## A family of monotone quantities along the inverse curvature flow in the Euclidean space

Monotone quantities along hypersurfaces evolving under the inverse curvature flow have many applications in geometry and relativity. In [HI], Huisken and Ilmanen applied the monotone increasing property of Hawking mass to give a proof of the Riemannian Penrose Inequality. In a recent paper [BHW], Brendle, Hung and Wang discovered a monotone decreasing quantity along the inverse mean curvature flow in Anti-Desitter-Schwarzschild manifolds and used it to establish a Minkowski-type inequality for star-shaped hypersurfaces.

In this note, which is a sequel to one of our previous posts, we provide a series of new monotone increasing quantities along smooth solutions to the inverse curvature flow [Gerhardt] in ${ \mathbb{R}^n}$:

 Theorem 1 Let ${ \Sigma }$ be a smooth, closed, embedded hypersurface in ${ \mathbb{R}^n}$ with positive normalized ${k}$-th mean curvature ${\sigma_k>0}$ (${k\ge 2}$), and is star-shaped with respect to a fixed point ${O}$. Let ${ I }$ be an open interval and ${ X: \Sigma \times I \rightarrow \mathbb{R}^n }$ be a smooth map satisfying $\displaystyle \frac{\partial X}{\partial t} = \frac{\sigma_{k-2}}{\sigma_{k-1}} \nu, \ \ \ \ \ (1)$ where ${\sigma_l}$ is the normalized ${l}$-th mean curvature of the surface ${ \Sigma_t = X (\Sigma, t) }$ and ${ \nu }$ is the outward unit normal vector to ${ \Sigma_t}$. Let ${ r = r (x)}$ be the distance from ${ x }$ to ${ O}$. Then the function $\displaystyle Q(t)=\left(\int_{\Sigma_t}\sigma_{k-2}d\mu\right)^{-\frac{n-k-1}{n-k+1}}\left(\int _{\Sigma_t}\sigma_{k-2}d\mu- \int_{\Sigma_t} \sigma_k r^2d\mu\right) \ \ \ \ \ (2)$ is monotone increasing and ${Q(t) }$ is a constant function if and only if ${\Sigma_t }$ is a round sphere for each ${t}$. Here ${ d \mu }$ denotes the volume form on a hypersurface.

The precise definition of ${\sigma_k}$ will be given in Section 1. Let us just remark that ${\displaystyle \int_{\Sigma}\sigma_0 d\mu= \mathrm{Area}(\Sigma)}$. So for example, when ${k=2}$, then our result states that if ${\Sigma}$ has positive scalar curvature, then

$\displaystyle \textrm{Area}(\Sigma_t)^{-\frac{n-3}{n-1}}\left(\textrm{Area}(\Sigma_t)- \int_{\Sigma_t} \sigma_2 r^2d\mu\right)$

is monotone increasing along the (normalized) mean curvature flow (cf. also [LWW]).

Our motivation is as follows. In [K], the author used the Minkowski type integral formulas to obtain a series of inequalities relating the weighted ${k}$-th mean curvature of a hypersurface. Indeed, it was shown in [K] that:

 Theorem 2 Suppose ${\Sigma}$ is a closed hypersurface embedded in ${\mathbb{R}^n}$ with ${\sigma_{k}>0}$ for some ${1\le k\le n-1}$. Then $\displaystyle \mathrm{Area}(\Sigma)\le \int_\Sigma \sigma_1 r\le \cdots\le \int_\Sigma\sigma_{k} r^{k}$ and $\displaystyle n\mathrm{Vol}(\Omega)\le \int_\Sigma r\le \int_\Sigma \sigma_1 r^2 \le \cdots \le \int_\Sigma \sigma_{k}r^{k+1}. \ \ \ \ \ (3)$ Here ${r=|X|}$ and ${\Omega}$ is the region enclosed by ${\Sigma}$. The equality occurs if and only if ${\Sigma}$ is a sphere centered at ${O}$.

On the other hand, in [KM], it was shown that a special case of the inequalities in (3) can be obtained by using the monotonic property of a related geometric quantity along the inverse mean curvature flow. The approach of using monotone quantities along various inverse curvature flow to derive geometric inequalities is a very fruitful and has attracted a lot of attention recently, see e.g. [BHW], [LG], and [LWW]. In view of the series of inequalities in Theorem 2, it is natural to ask: is there an associated series of geometric quantities which are monotone increasing along some inverse curvature flow in ${\mathbb{R}^n}$? In this note, we give an affirmative answer to this question by showing that for each ${k\ge 2}$, there is a quantity ${Q(t)}$ which is monotone along the ${\frac{\sigma_{k-2}}{\sigma_{k-1}}}$ flow, as long as the initial hypersurface is star-shaped and ${k}$-convex, i.e. ${\sigma_k>0}$.

