# Blaschke-Santaló type inequalities and quermassintegral inequalities in
space forms

Yingxiang Hu  School of Mathematics, Beihang University, Beijing 100191, P.R.
China<EMAIL_ADDRESS>and  Haizhong Li  Department of Mathematical
Sciences, Tsinghua University, Beijing 100084, P.R. China
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove a family of identities for closed and strictly convex
hypersurfaces in the sphere and hyperbolic/de Sitter space. As applications,
we prove Blaschke-Santaló type inequalities in the sphere and hyperbolic/de
Sitter space, which generalizes the previous work of Gao, Hug and Schneider
[GHS03]. We also prove the quermassintegral inequalities in hyperbolic/de
Sitter space.

###### Key words and phrases:

Blaschke-Santaló type inequality, Quermassintegral inequality, Duality, Dual
flows, Space forms

###### 2020 Mathematics Subject Classification:

53C21, 53C24, 53C65

##  1\. Introduction

A convex body $K$ in Euclidean space $\mathbb{R}^{n+1}$ is a compact convex
set with non-empty interior. For any interior point $z\in\operatorname{int}K$,
the dual body of $K$ with respect to $z$ is defined by

$\displaystyle K_{z}^{\ast}=\\{y+z|y\in\mathbb{R}^{n+1}:y\cdot(x-z)\leq
1,\forall~{}x\in K\\}.$

As noted by Santaló [Sant1949], there exists a unique point
$s=s(K)\in\operatorname{int}K$ such that
$\operatorname{Vol}(K_{s}^{\ast})\leq\operatorname{Vol}(K_{z}^{\ast})$ for all
$z\in\operatorname{int}K$. The famous Blaschke-Santaló inequality says that

$\displaystyle\operatorname{Vol}(K)\cdot\operatorname{Vol}(K_{s}^{\ast})\leq|\mathbb{B}^{n+1}|^{2},$
(1.1)

where $|\mathbb{B}^{n+1}|$ is the volume of the unit ball in
$\mathbb{R}^{n+1}$. Equality holds if and only if $K$ is an ellipsoid. This
inequality in $\mathbb{R}^{3}$ was first proved by Blaschke [Blaschke1917],
and later it is generalized to higher dimensions by Santaló [Sant1949] with
the equality case characterized by Petty [Petty1985]. For sake of its
importance, more proofs of the Blaschke-Santaló inequality were given by Hug
[Hug1996], Meyer and Pajor [Meyer-Pojor1990]. Andrews [And96] studied the
affine normal flow, which also provides a proof of the Blaschke-Santaló
inequality for smooth convex hypersurfaces. By using Steiner symmetrization,
Lutwak and Zhang [Lutwak-Zhang1997] extended the Blaschke-Santaló inequality
to the $L_{p}$ Blaschke-Santaló inequality for star bodies.

The volume functional can be considered as one of the quermassintegrals, which
are the coefficients in the Steiner formula:

$\displaystyle\operatorname{Vol}(K+tB)=\sum_{i=0}^{n+1}\binom{n+1}{i}W_{i}(K)t^{i},$

where $K+tB=\\{x\in\mathbb{R}^{n+1}:d(K,x)\leq t\\}$ is the outer parallel
body of $K$ at distance $t\geq 0$. If $K$ is smooth, then the
quermassintegrals can be also expressed in terms of curvature integrals of its
boundary and the enclosed volume as follows:

$\begin{split}&W_{0}(K)=\operatorname{Vol}(K),\quad
W_{k+1}(K)=\frac{1}{n+1}\int_{\partial K}E_{k}d\mu,\quad
k=0,1,\cdots,n,\end{split}$ (1.2)

where $E_{k}$ is the normalized $k$th mean curvature and $d\mu$ is the area
element of $\partial K$, respectively. For simplicity, we define the $k$th
mean radius of the smooth bounded domain $K$ as follows:

$\displaystyle\zeta_{k}(K)=f_{k}^{-1}(W_{k}(K)),$ (1.3)

where $f_{k}:[0,\infty)\rightarrow\mathbb{R}_{+}$ is a monotone function
defined by

$f_{k}(r)=W_{k}(B_{r})=\frac{1}{n+1}\int_{\partial
B_{r}}E_{k-1}d\mu=\frac{\omega_{n}}{n+1}r^{n+1-k},$

the $k$th quermassintegral for the ball of radius $r$, $\omega_{n}$ is the
area of the unit sphere $\mathbb{S}^{n}\subset\mathbb{R}^{n+1}$ and
$f_{k}^{-1}$ is the inverse of $f_{k}$. In terms of the mean radius, the
Blaschke-Santaló inequality (1.1) becomes

$\displaystyle\zeta_{0}(K)\cdot\zeta_{0}(K_{s}^{\ast})\leq 1.$ (1.4)

The following inequalities can be considered as a natural extension of the
Blaschke-Santaló inequality (1.4) in Euclidean space.

######  Theorem A.

Let $K$ be a convex body in $\mathbb{R}^{n+1}$ and $K_{z}^{\ast}$ is its dual
body of $K$ with respect to any point $z\in\operatorname{int}K$. Then for
$0\leq i,j\leq n$ satisfying $i+j\geq n$, there holds

$\displaystyle
W_{i}(K)^{n+1-j}W_{j}(K_{z}^{\ast})^{n+1-i}\geq|\mathbb{B}^{n+1}|^{2n+2-i-j},$
(1.5)

or equivalently,

$\displaystyle\zeta_{i}(K)\cdot\zeta_{j}(K_{z}^{\ast})\geq 1.$ (1.6)

Equality holds if and only if $K$ is a ball centered at $z$.

For $i+j=n$, this result is due to Firey [Firey1973] and for $i=j=n$ to Lutwak
[Lutwak1975]. The general result was deduced from Firey’s case, independently
by Heil [Heil1976] and Lutwak [Lutwak1976]. Further inequalities of this type
were proved by Ghandehari [Ghandehari1991], see also Schneider’s book
[Schneider2014, P. 393].

In the sphere or hyperbolic space, the Steiner formula also makes sense, but
leads to different functionals. An investigation of such functionals within
integral geometry can be found in Santaló’s book [Sant2004]. In the smooth
category, convex bodies and their dual bodies in the sphere, as well as the
counterparts in hyperbolic/de Sitter space, are well-defined in terms of the
Gauss map (see §2.2 for details). The motivation of this paper is to extend
Blaschke-Santaló’s inequality (1.4) and the inequalities (1.6) in the sphere
and hyperbolic/de Sitter space. The quermassintegrals $W_{k}$ in Riemannian
space forms and the upper branch of de Sitter space are introduced in (3.1),
(3.2) respectively, and the mean radius $\zeta_{k}$ can be similarly defined
by (3.3).

The first result of this paper is a complete family of identities for convex
bodies and their dual bodies in the sphere, where the dual body of a convex
body in the sphere is defined by (2.14).

######  Theorem 1.1.

Let $K$ be a smooth bounded and strictly convex domain in $\mathbb{S}^{n+1}$
and $K^{\ast}$ the dual body of $K$. Then the following identities hold:

$\displaystyle\tan\left(\zeta_{n-k}(K)\right)\cdot\tan\left(\zeta_{k}(K^{\ast})\right)=1,\quad
k=0,1,\cdots,n.$ (1.7)

We also prove a complete family of identities for convex bodies in hyperbolic
space and their dual bodies in de Sitter space, where the dual body in de
Sitter space is defined by (2.19).

######  Theorem 1.2.

Let $K$ be a smooth bounded and strictly convex domain in $\mathbb{H}^{n+1}$
and $K^{\ast}$ the dual body of $K$ in $\mathbb{S}^{n,1}_{+}$. Then the
following identities hold:

$\displaystyle\coth(\zeta_{k}(K))\cdot\tanh(\zeta_{n-k}(K^{\ast}))=1,\quad
k=0,1,\cdots,n.$ (1.8)

As a direct application of Theorem 1.1, we obtain the following Blaschke-
Santaló type inequalities in the sphere.

######  Theorem 1.3.

Let $K$ be a smooth bounded and strictly convex domain in $\mathbb{S}^{n+1}$
and $K^{\ast}$ the dual body of $K$.

  1. (i)

For $0<k\leq n$, there hold

$\displaystyle\tan\left(\zeta_{n}(K)\right)\cdot\tan\left(\zeta_{k}(K^{\ast})\right)\geq$
$\displaystyle 1,$ (1.9)
$\displaystyle\tan\left(\zeta_{n-k}(K)\right)\cdot\tan\left(\zeta_{0}(K^{\ast})\right)\leq$
$\displaystyle 1.$ (1.10)

  2. (ii)

For $0\leq l\leq n$, if $0\leq 2i\leq l$ and $0\leq 2j\leq n-l$, then there
hold

$\displaystyle\tan\left(\zeta_{n-l+2i}(K)\right)\cdot\tan\left(\zeta_{l}(K^{\ast})\right)\geq$
$\displaystyle 1,$ (1.11)
$\displaystyle\tan\left(\zeta_{n-l-2j}(K)\right)\cdot\tan\left(\zeta_{l}(K^{\ast})\right)\leq$
$\displaystyle 1.$ (1.12)

Equality holds if and only if $K$ is a geodesic ball.

######  Remark 1.

It should be mentioned that Gao, Hug and Schneider [GHS03, P. 166] proved the
following Blaschke-Santaló type inequality in the sphere: Let $K$ be a convex
body in $\mathbb{S}^{n+1}$ and $K^{\ast}$ the dual body of $K$. Let $B_{r}$ be
a geodesic ball with $\operatorname{Vol}(K)=\operatorname{Vol}(B_{r})$ and
$B_{r}^{\ast}=B_{\frac{\pi}{2}-r}$ the dual body of $B_{r}$. Then
$\operatorname{Vol}(K^{\ast})\leq\operatorname{Vol}(B_{r}^{\ast})$, with
equality holds if and only if $K$ is a geodesic ball. This inequality is
equivalent to

$\displaystyle\tan\left(\zeta_{0}(K)\right)\cdot\tan(\zeta_{0}(K^{\ast}))\leq
1,$

which is a special case ($k=n$) of the inequality (1.10).

Similarly, we apply Theorem 1.2 to obtain the complete family of Blaschke-
Santaló type inequalities in hyperbolic/de Sitter space.

######  Theorem 1.4.

Let $K$ be a smooth bounded h-convex111A smooth bounded domain in hyperbolic
space is called h-convex (which is short for horospherically convex) if the
principal curvatures of its boundary satisfy $\kappa_{i}\geq 1$ for all $i$.
domain in $\mathbb{H}^{n+1}$ and $K^{\ast}$ the dual body of $K$ in
$\mathbb{S}^{n,1}_{+}$. Then for any $0\leq k,l\leq n$, there hold

$\displaystyle\coth(\zeta_{k}(K))\cdot\tanh(\zeta_{l}(K^{\ast}))\geq
1,\quad\text{if $k+l<n$},$ (1.13)

and

$\displaystyle\coth(\zeta_{k}(K))\cdot\tanh(\zeta_{l}(K^{\ast}))\leq
1,\quad\text{if $k+l>n$}.$ (1.14)

Equality holds if and only if $K$ is a geodesic ball.

Another application of Theorem 1.2 is to prove geometric inequalities in
hyperbolic/de Sitter space. We first obtain the following quermassintegral
inequalities in hyperbolic space.

######  Theorem 1.5.

Let $K$ is a smooth bounded and strictly convex domain in $\mathbb{H}^{n+1}$.
Then there hold

$\displaystyle W_{n-1}(K)\geq f_{n-1}\circ f^{-1}_{l}(W_{l}(K)),\quad 0\leq
l<n-1.$ (1.15)

Equality holds if and only if $K$ is a geodesic ball.

