.
  * Gerhardt [2006] Claus Gerhardt.  _Curvature Problems_ , volume 39 of _Series in Geometry and topology_.  International Press, 2006.
  * Guan [2014] Pengfei Guan.  _Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs_ , pages 47–94.  Springer International Publishing, Cham, 2014.
  * Guan and Li [2009] Pengfei Guan and Junfang Li.  The quermassintegral inequalities for $k$-convex starshaped domains.  _Advances in Mathematics_ , 221:1725–1732, 2009.
  * Guan and Li [2015] Pengfei Guan and Junfang Li.  A mean curvature type flow in space forms.  _International Mathematics Research Notices_ , 2015(13):4716–4740, 2015.
  * Guan and Li [2021] Pengfei Guan and Junfang Li.  Isoperimetric type inequalities and hypersurface flows.  _Journal of Mathematical Study_ , 54:56–80, 2021.
  * Hu et al. [2022] Yingxiang Hu, Haizhong Li, and Yong Wei.  Locally constrained curvature flows and geometric inequalities in hyperbolic space.  _Mathematische Annalen_ , 382:1425–1474, 2022.
  * Krylov [1987] Nicolai V. Krylov.  _Nonlinear Elliptic and Parabolic Equations of the Second Order_ , volume 7.  Springer, 1987.
  * Kwong and Wei [2023] Kwok-Kun Kwong and Yong Wei.  Geometric inequalities involving three quantities in warped product manifolds.  _Advances in Mathematics_ , 430:109213, 2023.
  * Kwong et al. [2022] Kwok-Kun Kwong, Yong Wei, Glen Wheeler, and Valentina-Mira Wheeler.  On an inverse curvature flow in two-dimensional space forms.  _Mathematische Annalen_ , 384:1–24, 2022.
  * Li and Xu [2022] Haizhong Li and Botong Xu.  Hyperbolic $p$-sum and horospherical $p$-Brunn-Minkowski theory in hyperbolic space.  _arXiv:2211.06875_ , 2022.
  * Nirenberg [1966] Louis Nirenberg.  An extended interpolation inequality.  _Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze_ , 20:733–737, 1966.
  * Rosa and Gioffrè [2018] Antonio De Rosa and Stefano Gioffrè.  Quantitative stability for anisotropic nearly umbilical hypersurfaces.  _The Journal of Geometric Analysis_ , 29:2318–2346, 2018\.
  * Sahjwani and Scheuer [2023] Prachi Sahjwani and Julian Scheuer.  Stability of the quermassintegral inequalities in hyperbolic space.  _The Journal of Geometric Analysis_ , 34, 2023.
  * Scheuer [2023] Julian Scheuer.  Stability from rigidity via umbilicity.  _arXiv:2103.07178v2_ , 2023.
  * Solanes [2005] Gil Solanes.  Integral geometry and the Gauss-Bonnet theorem in constant curvature spaces.  _Transactions of the American Mathematical Society_ , 358:1105–1115, 2005.
  * Wang and Xia [2014] Guofang Wang and Chao Xia.  Isoperimetric type problems and Alexandrov-Fenchel type inequalities in the hyperbolic space.  _Advances in Mathematics_ , 259:532–556, 2014.
  * Wei and Xiong [2022] Yong Wei and Changwei Xiong.  A fully nonlinear locally constrained anisotropic curvature flow.  _Nonlinear Analysis_ , 217:112760, 2022.