As an application of Theorem 1, we derive a series of sharp inequalities for star-shaped hypersurfaces in ${ \mathbb{R}^n}$, which partially generalizes Theorem 2:

 Theorem 3 Let ${\Sigma }$ be a smooth, closed hypersurface embedded in ${\mathbb R^n}$ with positive ${\sigma_k}$ (${k\ge 2}$) and is star-shaped with respect to ${O}$. Then $\displaystyle \int_\Sigma \sigma_{k-2}d\mu\le\int_\Sigma \sigma_k r^2d\mu \ \ \ \ \ (4)$ where ${r }$ is the distance to a fixed point ${O }$ and ${\sigma_k}$ is the normalized ${k}$-th mean curvature of ${ \Sigma}$. Furthermore, equality in (4) holds if and only if ${\Sigma}$ is a sphere centered at ${O}$.

The rest of this note is organized as follows: in Section 1, we will collect all the necessary definitions and preliminary results, and we will prove our main theorem in Section 2. I would also like to thank Prof. Pengzi Miao for discussion.

1. Preliminaries

Let us fix some notations. Let ${\Sigma\subset \mathbb R^n}$ be a hypersurface and ${m=n-1}$. Let ${\lbrace e_i\rbrace_{i=1}^{m}}$ be a local orthonormal frame on ${\Sigma}$ and let ${\nu=e_n}$ be the unit outward normal of ${\Sigma}$. Let ${\overline \nabla }$ and ${\nabla }$ be the connections on ${\mathbb R^n}$ and ${\Sigma}$ respectively. We define the the shape operator by ${A=\overline \nabla \nu: T\Sigma \rightarrow T\Sigma}$ and ${A_i^j}$ is defined by ${A(e_i)=\sum_{j=1}^m A_i^j e_j}$. By abusing of notation, we will also denote the second fundamental form ${\langle A(u), v\rangle}$ by ${A(u,v)}$. For simplicity, we will denote the dot product in ${\mathbb{R}^n}$ by ${\cdot}$.

We define the ${k}$-th mean curvature ${H_k}$ and the normalized ${k}$-th mean curvature ${\sigma_k}$ of ${\Sigma}$ by

$\displaystyle H _k = \frac 1{k!}\sum_{\substack{1\le i_1,\cdots, i_k\le m\\ 1\le j_1, \cdots, j_k\le m}} \epsilon_{j_1\cdots j_k}^{i_1\cdots i_k}A_{i_1}^{j_1}\cdots A_{i_k}^{j_k}\textrm{ and } \sigma_k= \frac{H_k}{{m\choose k}}$

respectively. Here ${\epsilon_{i_1 \cdots i_k}^{j_1\cdots j_k}}$ is zero if ${i_k=i_l}$ or ${j_k=j_l}$ for some ${k\ne l}$, or if ${\lbrace i_1, \cdots, i_k\rbrace \ne \lbrace j_1, \cdots, j_k\rbrace}$ as sets, otherwise it is defined as the sign of the permutation ${(i_1, \cdots, i_k)\mapsto (j_1, \cdots, j_k)}$. For convenience, we also define ${\sigma_0=1}$ and ${\sigma_{-1}=X\cdot \nu}$. The reason for this definition is that the Hsiung-Minkowski formula (Theorem 7) holds for ${\sigma_{-1}}$ (trivially).

Following [Reilly], we define the ${k}$-th Newton transformation ${T_k: T\Sigma\rightarrow T\Sigma}$ of ${A}$ as

$\displaystyle {(T_k)}_j^{\,i}= \frac 1 {k!} \sum_{\substack{1\le i_1,\cdots, i_k\le m\\ 1\le j_1, \cdots, j_k\le m}} \epsilon^{i i_1 \ldots i_k}_{j j_1 \ldots j_k} A_{i_1}^{j_1}\cdots A_{i_k}^{j_k}.$

For ${k=0}$, ${T_0= \textrm{I}}$ is the identity. It is not hard to see that if ${\lbrace e_i\rbrace_{i=1}^m}$ are the principal directions, then

$\displaystyle T_k(e_i)=\sum_{\substack{i_1<\cdots

where ${\{\lambda_i\}_{i=1}^m}$ are the principal curvatures.