######  Remark 2.

The full quermassintegral inequalities in hyperbolic space

$\displaystyle W_{k}(K)\geq f_{k}\circ f_{l}^{-1}(W_{l}(K)),\quad 0\leq
l<k\leq n$ (1.16)

were first proved by Wang and Xia [Wang-Xia14] for smooth bounded h-convex
domains, see also a new proof by the authors of this paper with Wei [Hu-Li-
Wei2020]. It is challenging to extend these inequalities to merely
$(k-1)$-convex222A smooth hypersurface is called $m$-convex if its principal
curvatures satisfy
$\kappa\in\Gamma_{m}^{+}=\\{\kappa\in\mathbb{R}^{n}~{}|~{}E_{i}(\kappa)>0,1\leq
i\leq m\\}$. $n$-convex is strictly convex and $1$-convex is usually refered
as mean convex. and starshaped hypersurfaces in hyperbolic space. In this
direction, the second author with Wei and Xiong [LWX14] proved the inequality
(1.16) with $k=3$ and $l=1$ for $2$-convex and star-shaped hypersurfaces.
Later, Andrews, Chen and Wei [ACW2018] proved the inequality (1.16) with
$0<k\leq n$ and $l=0$ for hypersurfaces with positive sectional curvature333A
smooth hypersurface in hyperbolic space has positive sectional curvature if
its principal curvatures satisfy $\kappa_{i}\kappa_{j}>1$ for all distant
$i,j$.. Recently, Brendle, Guan and Li [BGL19] (see also [GL21]) proved the
inequality (1.16) with $k=2$ and $l=1$ for mean convex and star-shaped
hypersurface, and the inequality (1.16) with $k=n$ and $0\leq l<n$ for
strictly convex hypersurfaces. The authors of this paper with Andrews proved
the inequality (1.16) with $k=n-1$ and $l=n-1-2i(0<2i<n)$ for strictly convex
hypersurfaces.

Several special cases of the Blaschke-Santaló type inequalities in Theorem 1.4
can be generalized to the strictly convex domains in hyperbolic/de Sitter
space.

######  Theorem 1.6.

Let $K$ be a smooth bounded and strictly convex domain in $\mathbb{S}^{n+1}$
and $K^{\ast}$ the dual body of $K$. Then there hold

$\displaystyle\coth(\zeta_{l}(K))\cdot\tanh(\zeta_{1}(K^{\ast}))\geq$
$\displaystyle 1,\quad 0\leq l<n-1,$ (1.17)

and

$\displaystyle\coth(\zeta_{k}(K))\cdot\tanh(\zeta_{0}(K^{\ast}))\geq$
$\displaystyle 1,\quad 0\leq k<n.$ (1.18)

Equality holds if and only if $K$ is a geodesic ball.

######  Remark 3.

The inequality (1.18) with $k=0$ is equivalent to the following Blachke-
Santaló type inequality which was previously proved by Gao, Hug and Schneider
[GHS03] via two-point symmetrization: Let $K$ be a convex body in hyperbolic
space and $K^{\ast}$ the dual body in de Sitter space. Let $B_{r}$ be a
geodesic ball with $\operatorname{Vol}(K)=\operatorname{Vol}(B_{r})$ and
$B_{r}^{\ast}$ the dual body of $B_{r}$. Then
$\operatorname{Vol}(K^{\ast})\leq\operatorname{Vol}(B_{r}^{\ast})$, with
equality holds if and only if $K$ is a geodesic ball.

Motivated by the quermassintegral inequalities in hyperbolic space, it is
natural to investigate the following quermassintegral inequalities in de
Sitter space.

######  Conjecture 1.1.

Let $K$ be a smooth bounded domain with $(k-1)$-convex and spacelike boundary
$\partial K$ in $\mathbb{S}_{+}^{n,1}$. Then there hold

$\displaystyle W_{k}(K)\leq f_{k}\circ f_{l}^{-1}(W_{l}(K)),\quad 0\leq
l<k\leq n.$ (1.19)

Equality holds if and only if $\partial K$ is isometric to a coordinate slice.

######  Remark 4.

Two special cases of Conjecture 1.1 have been completely solved:

  1. (i)

The inequality (1.19) with $k=1$, $l=0$ was verified by Lambert and Scheuer
[Lambert-Scheuer2021], who proved the isoperimetric inequalities for spacelike
domains in generalized Robertson-Walker spaces including de Sitter space.

  2. (ii)

The inequality (1.19) with $k=2$, $l=1$ was recently proved by Scheuer
[Scheuer19].

As an application of Theorem 1.2, we prove the full quermassintegral
inequalities (1.19) under the stronger assumption.

######  Theorem 1.7.

Let $K$ be a smooth bounded domain with spacelike boundary $\partial K$ in
$\mathbb{S}_{+}^{n,1}$. Assume that the principal curvatures of $\partial K$
satisfy $0<\kappa_{i}\leq 1$. Then there hold

$\displaystyle W_{k}(K)\leq f_{k}\circ f_{l}^{-1}(W_{l}(K)),\quad 0\leq
l<k\leq n.$ (1.20)

Equality holds if and only if $\partial K$ is isometric to a coordinate slice.

The inequality (1.20) with $1<k\leq n$ and $l=1$ holds for strictly convex
hypersurfaces in $\mathbb{S}_{+}^{n,1}$.

######  Theorem 1.8.

Let $K$ be a smooth bounded domain with strictly convex and spacelike boundary
in $\mathbb{S}^{n,1}_{+}$. Then there hold

$\displaystyle W_{k}(K)\leq f_{k}\circ f_{1}^{-1}(W_{1}(K)),\quad 1<k\leq n,$
(1.21)

and

$\displaystyle W_{n-1}(K)\leq$ $\displaystyle f_{n-1}\circ
f_{n-2}^{-1}(W_{n-2}(K)),$ (1.22) $\displaystyle W_{n-1}(K)\leq$
$\displaystyle f_{n-1}\circ f_{n-3}^{-1}(W_{n-3}(K)).$ (1.23)

Equality holds if and only if $\partial K$ is isometric to a coordinate slice.

######  Remark 5.

Beyond the quermassintegral inequalities, it is not difficult to apply Theorem
1.2 to transform all geometric inequalities in hyperbolic space (e.g. [GeWW14,
HL19, Hu-Li-2022, Wang-Xia14, Hu-Li-Wei2020]) to their counterparts in de
Sitter space under appropriate convexity assumptions. This idea was previously
utilized by Andrews and the authors of this paper, see [AHL2020, Theorem 1.4,
Remark 6.4].

The paper is organized as follows. In §2, we collect the preliminaries on
starshaped/spacelike hypersurface in Riemannian/Lorentzian warped product,
duality and dual flows in the sphere and in hyperbolic/de Sitter space. In §3,
we give the definition of quermassintegrals and their variational formulas in
Riemannian space forms and in de Sitter space. In §4, we give the proofs of
Theorem 1.1 and Theorem 1.2. In §5, we give the proofs of Theorems 1.3–1.8.

######  Acknowledgments.

This work was supported by National Key R and D Program of China
2021YFA1001800, NSFC grant No.12101027, NSFC grant No.11831005 and NSFC grant
No.12126405.

##  2\. Preliminaries

###  2.1. Hypersurfaces in semi-Riemannian manifolds

Let $(N,\overline{g})$ be an $(n+1)$-dimensional semi-Riemannian manifold. The
coordinates $(x^{\alpha})_{0\leq\alpha\leq n}$ in $N$ are labelled from $0$ to
$n$. In general, the coordinate $x^{0}$ refers to the radial distance to a
fixed point or the time function. Let $(\xi^{i})$ be the local coordinates
system of an open neighbourhood $U$ in $M$, where $M$ is a smooth hypersurface
in $N$. Locally, $M$ can be represented by a map $X=X(\xi)=(x^{\alpha}(\xi))$,
and the induced metric is given by

$\displaystyle
g_{ij}=\overline{g}(X_{i},X_{j}),\quad\text{where}~{}X_{i}=\frac{\partial
X}{\partial\xi^{i}}.$

Since $M$ has codimension $1$, its normal space is spanned by a single vector
$\nu$ at each point $x\in M$. We can always define a continuous normal vector
field $\nu$ by requiring that

$\displaystyle\det\left(X_{1},\cdots,X_{n},\nu\right)>0,\quad\forall\xi\in U.$

Moreover, the normal vector $\nu$ is normalized such that

$\displaystyle\overline{g}(\nu,\nu)=\sigma=\pm 1.$

The geometry of a hypersurface $M$ in a semi-Riemannian manifold $N$ is
governed by the following basic equations (see [Gerh06,
(1.1.6),(1.1.21),(1.1.36),(1.1.37)]):

  1. (i)

The Gauss formula is

$\displaystyle X_{ij}=-\sigma h_{ij}\nu,$ (2.1)

where $X_{ij}$ is the second covariant derivatives of $X$ with respect to the
induced metric $g$, and $h_{ij}$ is called the second fundamental form of $M$
with respect to $-\sigma\nu$.

  2. (ii)

The Weingarten equation is

$\displaystyle\nu_{i}=h_{i}^{k}X_{k},$ (2.2)

where $h_{i}^{k}=g^{kl}h_{il}$ is the Weingarten matrix, with its eigenvalues
$\kappa=(\kappa_{1},\cdots,\kappa_{n})$ being called the principal curvatures
of $M$ in $N$.

  3. (iii)

The Codazzi equation is

$\displaystyle
h_{ij;k}-h_{ik;j}=\overline{R}_{\alpha\beta\gamma\delta}\nu^{\alpha}X_{i}^{\beta}X_{j}^{\gamma}X_{k}^{\delta},$
(2.3)

where $h_{ij;k}=\nabla_{k}h_{ij}$ and $\overline{R}_{\alpha\beta\gamma\delta}$
is the components of Riemann curvature tensor.

  4. (iv)

The Gauss equation is

$\displaystyle
R_{ijkl}=\sigma(h_{ik}h_{jl}-h_{jk}h_{il})+\overline{R}_{\alpha\beta\gamma\delta}X_{i}^{\alpha}X_{j}^{\beta}X_{k}^{\gamma}X_{l}^{\delta}.$
(2.4)

######  Remark 6.

The choice of the normal $\nu$ in the Gaussian formula (2.1) is free, i.e., we
could just as well have replaced $\nu$ by $-\nu$, then the principal
curvatures $\kappa_{i}$ would have been replaced by $-\kappa_{i}$. We make the
following convention on the choice of $\nu$:

  1. (1)

If $M$ is a closed oriented hypersurface and the ambient space is Riemannian,
then in this case we always choose $\nu$ as the outward normal to $M$. Then
the starshapedness of $M$ is equivalent to $\overline{g}(\partial_{r},\nu)>0$.

  2. (2)

If $M$ is spacelike (i.e., the induced metric $g$ of $M$ is Riemannian) and
the ambient space is Lorentzian, and the coordinate system $(x^{\alpha})$ is
supposed to be future directed, i.e., the time function $x^{0}$ is increasing
on future directed curves. Then in this case we always choose $\nu$ to be also
future directed (timelike) normal, which is equivalent to
$\overline{g}(\partial_{r},\nu)<0$.

In the sequel, the Euclidean space $\mathbb{R}^{n+2}$ is the
$(n+2)$-dimensional vector space equipped with the bilinear form

$\displaystyle\langle v,w\rangle=\sum_{\alpha=0}^{n+1}v^{\alpha}w^{\alpha},$

while the Minkowski space $\mathbb{R}^{n+1,1}$ is the $(n+2)$-dimensional
vector space equipped with the bilinear form

$\displaystyle\langle
v,w\rangle=-v^{0}w^{0}+\sum_{\alpha=1}^{n+1}v^{\alpha}w^{\alpha}.$

The Riemannian space form $\mathbb{N}^{n+1}(\epsilon)$ is the Riemannian
manifold with constant curvature $\epsilon$:

$\displaystyle\left\\{\begin{aligned} &\text{if
$\epsilon=1$},\quad\mathbb{S}^{n+1}=\\{y\in\mathbb{R}^{n+2}:~{}\langle
y,y\rangle=1\\};\\\ &\text{if $\epsilon=0$},\quad\mathbb{R}^{n+1};\\\
&\text{if
$\epsilon=-1$},\quad\mathbb{H}^{n+1}=\\{y\in\mathbb{R}^{n+1,1}:~{}\langle
y,y\rangle=-1,y^{0}>0\\}.\end{aligned}\right.$

Similarly, let $\mathbb{N}^{n,1}(\epsilon)$ be the following Lorentzian
manifold of constant curvature $\epsilon$ (cf. [Oneil1983, P. 110,228]):

$\displaystyle\left\\{\begin{aligned} &\text{if
$\epsilon=1$},\quad\mathbb{S}^{n,1}=\\{y\in\mathbb{R}^{n+1,1}:~{}\langle
y,y\rangle=1\\};\\\ &\text{if $\epsilon=0$},\quad\mathbb{R}^{n,1};\\\
&\text{if
$\epsilon=-1$},\quad\mathbb{H}^{n,1}=\\{y\in\mathbb{R}^{n,2}:~{}\langle
y,y\rangle=-1\\},\end{aligned}\right.$

where $\mathbb{R}^{n,2}$ is the $(n+2)$-dimensional vector space endowed with
the bilinear form

$\displaystyle\langle
v,w\rangle=-v^{0}w^{0}-v^{1}w^{1}+\sum_{\alpha=2}^{n+1}v^{\alpha}w^{\alpha}.$

Here $\mathbb{S}^{n,1}$ and $\mathbb{H}^{n,1}$ are called de Sitter space and
anti de Sitter space, respectively. The upper branch of de Sitter space is
defined by

$\displaystyle\mathbb{S}^{n,1}_{+}=\\{y\in\mathbb{R}^{n+1,1}:~{}\langle
y,y\rangle=1,~{}y^{0}>0\\}.$

The Rimannian space form $\mathbb{N}^{n+1}(\epsilon)$ and de Sitter space
$\mathbb{S}^{n,1}$ can be expressed as a warped product manifold with the base
manifold $\mathbb{S}^{n}$. Let $N=I\times\mathbb{S}^{n}$ be a Riemannian or
Lorentzian warped product with metric

$\displaystyle\overline{g}=\sigma dr^{2}+\lambda^{2}(r)g_{\mathbb{S}^{n}},$
(2.5)

where $g_{\mathbb{S}^{n}}=\sigma_{ij}d\theta^{i}d\theta^{j}$ is the round
metric of $\mathbb{S}^{n}\subset\mathbb{R}^{n+1}$ and

  1. (1)

if $N=\mathbb{S}^{n+1}$, then $I=[0,\pi)$, $\sigma=1$ and $\lambda(r)=\sin r$;

  2. (2)

if $N=\mathbb{H}^{n+1}$, then $I=[0,\infty)$, $\sigma=1$ and $\lambda(r)=\sinh
r$;

  3. (3)

if $N=\mathbb{S}^{n,1}$, then $I=(-\infty,\infty)$, $\sigma=-1$ and
$\lambda(r)=\cosh r$, see e.g. [Scheuer19, Lemma 2.1].

If $M$ is a compact, starshaped hypersurface in $\mathbb{N}^{n+1}(\epsilon)$,
then it can be represented as a smooth radial graph over $\mathbb{S}^{n}$,
i.e., in the coordinates

$\displaystyle
M=\\{(\rho(\theta),\theta):\theta\in\mathbb{S}^{n}\\}=\\{(\rho(\theta(\xi)),\theta(\xi)):\xi\in
M\\},$ (2.6)

where $\rho:\mathbb{S}^{n}\rightarrow\mathbb{R}^{+}$ is a smooth function.

Similarly, it follows from [Gerh06, Proposition 1.6.3, Remark 1.6.4] that if
$M$ is a compact, spacelike hypersurface in $\mathbb{S}_{+}^{n,1}$, then $M$
is also a smooth graph over $\mathbb{S}^{n}$ as (2.6), since the compact
Cauchy hypersurface $\mathcal{S}_{0}$ can be chosen as the slice
$\\{y^{0}=0\\}$ which is isometric to $\mathbb{S}^{n}$. Here we restrict the
de Sitter space $\mathbb{S}^{n,1}$ to be its upper branch
$\mathbb{S}^{n,1}_{+}$ and hence $\rho>0$, which guarantees that a bounded
domain can be enclosed by a spacelike hypersurface and $\\{y^{0}=0\\}$. Along
the hypersurface $M$ in $\mathbb{S}^{n,1}_{+}$, the future directed unit
timelike normal is given by

$\displaystyle\nu=\frac{1}{v}\left(1,-\frac{1}{\cosh^{2}\rho}\sigma^{ik}\rho_{k}\right),$
(2.7)

where $\rho_{k}=\partial_{\theta^{k}}\rho$ is the covariant derivative on
$\mathbb{S}^{n}$ and

$\displaystyle
v^{2}=1-\cosh^{-2}\rho\sigma^{ij}\rho_{i}\rho_{j}=1-\cosh^{-2}\rho|D\rho|^{2}.$
(2.8)

The induced metric and the area element of $M$ are given by

$\displaystyle g_{ij}=-\rho_{i}\rho_{j}+\cosh^{2}\rho\sigma_{ij},$

and

$\displaystyle d\mu=\sqrt{\det(g_{ij})}d\theta=\cosh^{n}\rho
v\sqrt{\det(\sigma_{ij})}d\theta.$ (2.9)

Along the variational vector field $\frac{\partial}{\partial t}X$, the
spacelike hypersurfaces $M_{t}$ can be parametrized by

$\displaystyle M_{t}=\\{(\rho(t,\theta),\theta):\theta\in\mathbb{S}^{n}\\},$

where $\rho(t,\cdot)$ is a smooth function defined on the unit sphere. Hence
the evolution of the spacelike hypersurfaces $M_{t}$ can be reduced to a
parabolic equation for the radial function. As long as the evolving
hypersurface exists and remains spacelike, the radial function $\rho$
satisfies

$\displaystyle\frac{\partial\rho}{\partial t}=\langle\frac{\partial}{\partial
t}X,\nu\rangle v,$ (2.10)

where $v$ is given by (2.8). Notice that the velocity vector of (2.10) is in
the direction of $\partial_{r}$. A direct calculation yields

$\displaystyle\frac{\partial}{\partial t}X-\frac{\partial\rho}{\partial
t}\partial_{r}=$ $\displaystyle(\frac{\partial}{\partial
t}X)^{\top}+\langle\frac{\partial}{\partial
t}X,\nu\rangle(\nu-v\partial_{r}),$

and $\langle\nu-v\partial_{r},\nu\rangle=0$. Therefore, the difference
$\frac{\partial}{\partial t}X-\frac{\partial\rho}{\partial t}\partial_{r}$ is
a time-dependent tangential vector field, and (2.10) follows by composing the
original flow with the reparametrization associated with this tangential
vector field.

###  2.2. Duality

In this subsection, we recall the duality relations in the sphere and in
hyperbolic/de Sitter space via the Gauss map. For more details, we refer the
readers to Gerhardt’s book [Gerh06, Chapters 9&10].

Duality in the sphere. Let $X:M_{0}\rightarrow M\subset\mathbb{S}^{n+1}$ be an
embedding of a closed, connected, strictly convex hypersurface. Then by the
Hadamard theorem in the sphere [DoCarmo-Warner1970], $M$ is embedded,
homeomorphic to $\mathbb{S}^{n}$, contained in an open hemisphere and it is
the boundary of a convex body $\hat{M}\subset\mathbb{S}^{n+1}$. Viewing $M$ as
a submanifold of codimension $2$ in $\mathbb{R}^{n+2}$, its Gaussian formula
is

$\displaystyle X_{ij}=-g_{ij}X-h_{ij}X^{\ast},$

where $X^{\ast}\in T_{X}\mathbb{R}^{n+2}$ represents the unit outward normal
$\nu\in T_{X}\mathbb{S}^{n+1}$. Then the map

$\displaystyle X^{\ast}:M_{0}\rightarrow M^{\ast}\subset\mathbb{S}^{n+1}$
(2.11)

is an embedding of a closed and strictly convex hypersurface $M^{\ast}$. We
call this map the Gauss map of $M$.

The duality in the sphere is characterized by the following theorem.

######  Theorem 2.1.

[Gerh06, Theorem 9.2.5, 9.2.9] Let $X:M_{0}\rightarrow
M\subset\mathbb{S}^{n+1}$ be a closed, connected, strictly convex
hypersurface. Then the Gauss map $X^{\ast}$ given by (2.11) is the embedding
of a closed, connected, strictly convex hypersurface
$M^{\ast}\subset\mathbb{S}^{n+1}$. Viewing $M^{\ast}$ as a submanifold of
codimension $2$ in $\mathbb{R}^{n+2}$, its Gaussian formula is

$\displaystyle X^{\ast}_{ij}=-g^{\ast}_{ij}X^{\ast}-h^{\ast}_{ij}X,$ (2.12)

where $g^{\ast}_{ij}$ and $h^{\ast}_{ij}$ are the metric and the second
fundamental form of the hypersurface $M^{\ast}\subset\mathbb{S}^{n+1}$ and
$X=X(\xi)$ is the embedding of $M$ which also represents the unit outward
normal of $M$. The second fundamental form $h^{\ast}_{ij}$ is defined with
respect to the unit inner normal vector.

The second fundamental forms of $M$, $M^{\ast}$ and the corresponding
principal curvatures $\kappa_{i},\kappa_{i}^{\ast}$ satisfy

$\displaystyle h_{ij}=h^{\ast}_{ij}=\langle
X^{\ast}_{i},X_{j}\rangle,\quad\kappa_{i}^{\ast}=\kappa_{i}^{-1},$ (2.13)

where $\langle\cdot,\cdot\rangle$ is the inner product in $\mathbb{R}^{n+2}$.

The hypersurface $M^{\ast}$ is also called the polar hypersurface of $M$,
which can be equivalently defined by

$M^{\ast}=\\{y\in\mathbb{S}^{n+1}:\sup_{x\in M}\langle x,y\rangle=0\\}.$

Let $\hat{M}$ be the (closed) convex body of $M\subset\mathbb{S}^{n+1}$. Then
the dual body of $\hat{M}$ is defined by

$\displaystyle\hat{M}^{\ast}=\\{y\in\mathbb{S}^{n+1}:\langle x,y\rangle\leq
0,~{}\forall~{}x\in\hat{M}\\}.$ (2.14)

By the Hadamard theorem, there exists a point $z\in\hat{M}$ such that
$\hat{M}\subset\mathcal{H}(z)$, where $\mathcal{H}(z)$ is the open hemisphere
centered at $z\in\mathbb{S}^{n+1}$. Then it follows from [Gerh06, Corollary
9.2.10] that the dual body $\hat{M}^{\ast}$ of $\hat{M}$ is a convex body in
$\mathcal{H}(-z)$, where $-z\in\mathbb{S}^{n+1}$ is the antipodel point of
$z$. Moreover, the dual body $\hat{M}^{\ast}$ is the convex body of
$M^{\ast}$. Furthermore, we have $(M^{\ast})^{\ast}=M$ and
$(\hat{M}^{\ast})^{\ast}=\hat{M}$.

Duality between hyperbolic/de Sitter space. Similar as the duality in the
sphere, let $X:M_{0}\rightarrow M\subset\mathbb{H}^{n+1}$ be an embedding of a
closed, connected, strictly convex hypersurface. Then the representation
$X^{\ast}\in T_{X}\mathbb{R}^{n+1,1}$ of the unit outward normal $\nu\in
T_{X}\mathbb{H}^{n+1}$ is also an embedding

$\displaystyle X^{\ast}:M_{0}\rightarrow M^{\ast}\subset\mathbb{S}^{n,1}$
(2.15)

of a strictly convex, closed and spacelike hypersurface $M^{\ast}$. We also
call $X^{\ast}$ the Gauss map of $M$. The duality in hyperbolic/de Sitter
space is characterized by the following two theorems.