From now on, we use the convention that repeated indices are summed from ${1}$ to ${m}$. We collect some basic properties of ${T_k}$ and ${H_k}$ (e.g. [BC] Lemma 2.1):

 Lemma 4 Let ${m=n-1}$, then on a hypersurface ${\Sigma}$ in ${\mathbb{R}^n}$, ${\mathrm{tr}(T_k)= (m-k)H_k=(m-k){m\choose k}\sigma_k}$. ${\mathrm{div}(T_k)=0}$. i.e. ${\displaystyle \nabla _{e_i}(T_{k})^i_{j}=0.}$ ${\displaystyle (T_k)_i^jA_j^i=(k+1) H_{k+1}. }$ ${ \displaystyle (T_{k-1})^l_j A^j_{i}A^{i}_l = H_k H_1-(k+1)H_{k+1} .}$

Let ${\Sigma}$ be a smooth hypersurface in ${\mathbb{R}^n}$ given by the embedding ${X_0}$. Let ${\Sigma_t}$ be the surface evolved by

$\displaystyle \begin{cases} \frac{\partial X}{\partial t}=F\nu\\ X(\cdot, 0)= X_0 \end{cases} \ \ \ \ \ (5)$

where ${F}$ is a function on ${\Sigma_t}$ which is independent of ${t}$. We have

 Lemma 5 Under the flow in (5), $\displaystyle \begin{array}{rl} g_{ij}'=&2FA_{ij}, \quad (g^{ij})'=-2FA^{ij}, \quad \nu'=-\nabla F, \quad \\ A_{ij}'=&-\nabla ^2_{ij} F+FA_{ik}A^k_j, \quad {A_i^j}'=-F_i^j-FA_{i}^kA_k^j, \quad d\mu'=FH_1d\mu,\\ H_m'=&(T_{m-1})_j^i\left(-(\nabla ^2F)_i^j-FA_{i}^kA_k^j\right)\\ =&-\nabla ^j\left((T_{m-1})_j^iF_i\right)-F\left(H_1H_m-(m+1)H_{m+1}\right). \end{array}$

Proof: These equations are standard, and we only prove some of them for illustration. We have

$\displaystyle \begin{array}{rl} \nu'=g^{ij}\langle \nu', \partial _i \rangle\partial _j =-g^{ij}\langle \nu, \overline \nabla _t\partial _i\rangle\partial _j =-g^{ij}\langle \nu, (F\nu)_i\rangle\partial _j =&-g^{ij}F_i\partial _j =-\nabla F. \end{array}$

$\displaystyle \begin{array}{rl} A_{ij}' =\langle \overline \nabla _i \nu, \partial _j\rangle' =\langle \overline \nabla _t\overline \nabla _i \nu, \partial _j\rangle+\langle \overline \nabla _i \nu, \overline \nabla _t\partial _j\rangle =&\langle \overline \nabla _i\overline \nabla _t \nu, \partial _j\rangle+\langle \overline \nabla _i \nu, \overline \nabla _j(F\nu)\rangle\\ =&-\langle \nabla _i\nabla F, \partial _j\rangle+F\langle \overline \nabla _i \nu, \overline \nabla _j\nu\rangle\\ =&-\nabla ^2_{ij} F+FA_{ik}A^k_j. \end{array}$

The formula ${ d\mu'=FH_1 d\mu}$ follows by the first variation formula for area. Finally, by Lemma 4,

$\displaystyle \begin{array}{rl} H_m'= \frac{\partial H_m}{\partial A_i^j}\partial _t A_i^j =&(T_{m-1})_j^i\left(-(\nabla ^2F)_i^j-FA_{i}^kA_k^j\right)\\ =&-\nabla ^j\left((T_{m-1})_j^iF_i\right)-F\left(H_1H_m-(m+1)H_{m+1}\right). \end{array}$