######  Theorem 2.2.

[Gerh06, Theorem 10.4.4] Let $X:M_{0}\rightarrow M\subset\mathbb{H}^{n+1}$ be
a closed, connected, strictly convex hypersurface. Then the Gauss map
$X^{\ast}$ given by (2.15) is an embedding of a closed, spacelike, strictly
convex hypersurface $M^{\ast}\subset\mathbb{S}^{n,1}$. Viewing $M^{\ast}$ as a
submanifold of codimension $2$ in $\mathbb{R}^{n+1,1}$, its Gaussian formula
is

$\displaystyle X^{\ast}_{ij}=-g^{\ast}_{ij}X^{\ast}-h^{\ast}_{ij}X,$ (2.16)

where $g^{\ast}_{ij}$ and $h^{\ast}_{ij}$ are the metric and the second
fundamental form of the hypersurface $M^{\ast}\subset\mathbb{S}^{n,1}$ and
$X=X(\xi)$ is the embedding of $M$ which also represents the future directed
normal vector of $M^{\ast}$. The second fundamental form $h^{\ast}_{ij}$ is
defined with respect to the future directed unit normal, where the time
orientation is inherited from $\mathbb{R}^{n+1,1}$.

The second fundamental forms of $M$, $M^{\ast}$ and the corresponding
principal curvatures $\kappa_{i},\kappa_{i}^{\ast}$ satisfy

$\displaystyle h_{ij}=h^{\ast}_{ij}=\langle
X^{\ast}_{i},X_{j}\rangle,\quad\kappa_{i}^{\ast}=\kappa_{i}^{-1}.$ (2.17)

where $\langle\cdot,\cdot\rangle$ is the inner product in
$\mathbb{R}^{n+1,1}$.

The hypersurface $M^{\ast}$ is also called the polar hypersurface of $M$,
which can be also equivalently defined by

$M^{\ast}=\\{y\in\mathbb{S}^{n,1}:\sup_{x\in M}\langle x,y\rangle=0\\}.$

Let $\hat{M}$ be the (closed) convex body of $M\subset\mathbb{H}^{n+1}$, which
is characterized by (see [Gerh06, Lemma 10.4.1])

$\displaystyle\hat{M}=\\{y\in\mathbb{H}^{n+1}:\langle y,X^{\ast}\rangle\leq
0,~{}\forall X\in M\\}.$ (2.18)

In the hyperboloid model of the hyperbolic space, the point
$e_{0}=(1,0,\cdots,0)\in\mathbb{R}^{n+1,1}$ is called the Beltrami point. The
Hadamard theorem in hyperbolic space ([Gerh06, Theorem 10.3.1]) states that
any strictly convex hypersurface $M\subset\mathbb{H}^{n+1}$ bounds a strictly
convex body $\hat{M}$ of the hyperbolic space. Due to the homogeneity of the
hyperbolic space, any interior point $z$ in $\hat{M}$ may act as a Beltrami
point by using a Lorenz boost $\mathbb{O}_{z}\in O(n+1,1)$ to bring $z$ to
$e_{0}$. By [Gerh06, Theorem 10.4.9], the Gauss maps provide a bijection
between the closed, strictly convex and connected hypersurfaces
$M\subset\mathbb{H}^{n+1}$ having the Beltrami point in the interior of their
convex bodies and the spacelike, closed, connected and strictly convex
hypersurfaces $M^{\ast}\subset\mathbb{S}^{n,1}_{+}$, where
$\mathbb{S}^{n,1}_{+}$ is the upper branch of the de Sitter space. Therefore,
the polar hypersurface $M^{\ast}$ of $M$ is contained in
$\mathbb{S}^{n,1}_{+}$. Moreover, the dual body of $\hat{M}$ can be defined by

$\displaystyle\hat{M}^{\ast}=\\{y\in\mathbb{S}^{n,1}:\langle x,y\rangle\leq
0,~{}\forall x\in\hat{M}\\}\subset\mathbb{S}^{n,1}_{+},$ (2.19)

which is the bounded domain enclosed by the slice $\\{y^{0}=0\\}$ and the
hypersurface $M^{\ast}$.

The reverse direction starting from a closed, spacelike, strictly convex
hypersurface in $\mathbb{S}^{n,1}_{+}$ will be stated as follows:

######  Theorem 2.3.

[Gerh06, Theorem 10.4.5] Let $X^{\ast}:M_{0}\rightarrow
M^{\ast}\subset\mathbb{S}_{+}^{n,1}$ be a closed, connected, spacelike and
strictly convex hypersurface. Viewing $M^{\ast}$ as a submanifold of
codimension $2$ in $\mathbb{R}^{n+1,1}$, its Gaussian formula is

$\displaystyle X^{\ast}_{ij}=-g^{\ast}_{ij}X^{\ast}+h^{\ast}_{ij}X,$ (2.20)

where $X^{\ast}=X^{\ast}(\xi)$ is the embedding, $X$ the future directed unit
normal, and $g^{\ast}_{ij}$ and $h^{\ast}_{ij}$ the induced metric and the
second fundamental form of the hypersurface $M^{\ast}\subset\mathbb{S}^{n,1}$.
We define the Gauss map as $X=X(\xi)$

$\displaystyle X:M_{0}\rightarrow M\subset\mathbb{H}^{n+1}.$

Then the Gauss map is the embedding of a closed, connected, strictly convex
hypersurface $M$ in $\mathbb{H}^{n+1}$. Viewing $M$ as a submanifold of
codimension $2$ in $\mathbb{R}^{n+1,1}$, then its Gaussian formula is

$\displaystyle X_{ij}=g_{ij}X-h_{ij}X^{\ast}.$

where $g_{ij}$, $h_{ij}$ be the induced metric and the second fundamental form
of $M$. The second fundamental forms of $M$, $M^{\ast}$ and the corresponding
principal curvatures $\kappa_{i},\kappa_{i}^{\ast}$ satisfy

$\displaystyle h_{ij}=h^{\ast}_{ij}=\langle
X_{i},X^{\ast}_{j}\rangle,\quad\kappa^{\ast}_{i}=\kappa_{i}^{-1},$ (2.21)

Combining Theorems 2.2 with 2.3, we conclude that $(M^{\ast})^{\ast}=M$ and
$(\hat{M}^{\ast})^{\ast}=\hat{M}$.

######  Remark 7.

For a convex body $K$ in Euclidean space, its dual body $K_{z}^{\ast}$ depends
on the choice of the interior point $z$ in $K$. For a strictly convex body $K$
in $\mathbb{S}^{n+1}$ or $\mathbb{H}^{n+1}$, or a spacelike and strictly
convex domain $K$ in $\mathbb{S}^{n,1}_{+}$, its dual body $K^{\ast}$ in
$\mathbb{S}^{n+1}$, $\mathbb{S}^{n,1}_{+}$ or $\mathbb{H}^{n+1}$ of $K$ is
defined by (2.14), (2.19) and (2.18), respectively. This definition is
independent of the choice of the interior point in $K$.

###  2.3. Dual flows

The dual flows for (pure) curvature flows in the sphere was first introduced
by Gerhardt [Ge15], and later it was also used by Yu [Yu2016] for curvature
flows in hyperbolic space. For general curvature flows, the deduction of the
dual flows can be found in [Bryan-Ivaki-Scheuer2020, (4.3),(4.4)].

Let $X_{0}:M_{0}^{n}\rightarrow(N^{n+1},\overline{g})$ be a smooth embedding
such that $M=X_{0}(M_{0})$ is a closed, strictly convex hypersurface, where
$N$ is one of $\mathbb{S}^{n+1}$, $\mathbb{H}^{n+1}$ and
$\mathbb{S}^{n,1}_{+}$. We consider the smooth family of strictly convex
embeddings $X:M_{0}\times[0,T)\rightarrow N$ satisfying

$\displaystyle\left\\{\begin{aligned} \frac{\partial}{\partial
t}X=&-\sigma\mathcal{F}\nu,\\\ X(\cdot,0)=&X_{0}(\cdot).\end{aligned}\right.$
(2.22)

where $\nu$ is the unit normal of $M_{t}=X(M_{0},t)$ and $\mathcal{F}$ is a
smooth and strictly monotone function of the principal curvatures of $M_{t}$.
Here the signature $\sigma=\overline{g}(\nu,\nu)=\langle\nu,\nu\rangle$, where
$\langle\cdot,\cdot\rangle$ represents the Euclidean or the Minkowski inner
product for the flows in $\mathbb{S}^{n+1}$ or $\mathbb{H}^{n+1}$,
$\mathbb{S}^{n,1}$, respectively. In all three cases, the pair $X$,
$X^{\ast}=\nu$ satisfies $\langle X,X^{\ast}\rangle=0$. It follows that

$\displaystyle\langle\frac{\partial}{\partial
t}X^{\ast},X\rangle=-\langle\frac{\partial}{\partial
t}X,X^{\ast}\rangle=-\langle-\sigma\mathcal{F}X^{\ast},X^{\ast}\rangle=\mathcal{F}.$

Due to $\langle X^{\ast},X^{\ast}_{i}\rangle=0$, the Weingarten equation
$X^{\ast}_{i}=h_{i}^{k}X_{k}$ (see also [Gerh06, Lemma 9.2.4, Lemma 10.4.3])
imply that

$\displaystyle 0=\langle\frac{\partial}{\partial
t}X^{\ast},X^{\ast}_{i}\rangle=\langle\frac{\partial}{\partial
t}X^{\ast},h_{i}^{k}X_{k}\rangle=-h_{i}^{k}\langle
X^{\ast},(-\sigma\mathcal{F}X^{\ast})_{k}\rangle=\sigma h_{i}^{k}\langle
X^{\ast},\mathcal{F}_{k}X^{\ast}\rangle=h_{i}^{k}\mathcal{F}_{k},$

where $\mathcal{F}_{k}=\frac{\partial\mathcal{F}}{\partial\xi^{k}}$. Note that
$X=(X^{\ast})^{\ast}=\nu^{\ast}$ and $X^{\ast}_{i}=h_{i}^{k}X_{k}$ span
$T_{X^{\ast}}\mathbb{S}^{n+1}$, $T_{X^{\ast}}\mathbb{H}^{n+1}$ or
$T_{X^{\ast}}\mathbb{S}^{n,1}$ respectively, we obtain

$\displaystyle\frac{\partial}{\partial t}X^{\ast}=\langle
X,X\rangle\mathcal{F}X+h_{i}^{k}\mathcal{F}_{k}(g^{\ast})^{il}X^{\ast}_{l}=\sigma^{\ast}\mathcal{F}\nu^{\ast}+h_{i}^{k}(g^{\ast})^{il}\mathcal{F}_{k}X^{\ast}_{l}=\sigma^{\ast}\mathcal{F}\nu^{\ast}+(b^{\ast})^{kl}\mathcal{F}_{k}X^{\ast}_{l},$

where $\sigma^{\ast}=\langle X,X\rangle$, $((b^{\ast})_{i}^{j})$ is the
inverse matrix of the Weingarten matrix
$\mathcal{W}^{\ast}=((h^{\ast})_{i}^{j})$ of $M^{\ast}$ and
$h_{i}^{k}=(b^{\ast})_{i}^{k}$ due to (2.13), (2.17) and (2.21). Let us define

$\displaystyle\mathcal{F}^{\ast}(\mathcal{W}^{\ast}):=-\mathcal{F}(\mathcal{W}).$
(2.23)

Let $N^{\ast}$ be dual ambient space of $N$, that is, $\mathbb{S}^{n+1}$,
$\mathbb{S}^{n,1}$ and $\mathbb{H}^{n+1}$, respectively. Then the flow of the
polar hypersurfaces is a smooth family of strictly convex embeddings
$X^{\ast}:M_{0}\times[0,T)\rightarrow N^{\ast}$ satisfying

$\displaystyle\left\\{\begin{aligned} \frac{\partial}{\partial
t}X^{\ast}=&-\sigma^{\ast}\mathcal{F}^{\ast}\nu^{\ast}-(b^{\ast})^{kl}\mathcal{F}^{\ast}_{k}X^{\ast}_{l},\\\
X^{\ast}(\cdot,0)=&X^{\ast}_{0}(\cdot).\end{aligned}\right.$ (2.24)

where $\mathcal{F}^{\ast}$ is evaluated at the Weingarten matrix
$\mathcal{W}^{\ast}$ of $M_{t}^{\ast}=X^{\ast}(M_{0},t)$, and
$X^{\ast}_{0}:M_{0}\rightarrow N^{\ast}$ is the smooth embedding of
$M^{\ast}=X_{0}^{\ast}(M_{0})$ which is determined by the Gauss map of the
hypersurface $M$. Therefore, a flow of the form (2.22) in the ambient spaces
$\mathbb{S}^{n+1}$, $\mathbb{H}^{n+1}$, $\mathbb{S}^{n,1}$ has a dual flow of
the form (2.24) in the ambient spaces $\mathbb{S}^{n+1}$, $\mathbb{S}^{n,1}$,
$\mathbb{H}^{n+1}$, respectively.

##  3\. Quermassintegrals

Let $K$ be a bounded domain with smooth boundary $\partial K$ in Riemannian
space forms $\mathbb{N}^{n+1}(\epsilon)$ with constant curvature
$\epsilon\in\\{-1,0,1\\}$. Then the quermassintegrals $W_{k}(K)$ can be
expressed by a linear combination of the curvature integrals and of the
enclosed volume as follows:

$\begin{split}&W_{0}(K)=\operatorname{Vol}(K),\quad
W_{1}(K)=\frac{1}{n+1}\operatorname{Area}(\partial K),\\\
&W_{k+1}(K)=\frac{1}{n+1}\int_{\partial
K}E_{k}d\mu+\epsilon\frac{k}{n+2-k}W_{k-1}(K),\quad k=1,\cdots,n,\end{split}$
(3.1)

where $E_{k}$ is the normalized $k$th mean curvature and $d\mu$ is the area
element of $\partial K$.

When the ambient space is the upper branch of the de Sitter space, then the
volume of the region $K$ enclosed by the slice $\\{y^{0}=0\\}$ and a spacelike
graph $\partial K=\\{(\rho(\theta),\theta):\theta\in\mathbb{S}^{n}\\}$ can be
expressed by (see [Makowski2013, Section 4])

$\displaystyle\operatorname{Vol}(K)=\int_{\mathbb{S}^{n}}\int_{0}^{\rho(\theta)}\sqrt{|\det(\overline{g}_{ij}(r,\theta))|}drd\theta=\int_{\mathbb{S}^{n}}\int_{0}^{\rho(\theta)}\cosh^{n}r\sqrt{\det(\sigma_{ij})}drd\theta,$

where $\overline{g}=-dr^{2}+\cosh^{2}rg_{\mathbb{S}^{n}}$. As the hypersurface
$\partial K$ is spacelike, the surface area of $\partial K$ is defined as
usual by its induced Riemannian metric. For simplicity, we also call $K$ to be
a spacelike domain in $\mathbb{S}^{n,1}_{+}$ if $\partial K$ is a spacelike
hypersurface. It is natural to define the quermassintegrals for a smooth
bounded domain $K$ with spacelike boundary $\partial K$ in the upper branch of
the de Sitter space. In this direction, Scheuer [Scheuer19] defined the second
quermassintegral as

$\displaystyle W_{2}(K)=\frac{1}{n+1}\int_{\partial
K}E_{1}d\mu-\frac{1}{n+1}\operatorname{Vol}(K).$

Inspired by the expression (3.1) of quermassintegrals in the Riemannian space
forms, the quermassintegrals for a spacelike domain $K$ in
$\mathbb{S}_{+}^{n,1}$ can be defined inductively by

$\begin{split}&W_{0}(K)=\operatorname{Vol}(K),\quad
W_{1}(K)=\frac{1}{n+1}\operatorname{Area}(\partial K),\\\
&W_{k+1}(K)=\frac{1}{n+1}\int_{\partial
K}E_{k}d\mu-\frac{k}{n+2-k}W_{k-1}(K),\quad k=1,\cdots,n.\end{split}$ (3.2)

In Riemannian space forms $\mathbb{N}^{n+1}(\epsilon)$, the definition (3.1)
coincides with the measure of totally geodesic $k$-dimensional subspaces in
$\mathbb{N}^{n+1}(\epsilon)$ which intersect $K$, which is the definition of
quermassintegrals from integral geometry, see [Sant2004] or [Solanes06,
Proposition 7]. It would be an interesting question whether one can establish
the integral geometric theory for smooth spacelike domains in
$\mathbb{S}_{+}^{n,1}$. For example, the Crofton formulas and Cauchy formulas
have been extended to Lorentzian constant curvature space, see [Solanes-
Teufel05].

######  Remark 8.

It is tempting to ask that whether or not there is a similar definition for
the quermassintegrals in the other Lorentzian space forms, i.e., the Minkowski
space $\mathbb{R}^{n,1}$ or the anti de Sitter space $\mathbb{H}^{n,1}$.
However, we are not able to define the surface area of the complete spacelike
hypersurfaces in $\mathbb{R}^{n,1}$ and $\mathbb{H}^{n,1}$. For example, the
slice $\\{y^{0}=0\\}$ in $\mathbb{R}^{n,1}$ or $\mathbb{H}^{n,1}$ is a
complete noncompact spacelike hypersurface, which it is isometric to
$\mathbb{R}^{n}$ or $\mathbb{H}^{n}$, respectively.

The $k$th mean radius of the smooth bounded domain $K$ in
$\mathbb{N}^{n+1}(\epsilon)$ or a spacelike domain $K$ in
$\mathbb{S}^{n,1}_{+}$ can be similarly defined by

$\displaystyle\zeta_{k}(K)=f_{k}^{-1}(W_{k}(K)),$ (3.3)

where $f_{k}:[0,\infty)\rightarrow\mathbb{R}_{+}$ is a smooth function
determined by $f_{k}(s)=W_{k}(B_{s})$ in $\mathbb{N}^{n+1}(\epsilon)$, where
$B_{s}$ is the geodesic ball of radius $s$; or $f_{k}(s)=W_{k}(\\{0\leq r\leq
s\\})$ in $\mathbb{S}^{n,1}_{+}$, where $\\{0\leq r\leq s\\}$ denotes the
region enclosed by the slice $\\{y^{0}=0\\}$ and the slice $\\{r=s\\}$ in
$\mathbb{S}_{+}^{n,1}$. In all above cases, it is easy to check that $f_{k}$
is monotone increasing, and $f_{k}^{-1}$ is the inverse of $f_{k}$.

The key feature of quermassintegrals is the following elegant variational
formulas in Riemannian space forms and in de Sitter space.

######  Lemma 3.1.

Assume that one of the following holds:

  1. (i)

Let $K_{t}$ be a family of smooth bounded domains in
$\mathbb{N}^{n+1}(\epsilon)$ such that $\partial K_{t}$ is a family of smooth
closed hypersurfaces.

  2. (ii)

Let $K_{t}$ be a family of smooth bounded domains in $\mathbb{S}^{n,1}_{+}$
such that $\partial K_{t}$ is a family of smooth closed spacelike
hypersurfaces.

Let $\frac{\partial}{\partial t}X$ be the variational vector field of
$\partial K_{t}$. Then

$\displaystyle\frac{d}{dt}W_{k}(K_{t})=\frac{n+1-k}{n+1}\int_{\partial
K_{t}}\langle\frac{\partial}{\partial t}X,\nu\rangle E_{k}d\mu_{t},\quad
k=0,1,\cdots,n.$ (3.4)

###### Proof.

For simplicity, we denote by $f=\langle\frac{\partial}{\partial
t}X,\nu\rangle$, where $\langle\cdot,\cdot\rangle$ denotes the Euclidean inner
product for $\mathbb{S}^{n+1}$, $\mathbb{R}^{n+1}$ and the Minkowski inner
product for $\mathbb{H}^{n+1}$, $\mathbb{S}^{n,1}_{+}$, respectively.

(i) By [Reilly73, Theorem A], the curvature integral of $\partial K_{t}$
evolves by

$\displaystyle\frac{d}{dt}\int_{\partial K_{t}}E_{k}d\mu_{t}=\int_{\partial
K_{t}}\left((n-k)E_{k+1}-\epsilon kE_{k-1}\right)fd\mu_{t},\quad
k=1,\cdots,n.$ (3.5)

The volume of $K_{t}$ evolves by (see e.g. [Barbosa-doCarmo-Eschenburg88])

$\displaystyle\frac{d}{dt}\operatorname{Vol}(K_{t})=\int_{\partial
K_{t}}fd\mu_{t}.$ (3.6)

Now the variational formula (3.4) can be proved by induction (see [Wang-Xia14,
Proposition 3.1]). In view of (3.5) and (3.6), it is true for $k=0,1$. Assume
that it is true for $k-1$, that is,

$\displaystyle\frac{d}{dt}W_{k-1}(K_{t})=\frac{n+1-(k-1)}{n+1}\int_{\partial
K_{t}}E_{k-1}fd\mu_{t},$ (3.7)

then by using (3.1), (3.5) and the inductive assumption (3.7), we can compute
that

$\displaystyle\frac{d}{dt}W_{k+1}(K_{t})=$
$\displaystyle\frac{1}{n+1}\frac{d}{dt}\int_{\partial
K_{t}}E_{k}d\mu_{t}+\epsilon\frac{k}{n+2-k}\frac{d}{dt}W_{k-1}(K_{t})$
$\displaystyle=$ $\displaystyle\frac{1}{n+1}\int_{\partial
K_{t}}\left((n-k)E_{k+1}-\epsilon
kE_{k-1}\right)fd\mu_{t}+\epsilon\frac{k}{n+1}\int_{\partial
K_{t}}E_{k-1}fd\mu_{t}$ $\displaystyle=$
$\displaystyle\frac{n+1-(k+1)}{n+1}\int_{\partial K_{t}}E_{k+1}fd\mu_{t}.$

(ii) When the ambient space is $\mathbb{S}_{+}^{n,1}$, along the variational
vector field $\frac{\partial}{\partial t}X$, the volume of the spacelike
domain $K_{t}$ evolves by

$\displaystyle\frac{d}{dt}\operatorname{Vol}(K_{t})=$
$\displaystyle\frac{d}{dt}\int_{\mathbb{S}^{n}}\int_{0}^{\rho(t,\theta)}\cosh^{n}r\sqrt{\det(\sigma_{ij})}drd\theta$
$\displaystyle=$
$\displaystyle\int_{\mathbb{S}^{n}}\cosh^{n}\rho(t,\theta)\frac{\partial\rho}{\partial
t}\sqrt{\det(\sigma_{ij})}d\theta$ $\displaystyle=$
$\displaystyle\int_{\mathbb{S}^{n}}\cosh^{n}\rho(t,\theta)fv\sqrt{\det(\sigma_{ij})}d\theta$
$\displaystyle=$ $\displaystyle\int_{\partial K_{t}}fd\mu,$

where we used (2.10) and (2.9). By using the similar calculation as in
[Reilly73, Theorem A], the curvature integral of $\partial K_{t}$ evolves by
(see e.g. [Solanes-Teufel05, Proposition 2])

$\displaystyle\frac{d}{dt}\int_{\partial K_{t}}E_{k}d\mu_{t}=\int_{\partial
K_{t}}\left((n-k)E_{k+1}+kE_{k-1}\right)fd\mu_{t},\quad k=1,\cdots,n.$ (3.8)

Then the remaining proof is the same as (i). ∎

For any smooth bounded domains in $\mathbb{N}^{n+1}(\epsilon)$, the
quermassintegral $W_{k}$ can be expressed as follows (see e.g. [Solanes06,
Corollary 8] for the case in $\mathbb{H}^{n+1}$):

  * •

for $1\leq m\leq n$ and $m$ being even,

$\displaystyle W_{m}(K)=$
$\displaystyle\frac{1}{n+1}\sum_{i=0}^{\frac{m}{2}-1}\epsilon^{i}\frac{(m-1)!!(n+1-m)!!}{(m-1-2i)!!(n+1-m+2i)!!}\int_{\partial
K}E_{m-1-2i}d\mu$
$\displaystyle+\epsilon^{\frac{m}{2}}\frac{(m-1)!!(n+1-m)!!}{(n+1)!!}\operatorname{Vol}(K);$
(3.9)

  * •

for $1\leq m\leq n$ and $m$ being odd,

$\displaystyle
W_{m}(K)=\frac{1}{n+1}\sum_{i=0}^{\frac{m-1}{2}}\epsilon^{i}\frac{(m-1)!!(n+1-m)!!}{(m-1-2i)!!(n+1-m+2i)!!}\int_{\partial
K}E_{m-1-2i}d\mu.$ (3.10)

Here the notation $k!!$ means the product of all odd(even) integers up to
odd(even) $k$. In view of (3.2), the quermassintegrals in
$\mathbb{S}^{n,1}_{+}$ coincide with those in $\mathbb{H}^{n+1}$, i.e.,

  * •

for $1\leq m\leq n$ and $m$ being even,

$\displaystyle W_{m}(K)=$
$\displaystyle\frac{1}{n+1}\sum_{i=0}^{\frac{m}{2}-1}(-1)^{i}\frac{(m-1)!!(n+1-m)!!}{(m-1-2i)!!(n+1-m+2i)!!}\int_{\partial
K}E_{m-1-2i}d\mu$
$\displaystyle+(-1)^{\frac{m}{2}}\frac{(m-1)!!(n+1-m)!!}{(n+1)!!}\operatorname{Vol}(K);$
(3.11)

  * •

for $1\leq m\leq n$ and $m$ being odd,

$\displaystyle
W_{m}(K)=\frac{1}{n+1}\sum_{i=0}^{\frac{m-1}{2}}(-1)^{i}\frac{(m-1)!!(n+1-m)!!}{(m-1-2i)!!(n+1-m+2i)!!}\int_{\partial
K}E_{m-1-2i}d\mu.$ (3.12)

##  4\. Proofs of Theorem 1.1 and Theorem 1.2

Along the flow (2.22), we have the following identities for convex bodies and
their dual bodies.

######  Lemma 4.1.

Assume that one of the following holds:

  1. (i)

Let $K_{t}$ be a family of smooth bounded domain with strictly convex boundary
$\partial K_{t}$ in $\mathbb{S}^{n+1}$, and $K_{t}^{\ast}$ the dual body of
$K_{t}$ in $\mathbb{S}^{n+1}$;

  2. (ii)

Let $K_{t}$ be a family of smooth bounded domain with strictly convex boundary
$\partial K_{t}$ in $\mathbb{H}^{n+1}$, and $K_{t}^{\ast}$ the dual body of
$K_{t}$ in $\mathbb{S}^{n,1}_{+}$;

  3. (iii)

Let $K_{t}$ be a family of smooth bounded domain with strictly convex and
spacelike boundary $\partial K_{t}$ in $\mathbb{S}^{n,1}_{+}$, and
$K_{t}^{\ast}$ the dual body of $K_{t}$ in $\mathbb{H}^{n+1}$.

Assume that $X:M_{0}\times[0,T)\rightarrow N$ is a family of smooth embeddings
satisfying (2.22) such that $X_{t}(M_{0})=\partial K_{t}$, then there holds

$\displaystyle\frac{d}{dt}\left[(k+1)W_{k}(K_{t})+(n+1-k)W_{n-k}(K_{t}^{\ast})\right]=0,\quad
k=0,\cdots,n.$ (4.1)

###### Proof.

Along the flow (2.22) in $N$, i.e.,

$\displaystyle\frac{\partial}{\partial t}X=-\sigma\mathcal{F}\nu,$

it follows from (3.4) that

$\displaystyle\frac{d}{dt}W_{k}(K_{t})=$
$\displaystyle\frac{n+1-k}{n+1}\int_{\partial
K_{t}}E_{k}(\kappa)\langle\frac{\partial}{\partial t}X,\nu\rangle d\mu_{t}$
$\displaystyle=$
$\displaystyle-\frac{n+1-k}{n+1}\int_{M_{t}}E_{k}(\kappa)\mathcal{F}d\mu_{t},$
(4.2)

where $\langle\cdot,\cdot\rangle$ denotes the Euclidean inner product for
$\mathbb{S}^{n+1}$ and the Minkowski inner product for $\mathbb{H}^{n+1}$ and
$\mathbb{S}^{n,1}_{+}$, respectively.

Corresponding to the flow (2.22), the dual flow (2.24) in the dual ambient
space $N^{\ast}$ is

$\displaystyle\frac{\partial}{\partial
t}X^{\ast}=-\sigma^{\ast}\mathcal{F}^{\ast}\nu^{\ast}-(b^{\ast})^{kl}\mathcal{F}^{\ast}_{k}X^{\ast}_{l}.$
(4.3)

It also follows from (3.4) that

$\displaystyle\frac{d}{dt}W_{n-k}(K^{\ast}_{t})=$
$\displaystyle\frac{n+1-(n-k)}{n+1}\int_{\partial
K^{\ast}_{t}}E_{n-k}(\kappa^{\ast})\langle\frac{\partial}{\partial
t}X^{\ast},\nu^{\ast}\rangle d\mu^{\ast}_{t}$ $\displaystyle=$
$\displaystyle-\frac{n+1-(n-k)}{n+1}\int_{\partial
K^{\ast}_{t}}E_{n-k}(\kappa^{\ast})\mathcal{F}^{\ast}d\mu^{\ast}_{t}$
$\displaystyle=$ $\displaystyle\frac{k+1}{n+1}\int_{\partial
K_{t}}E_{k}(\kappa)\mathcal{F}d\mu_{t},$ (4.4)

where we used (2.23) and

$\displaystyle
E_{n-k}(\kappa^{\ast})=\frac{E_{k}(\kappa)}{E_{n}(\kappa)},\quad
d\mu^{\ast}_{t}=E_{n}(\kappa)d\mu_{t}$

due to the duality relations (2.13), (2.17) and (2.21). Therefore, the
identity (4.1) follows from (4) and (4). ∎

###### Proof of Theorem 1.1.

Let $M=X_{0}(M_{0})$ be a strictly convex hypersurface in $\mathbb{S}^{n+1}$
and $\hat{M}$ the convex body of $M$. By Lemma 4.1, the functional
$(k+1)W_{k}(\hat{M}_{t})+(n+1-k)W_{n-k}(\hat{M}_{t}^{\ast})$ remains invariant
for any flow preserving the strict convexity. To determine the value of this
functional, we consider the harmonic mean curvature flow in the sphere, i.e.,

$\displaystyle\frac{\partial}{\partial t}X=-\frac{E_{n}}{E_{n-1}}\nu.$

Then by the convergence result of contracting curvature flows in the sphere
due to Gerhardt [Ge15], we conclude that $\hat{M}_{t}$ shrinks to a fixed
point $z\in\mathbb{S}^{n+1}$, while the dual body $\hat{M}^{\ast}_{t}$ expands
to the half ball $\mathcal{H}(-z)$ as $t\rightarrow T^{\ast}$. Here
$T^{\ast}<\infty$ is both the maximal existence time of the contracting
curvature flow and its dual flow. Moreover, the flow hypersurfaces $M_{t}$ and
its polar hypersurfaces $M^{\ast}_{t}$ are strictly convex for all
$t\in[0,T^{\ast})$.

Since $\int_{\partial B_{r}}E_{m}d\mu=\omega_{n}\sin^{n-m}r\cos^{m}r$, we have

$\displaystyle\lim_{r\rightarrow 0^{+}}\int_{\partial
B_{r}}E_{m}d\mu=\left\\{\begin{aligned} &0,\quad&\text{if $0\leq m<n$};\\\
&\omega_{n},\quad&\text{if $m=n$},\end{aligned}\right.$

and

$\displaystyle\lim_{r\rightarrow(\frac{\pi}{2})^{-}}\int_{\partial
B_{r}}E_{m}d\mu=\left\\{\begin{aligned} &0,\quad&\text{if $0<m\leq n$};\\\
&\omega_{n},\quad&\text{if $m=0$}.\end{aligned}\right.$

For $1\leq m\leq n$, in view of (• ‣ 3)– (3.10), we have

$\displaystyle\lim_{r\rightarrow 0^{+}}W_{m}(B_{r})=0,$

and

$\displaystyle\lim_{r\rightarrow(\frac{\pi}{2})^{-}}W_{m}(B_{r})=\left\\{\begin{aligned}
&\frac{\omega_{n}}{n+1}\frac{(m-1)!!(n+1-m)!!}{n!!},\quad\text{if $m$ is
odd};\\\
&\frac{(m-1)!!(n+1-m)!!}{(n+1)!!}\operatorname{Vol}(B_{\pi/2}),\quad\text{if
$m$ is even}.\end{aligned}\right.$

Using the fact that
$\operatorname{Vol}(B_{\pi/2})=\frac{1}{2}|\mathbb{S}^{n+1}|$, and
$\omega_{i}=|\mathbb{S}^{i}|=\frac{(i+1)\pi^{(i+1)/2}}{\Gamma((i+1)/2+1)}$,
where $\Gamma(\cdot)$ is the Gamma function. A direct calculation gives

$\displaystyle\frac{\omega_{n+1}}{\omega_{n}}=\frac{(n+2)\pi^{(n+2)/2}}{\Gamma(\frac{n+2}{2}+1)}\cdot\frac{\Gamma(\frac{n+1}{2}+1)}{(n+1)\pi^{(n+1)/2}}=\left\\{\begin{aligned}
&\frac{2(n-1)!!}{n!!},\quad\text{if $n$ is odd},\\\
&\frac{\pi(n-1)!!}{n!!},\quad\text{if $n$ is even}.\end{aligned}\right.$

Then we have

$\displaystyle\lim_{r\rightarrow(\frac{\pi}{2})^{-}}W_{m}(B_{r})=\left\\{\begin{aligned}
&\frac{\pi}{2}\frac{\omega_{n}}{n+1}\frac{(m-1)!!(n+1-m)!!}{n!!},\quad&\text{if
$n$ is even and $m$ is even};\\\
&\frac{\omega_{n}}{n+1}\frac{(m-1)!!(n+1-m)!!}{n!!},\quad&\text{otherwise}.\end{aligned}\right.$

Therefore, we get

$\displaystyle(k+1)W_{k}(\hat{M})+(n+1-k)W_{n-k}(\hat{M}^{\ast})$
$\displaystyle=$ $\displaystyle\lim_{r\rightarrow
0^{+}}\left[(k+1)W_{k}(B_{r})+(n+1-k)W_{n-k}(B_{\frac{\pi}{2}-r})\right]$
$\displaystyle=$
$\displaystyle(n+1-k)\lim_{r\rightarrow(\frac{\pi}{2})^{-}}W_{n-k}(B_{r})$
$\displaystyle=$ $\displaystyle\left\\{\begin{aligned}
&\frac{\pi}{2}\frac{\omega_{n}}{n+1}\frac{(n-k+1)!!(k+1)!!}{n!!},\quad&\text{if
$n$ is even and $k$ is even};\\\
&\frac{\omega_{n}}{n+1}\frac{(n-k+1)!!(k+1)!!}{n!!},\quad&\text{otherwise}.\end{aligned}\right.$

In particular, for any pairs $(\hat{M},\hat{M}^{\ast})$ of strictly convex
bodies and their dual bodies in $\mathbb{S}^{n+1}$, the following identities
hold

$\displaystyle(k+1)W_{k}(\hat{M})+(n+1-k)W_{n-k}(\hat{M}^{\ast})=C_{n,k},\quad
k=0,1,\cdots,n,$

where $C_{n,k}$ are constants depending only on $n,k$. If the geodesic ball
$B_{r}$ satisfies $W_{k}(B_{r})=W_{k}(\hat{M})$, then
$W_{n-k}(B_{\frac{\pi}{2}-r})=W_{n-k}(\hat{M}^{\ast})$. Therefore, we get

$\displaystyle\zeta_{k}(\hat{M})+\zeta_{n-k}(\hat{M}^{\ast})=\frac{\pi}{2},\quad
0\leq k\leq n,$

which is equivalent to (1.7). This completes the proof of Theorem 1.1. ∎

###### Proof of Theorem 1.2.

Let the initial hypersurface $M=X_{0}(M_{0})$ be strictly convex in
$\mathbb{H}^{n+1}$ and $\hat{M}$ the convex body of $M$. Similar as above, we
consider the harmonic mean curvature flow in $\mathbb{H}^{n+1}$:

$\displaystyle\frac{\partial}{\partial t}X=-\frac{E_{n}}{E_{n-1}}\nu,$

By the convergence result of Yu [Yu2016, Theorem 1.2], the contracting
hypersurface $M_{t}=X_{t}(M_{0})$ in $\mathbb{H}^{n+1}$ shrinks to a point
$x_{0}$ as $t\rightarrow T^{\ast}$, where $T^{\ast}<\infty$ is the maximal
existence time of this flow. By using a Lorenz boost, we may assume this point
$x_{0}$ is the Beltrami point. Then the polar hypersurface $M_{t}^{\ast}$ of
the dual flow is contained in $\mathbb{S}^{n,1}_{+}$ and converge to the
coordinate slice $\\{r=0\\}$ as $t\rightarrow T^{\ast}$. Moreover, the flow
hypersurface $M_{t}$ in $\mathbb{H}^{n+1}$ is strictly convex, while its polar
hypersurface $M^{\ast}_{t}$ in $\mathbb{S}^{n,1}_{+}$ is strictly convex and
spacelike for all $t\in[0,T^{\ast})$. Let $\hat{M}_{t}$ be the convex body of
$M_{t}$ in $\mathbb{H}^{n+1}$, and $\hat{M}_{t}^{\ast}$ its dual body in
$\mathbb{S}^{n,1}_{+}$, respectively.

By Lemma 4.1, the functional
$(k+1)W_{k}(\hat{M}_{t})-(n+1-k)W_{n-k}(\hat{M}_{t}^{\ast})$ remains invariant
for any flow which preserves the strict convexity. To determine the value of
this functional, we first observe that

$\displaystyle\int_{\partial B_{r}}E_{m}(\kappa)d\mu=$
$\displaystyle\omega_{n}\sinh^{n-m}r\cosh^{m}r,$
$\displaystyle\int_{\\{r=r^{\ast}\\}}E_{m}(\kappa^{\ast})d\mu^{\ast}=$
$\displaystyle\omega_{n}\cosh^{n-m}r^{\ast}\sinh^{m}r^{\ast}.$

It follows that

$\displaystyle\lim_{r\rightarrow 0^{+}}\int_{\partial
B_{r}}E_{m}(\kappa)d\mu=\left\\{\begin{aligned} &0,\quad&\text{if $0\leq
m<n$};\\\ &\omega_{n},\quad&\text{if $m=n$},\end{aligned}\right.$

and

$\displaystyle\lim_{r^{\ast}\rightarrow
0^{+}}\int_{\\{r=r^{\ast}\\}}E_{m}(\kappa^{\ast})d\mu^{\ast}=\left\\{\begin{aligned}
&0,\quad&\text{if $0<m\leq n$};\\\ &\omega_{n},\quad&\text{if
$m=0$}.\end{aligned}\right.$

For $1\leq m\leq n$, in view of (• ‣ 3)–(3.12), we have

$\displaystyle\lim_{r\rightarrow 0^{+}}W_{m}(B_{r})=0,$

and

$\displaystyle\lim_{r^{\ast}\rightarrow
0^{+}}W_{m}(\\{r=r^{\ast}\\})=\left\\{\begin{aligned}
&(-1)^{\frac{m-1}{2}}\frac{\omega_{n}}{n+1}\frac{(m-1)!!(n+1-m)!!}{n!!},\quad&\text{if
$m$ is odd};\\\ &0,\quad&\text{if $m$ is even}.\end{aligned}\right.$

Therefore, we get

$\displaystyle(k+1)W_{k}(\hat{M})+(n+1-k)W_{n-k}(\hat{M}^{\ast})$
$\displaystyle=$ $\displaystyle(k+1)\lim_{r\rightarrow
0^{+}}W_{k}(B_{r})+(n+1-k)\lim_{r^{\ast}\rightarrow 0^{+}}W_{n-k}(\\{0\leq
r\leq r^{\ast}\\})$ $\displaystyle=$ $\displaystyle\left\\{\begin{aligned}
&(-1)^{\frac{n-k-1}{2}}\frac{\omega_{n}}{n+1}\frac{(n-k+1)!!(k+1)!!}{n!!},\quad&\text{if
$n-k$ is odd};\\\ &0,\quad&\text{otherwise}.\end{aligned}\right.$

This shows that the following identities

$\displaystyle(k+1)W_{k}(\hat{M})+(n+1-k)W_{n-k}(\hat{M}^{\ast})=C_{n,k},\quad
k=0,1,\cdots,n$

hold for all pairs $(\hat{M},\hat{M}^{\ast})$ of strictly convex bodies in
$\mathbb{H}^{n+1}$ and their dual bodies in $\mathbb{S}^{n,1}_{+}$, where
$C_{n,k}$ are constants depending only on $n$, $k$. If $B_{r}$ is a geodesic
ball in $\mathbb{H}^{n+1}$ such that $W_{k}(B_{r})=W_{k}(\hat{M})$, then
$W_{n-k}(\hat{M}^{\ast})=W_{n-k}(\\{0\leq r\leq r^{\ast}\\})$, where $\\{0\leq
r\leq r^{\ast}\\}$ is the dual body of $B_{r}$. Note that the geodesic sphere
$\partial B_{r}$ in $\mathbb{S}^{n+1}$ is strictly convex, while
$\\{r=r^{\ast}\\}$ in $\mathbb{S}^{n,1}_{+}$ is its polar hypersurface.
Moreover, the principal curvatures of $\partial B_{r}$ and $\\{r=r^{\ast}\\}$
are

$\displaystyle\kappa_{i}=\coth r,\quad\kappa^{\ast}_{i}=\tanh r^{\ast}.$

In view of the duality (2.17) and (2.21) in hyperbolic/de Sitter space, we
obtain

$\displaystyle\coth\zeta_{k}(K)\cdot\tanh\zeta_{n-k}(K^{\ast})=1,\quad 0\leq
k\leq n.$

This completes the proof of Theorem 1.2. ∎

##  5\. Proofs of Theorems 1.3–1.8

The quermassintegral inequalities [Alexandrov1937, Alexandrov1938] for convex
domains in Euclidean space are fundamental in classical geometry. There have
been some interests in extending the original quermassintegral inequalities to
non-convex domains (see [Trudinger1994]). In [GL09], Guan and Li extend these
inequalities to non-convex starshaped domains in $\mathbb{R}^{n+1}$. There are
generalizations of such inequalities to other ambient spaces, such as the
sphere [Makowski-Scheuer2016, CGLS21, Chen-Sun21, GL21, Wei-Xiong2015], the
hyperbolic space [ACW2018, AHL2020, BGL19, HL19, Hu-Li-2022, Hu-Li-Wei2020,
Wang-Xia14, Wei-Xiong2015, GeWW14, GL21] and the de Sitter space [Scheuer19,
Lambert-Scheuer2021]. We collect these results as follows.

######  Lemma 5.1.

If $K$ is a smooth bounded and strictly convex domain in $\mathbb{S}^{n+1}$,
then the following inequalities hold:

  1. (i)

[BGL19, GL21]

$\displaystyle W_{n}(K)\geq f_{n}\circ f_{l}^{-1}(W_{l}(K)),\quad 0\leq l<n,$
(5.1)

  2. (ii)

[CGLS21, Chen-Sun21]

$\displaystyle W_{k}(K)\geq f_{k}\circ f_{0}^{-1}(W_{0}(K)),\quad 0\leq k<n,$
(5.2)

  3. (iii)

[Chen-Sun21]

$\displaystyle W_{m+1}(K)\geq f_{m+1}\circ f_{m-1}^{-1}(W_{m-1}(K)),\quad
1\leq m\leq n-1.$ (5.3)

Equality holds if and only if $K$ is a geodesic ball.

######  Remark 9.

In view of the identities (1.7) in Theorem 1.1, the inequality (5.1) and the
inequality (5.2) are equivalent.

######  Lemma 5.2.

  1. (a)

Let $K$ be a smooth bounded domain in $\mathbb{H}^{n+1}$.

    1. (i)

[Wang-Xia14, Hu-Li-Wei2020] If $\partial K$ is h-convex, then

$\displaystyle W_{k}(K)\geq f_{k}\circ f_{l}^{-1}(W_{l}(K)),\quad 0\leq
l<k\leq n.$ (5.4)

    2. (ii)

[AHL2020, BGL19] If $\partial K$ is strictly convex, then

$\displaystyle W_{n}(K)\geq f_{n}\circ f_{l}^{-1}(W_{l}(K)),\quad 0\leq l<n,$
(5.5)

and

$\displaystyle W_{n-1}(K)\geq f_{n-1}\circ
f_{n-1-2i}^{-1}(W_{n-1-2i}(K)),\quad 0<2i<n.$ (5.6)

    3. (iii)

[BGL19, GL21] If $\partial K$ is mean convex and starshaped, then

$\displaystyle W_{2}(K)\geq f_{2}\circ f_{1}^{-1}(W_{1}(K)).$ (5.7)

    4. (iv)

[LWX14] If $\partial K$ is $2$-convex and starshaped, then

$\displaystyle W_{3}(K)\geq f_{3}\circ f_{1}^{-1}(W_{1}(K)).$ (5.8)

Equality holds if and only if $K$ is a geodesic ball.

  2. (b)

Let $K$ is a smooth bounded domain in $\mathbb{S}^{n,1}_{+}$.

    1. (i)

[Lambert-Scheuer2021] If $\partial K$ is spacelike, then

$\displaystyle W_{0}(K)\geq f_{0}\circ f_{1}^{-1}(W_{1}(K)).$ (5.9)

    2. (ii)

[Scheuer19] If $\partial K$ is mean convex and spacelike, then

$\displaystyle W_{1}(K)\geq f_{1}\circ f_{2}^{-1}(W_{2}(K)).$ (5.10)

Equality holds if and only if $\partial K$ is isometric to a coordinate slice.

###### Proof of Theorem 1.3.

By Theorem 1.1, we have

$\displaystyle\tan\left(\zeta_{n-k}(K)\right)\cdot\tan\left(\zeta_{k}(K^{\ast})\right)=1,\quad
0\leq k\leq n,$ (5.11)

for smooth bounded and strictly convex domain $K\subset\mathcal{H}(z)$ and its
dual body $K^{\ast}\subset\mathcal{H}(-z)$. Notice that $K^{\ast}$ is also a
smooth bounded and strictly convex domain in the sphere. Then the inequalities
(1.9) and (1.10) follow from the identity (5.11) and the quermassintegral
inequalities (5.1) and (5.2) in the sphere:

$\displaystyle\zeta_{k}(K^{\ast})\geq\zeta_{0}(K^{\ast}),\quad\zeta_{n-k}(K)\leq\zeta_{n}(K),\quad
0<k\leq n.$ (5.12)

Similarly, the inequalities (1.11) and (1.12) follows from the identities
(5.11) and the quermassintegral inequalities (5.3). If equality holds in any
one of (1.9)–(1.12), then the equality also holds in the quermassintegral
inequalities (5.1), (5.2) or (5.3), which implies that $K$ (or $K^{\ast}$) is
a geodesic ball in $\mathbb{S}^{n+1}$. This completes the proof of Theorem
1.3. ∎

###### Proof of Theorem 1.4.

By Theorem 1.2, we have the following identities

$\displaystyle\coth(\zeta_{k}(K))\cdot\tanh(\zeta_{n-k}(K^{\ast}))=1,\quad
0\leq k\leq n,$ (5.13)

for any smooth bounded and strictly convex domain $K$ in $\mathbb{H}^{n+1}$
and its dual body $K^{\ast}$ in $\mathbb{S}^{n,1}_{+}$. For any $0\leq k,l\leq
n$ with $k+l<n$, then $0\leq k<n-l$ and the quermassintegral inequalities
(5.4) for h-convex domains in $\mathbb{H}^{n+1}$ imply that

$\displaystyle\zeta_{k}(K)\leq\zeta_{n-l}(K).$ (5.14)

Note that $\coth s$ is monotone decreasing in $s$, we get

$\displaystyle\coth(\zeta_{k}(K))\cdot\tanh(\zeta_{l}(K^{\ast}))\geq\coth(\zeta_{n-l}(K))\cdot\tanh(\zeta_{l}(K^{\ast}))=1,$

which yields the desired inequality (1.13). The inequality (1.14) can be
proved similarly. If the equality holds in (1.13) or (1.14), then the equality
also holds in (5.14), which implies that $K$ is a geodesic ball in
$\mathbb{H}^{n+1}$. This completes the proof of Theorem 1.4. ∎

###### Proof of Theorem 1.5.

By (5.6), for any smooth bounded and strictly convex domain $K$ in
$\mathbb{H}^{n+1}$, there hold

$\displaystyle W_{n-1-2i}(K)\leq f_{n-1-2i}\circ
f^{-1}_{n-1}(W_{n-1}(K)),\quad 0<2i<n.$

To complete the proof, it remains to show that

$\displaystyle W_{n-2k}(K)\leq f_{n-2k}\circ f^{-1}_{n-1}(W_{n-1}(K)),\quad
0<2k\leq n.$ (5.15)

We prove (5.15) by induction. For $k=1$, by taking an interior point of $K$ as
the Beltrami point, its dual $K^{\ast}$ is a strictly convex and spacelike
domain in $\mathbb{S}^{n,1}_{+}$. Then the quermassintegral inequality (5.10)
in $\mathbb{S}_{+}^{n,1}$ implies that

$\tanh(\zeta_{2}(K^{\ast}))\leq\tanh(\zeta_{1}(K^{\ast})).$

Combining this with the identities (5.13), i.e.,

$\coth(\zeta_{n-2}(K))\cdot\tanh(\zeta_{2}(K^{\ast}))=\coth(\zeta_{n-1}(K))\cdot\tanh(\zeta_{1}(K^{\ast})),$

we obtain

$\displaystyle\zeta_{n-2}(K)\leq\zeta_{n-1}(K),$

in view of the monotonicity of $\coth s$. This verifies (5.15) for $k=1$.

Assume that (5.15) is true for $k-1$, i.e.,

$\displaystyle W_{n-2k+2}(K)\leq f_{n-2k+2}\circ f^{-1}_{n-1}(W_{n-1}(K)),$
(5.16)

we show that (5.15) also holds for $k$. Along the harmonic mean curvature flow
starting from $\partial K$ in $\mathbb{H}^{n+1}$, we know that the flow
hypersurface $M_{t}=\partial K_{t}$ is strictly convex and it shrinks to a
round point as $t\rightarrow T^{\ast}$. Then by (3.4), we have

$\displaystyle\frac{d}{dt}W_{n-1}(K_{t})=$
$\displaystyle-\frac{2}{n+1}\int_{M_{t}}E_{n}d\mu$ $\displaystyle=$
$\displaystyle-2W_{n+1}(K_{t})-nW_{n-1}(K_{t})$ $\displaystyle=$
$\displaystyle\left.\left(-2W_{n+1}(B_{r})-nW_{n-1}(B_{r})\right)\right|_{r=f_{n-1}^{-1}(W_{n-1}(K_{t}))}$
$\displaystyle=$ $\displaystyle-\left.\frac{2}{n+1}\int_{\partial
B_{r}}E_{n}d\mu\right|_{r=f_{n-1}^{-1}(W_{n-1}(K_{t}))}$ $\displaystyle=$
$\displaystyle-\left.\frac{2}{n+1}\omega_{n}\cosh^{n}r\right|_{r=f_{n-1}^{-1}(W_{n-1}(K_{t}))},$
(5.17)

where we used (3.1) and

$W_{n+1}(K_{t})=W_{n+1}(B_{r})|_{r=f_{n-1}^{-1}(W_{n-1}(K_{t}))}=\frac{\omega_{n}}{n+1}$

due to Gauss-Bonnet-Chern theorem (see [Chern1944, Chern1945], see also
[Solanes06]). For $k\geq 2$, we also have

$\displaystyle\frac{d}{dt}W_{n-2k}(K_{t})=$
$\displaystyle-\frac{2k+1}{n+1}\int_{M_{t}}E_{n-2k}\frac{E_{n}}{E_{n-1}}d\mu$
$\displaystyle\geq$ $\displaystyle-\frac{2k+1}{n+1}\int_{M_{t}}E_{n-2k+1}d\mu$
$\displaystyle=$
$\displaystyle-(2k+1)W_{n-2k+2}(K_{t})-(n+1-2k)W_{n-2k}(K_{t})$
$\displaystyle\geq$ $\displaystyle-(2k+1)f_{n-2k+2}\circ
f^{-1}_{n-1}(W_{n-1}(K_{t}))-(n+1-2k)W_{n-2k}(K_{t}),$ (5.18)

where we used the Newton-MacLaurin inequality, (3.1) and the induction
assumption (5.16). Note that $f_{m}(r)=W_{m}(B_{r})$, we have

$\displaystyle f^{\prime}_{m}(r)=\frac{d}{dr}W_{m}(B_{r})=$
$\displaystyle\frac{n+1-m}{n+1}\int_{\partial B_{r}}E_{m}d\mu$
$\displaystyle=$
$\displaystyle\frac{n+1-m}{n+1}\omega_{n}\sinh^{n-m}r\cosh^{m}r,\quad 0\leq
m\leq n.$ (5.19)

Using (5) and (5), we get

$\displaystyle\frac{d}{dt}f_{n-2k}\circ f_{n-1}^{-1}(W_{n-1}(K_{t}))$
$\displaystyle=$
$\displaystyle\left.\frac{f^{\prime}_{n-2k}(r)}{f^{\prime}_{n-1}(r)}\right|_{r=f_{n-1}^{-1}(W_{n-1}(K_{t}))}\cdot\frac{d}{dt}W_{n-1}(K_{t})$
$\displaystyle=$
$\displaystyle\frac{2k+1}{2}\left.\frac{\sinh^{2k-1}r}{\cosh^{2k-1}r}\cdot\left(-\frac{2}{n+1}\omega_{n}\cosh^{n}r\right)\right|_{r=f_{n-1}^{-1}(W_{n-1}(K_{t}))}$
$\displaystyle=$ $\displaystyle-\left.\frac{2k+1}{n+1}\int_{\partial
B_{r}}E_{n+1-2k}d\mu\right|_{r=f_{n-1}^{-1}(W_{n-1}(K_{t}))}$ $\displaystyle=$
$\displaystyle-(2k+1)f_{n+2-2k}\circ
f_{n-1}^{-1}(W_{n-1}(K_{t}))-(n+1-2k)f_{n-2k}\circ
f_{n-1}^{-1}(W_{n-1}(K_{t})),$

where we also used (3.1) in the last equality. Combining this with (5), we
obtain

$\displaystyle\frac{d}{dt}\left(f_{n-2k}\circ
f_{n-1}^{-1}(W_{n-1}(K_{t}))-W_{n-2k}(K_{t})\right)$ $\displaystyle\leq$
$\displaystyle-(n+1-2k)\left(f_{n-2k}\circ
f_{n-1}^{-1}(W_{n-1}(K_{t}))-W_{n-2k}(K_{t})\right),$

and hence

$\displaystyle\frac{d}{dt}\left[e^{(n+1-2k)t}\left(f_{n-2k}\circ
f_{n-1}^{-1}(W_{n-1}(K_{t}))-W_{n-2k}(K_{t})\right)\right]\leq 0.$

As the flow hypersurface $M_{t}$ shrinks to a round point as $t\rightarrow
T^{\ast}$, and the quermassintegral is monotone under the set inclusion, we
have (see [AHL2020, Lemma 3.3])

$\displaystyle\lim_{t\rightarrow T^{\ast}}W_{m}(K_{t})=0,\quad 0\leq m\leq n.$

It follows that

$\displaystyle f_{n-2k}\circ f_{n-1}^{-1}(W_{n-1}(K))-W_{n-2k}(K)$
$\displaystyle\geq$ $\displaystyle e^{(n+1-2k)T^{\ast}}\lim_{t\rightarrow
T^{\ast}}\left[f_{n-2k}\circ
f_{n-1}^{-1}(W_{n-1}(K_{t}))-W_{n-2k}(K_{t})\right]$ $\displaystyle=$
$\displaystyle 0,$

which verifies (5.15) for $k\geq 2$. The equality case can be similarly
characterized. This completes the proof of Theorem 1.5. ∎

###### Proof of Theorem 1.6.

The quermassintegral inequalities (1.15) in Theorem 1.5 and (5.5) in Lemma 5.2
for strictly convex domains in $\mathbb{H}^{n+1}$ imply that

$\displaystyle\zeta_{l}(K)\leq$ $\displaystyle\zeta_{n-1}(K),\quad 0\leq
l<n-1,$

and

$\displaystyle\zeta_{k}(K)\leq$ $\displaystyle\zeta_{n}(K),\quad 0\leq k<n.$

In view of (5.13) and the monotonicity of $\cosh s$, we get

$\displaystyle\coth(\zeta_{l}(K))\cdot\tanh(\zeta_{1}(K^{\ast}))\geq\coth(\zeta_{n-1}(K))\cdot\tanh(\zeta_{1}(K^{\ast}))=1,\quad
0\leq l<n-1,$

and

$\displaystyle\coth(\zeta_{k}(K))\cdot\tanh(\zeta_{0}(K^{\ast}))\geq\coth(\zeta_{n}(K))\cdot\tanh(\zeta_{0}(K^{\ast}))=1,\quad
0\leq k<n,$

which are the desired inequalities (1.17) and (1.18), respectively. If the
equality holds in (1.17) or (1.18), then the equality also holds in (1.15) or
(5.5), which implies that $K$ is a geodesic ball in $\mathbb{H}^{n+1}$. This
completes the proof of Theorem 1.6. ∎

###### Proof of Theorem 1.7.

For any smooth bounded spacelike domain $K$ in $\mathbb{S}^{n,1}_{+}$ with its
principal curvatures satisfy $0<\kappa_{i}\leq 1$, then its dual body
$K^{\ast}$ is a smooth bounded and h-convex domain in hyperbolic space. In
view of duality in hyperbolic/de Sitter space, we have $(K^{\ast})\ast=K$. The
quermassintegral inequality (5.4) for smooth bounded and h-convex domains in
hyperbolic space implies that

$\displaystyle\zeta_{l}(K^{\ast})\leq\zeta_{k}(K^{\ast}),\quad 0\leq l<k\leq
n.$

Then by the monotonicity of $\coth s$, we get

$\coth(\zeta_{l}(K^{\ast}))\geq\coth(\zeta_{k}(K^{\ast})),\quad 0\leq l<k\leq
n.$

In view of the identities (1.8), i.e.,

$\coth(\zeta_{l}(K^{\ast}))\cdot\tanh(\zeta_{n-l}(K))=\coth(\zeta_{k}(K^{\ast}))\cdot\tanh(\zeta_{n-k}(K)),\quad
0\leq l<k\leq n,$

we obtain

$\displaystyle\zeta_{n-l}(K)\leq\zeta_{n-k}(K),\quad 0\leq l<k\leq n,$

which is equivalent to (1.20). The equality case can be similarly
characterized. This completes the proof of Theorem 1.7. ∎

###### Proof of Theorem 1.8.

For any smooth bounded spacelike and strictly convex domain $K$ in
$\mathbb{S}^{n,1}_{+}$, its dual body $K^{\ast}$ is a smooth bounded and
strictly convex domain in hyperbolic space. Then by the monotonicity of $\coth
s$ and the quermassintegral inequalities (1.15), we obtain

$\displaystyle\coth(\zeta_{n-1}(K^{\ast}))\leq\coth(\zeta_{k}(K^{\ast})),\quad
0\leq k<n-1.$

In view of the identities (1.8), we have

$\displaystyle\coth(\zeta_{n-1}(K^{\ast}))\cdot\tanh(\zeta_{1}(K))=\coth(\zeta_{k}(K^{\ast}))\cdot\tanh(\zeta_{n-k}(K)),\quad
0\leq k<n-1.$

It follows that

$\displaystyle\tanh(\zeta_{1}(K))\geq\tanh(\zeta_{n-k}(K)),\quad 0\leq k<n-1,$

which is equivalent to (1.21) due to the monotonicity of $\tanh s$. Applying
the quermassintegral inequalities (5.7) and (5.8) to strictly convex
hypersurfaces in hyperbolic space, the inequalities (1.22) and (1.23) can be
proved similarly. Moreover, the equality case in (1.21)–(1.23) can be
similarly characterized. This completes the proof of Theorem 1.8. ∎

## References