Here we have used the fact that

$\displaystyle \begin{array}{rl} \frac{\partial }{\partial A_i^j} H_m = \frac{\partial }{\partial A_i^j} \left(\frac{1}{m!}\epsilon^{i_1\cdots i_m}_{j_1\cdots j_m}A_{i_1}^{j_1}\cdots A_{i_m}^{j_m}\right) = \frac{m}{m!}\epsilon^{ii_1\cdots i_{m-1}}_{jj_1\cdots j_{m-1}}A_{i_1}^{j_1}\cdots A_{i_{m-1}}^{j_{m-1}} =(T_{m-1})^i_j. \end{array}$

$\Box$

 Proposition 6 Under the flow in (5), for any function ${f}$ defined locally around ${\Sigma_t}$, $\displaystyle \begin{array}{rl} \left(\int _\Sigma H_k f d\mu\right)'=\int _\Sigma(-(T_{k-1})_j^iF(\nabla ^2 f)_i^j +(k+1)H_{k+1}Ff+ H_k \langle \overline \nabla f , F\nu\rangle)d\mu. \end{array}$

Proof: By Lemma 5,

$\displaystyle \begin{array}{rl} \left(\int_\Sigma H_k f d\mu \right)' =& \int_\Sigma \left(H_k ' f + H_k f' +H_k fFH_1\right)d\mu\\ =&\int_\Sigma (-\nabla ^j((T_{k-1})_j^iF_i)f-Ff\left(H_1H_k-(k+1)H_{k+1}\right)\\ &+ H_k \langle \overline \nabla f , F\nu\rangle +H_k fFH_1)d\mu\\ =&\int_\Sigma \left((T_{k-1})_j^iF_i f^j +(k+1)H_{k+1}Ff+ H_k \langle \overline \nabla f , F\nu\rangle\right)d\mu\\ =&\int_\Sigma \left(-(T_{k-1})_j^iF (\nabla ^2f)_i^j +(k+1)H_{k+1}Ff+ H_k \langle \overline \nabla f , F\nu\rangle\right)d\mu. \end{array}$

$\Box$

The inverse curvature flow described in Theorem 1 actually exists for all time, given that the initial hypersurface is star-shaped:

 Theorem 6 ([Gerhardt] and [Urbas]) } Let ${\Sigma }$ be a smooth, closed hypersurface in ${ \mathbb{R}^{n}}$ with positive ${\sigma_k}$, given by a smooth embedding ${X_0 : \mathbb{S}^{n-1} \rightarrow \mathbb{R}^{n} }$. Suppose ${ \Sigma}$ is star-shaped with respect to a point ${ P }$. Then the initial value problem $\displaystyle \begin{cases} \frac{ \partial X}{ \partial t} = \frac{\sigma_{k-2}}{ \sigma_{k-1} } \nu \\ X (\cdot, 0) = X_0 ( \cdot) \end{cases} \ \ \ \ \ (6)$ has a unique smooth solution ${ X : \mathbb{S}^{n-1} \times [0, \infty) \rightarrow \mathbb{R}^{n} }$, where ${ \nu }$ is the unit outer normal vector to ${\Sigma_t = X (\mathbb{S}^{n-1}, t) }$ and ${\sigma_l}$ is the normalized ${l}$-th mean curvature of ${\Sigma_t}$. Moreover, ${\Sigma_t}$ is star-shaped with respect to ${P}$ and the rescaled hypersurface ${ \widetilde{\Sigma_t }}$, parametrized by ${ \widetilde{X}(\cdot , t) = e^{- t } X( \cdot, t) }$, converges to a sphere centered at ${P}$ in the ${ C^\infty}$ topology as ${ t \rightarrow \infty}$.

Finally, we also need the Hsiung-Minkowski formula ([Hsiung], [K]):

 Theorem 7 ([K] Cor. 3.1) Corollary 3.1} Suppose ${\Sigma}$ is a closed hypersurface in ${\mathbb{R}^n}$, and ${f}$ is a smooth function on ${\Sigma}$. Then $\displaystyle \int_\Sigma f \sigma_k d\mu =\int_\Sigma f\sigma_{k+1}\nu\cdot X d\mu- \frac{1}{(n-1-k){{n-1}\choose k}}\int_\Sigma \langle T_k(\nabla f), X^T\rangle d\mu.$ Here ${X}$ is the position vector, ${X^T}$ is its tangential component and ${\nu}$ is the unit outward normal of ${\Sigma}$.

2. Proof of the main results

Proof of Theorem 1: By Lemma 5, under the flow in (5),

$\displaystyle \left(\int_\Sigma H_k fd\mu\right)' =\int_\Sigma \left(-(T_{k-1})_j^iF (\nabla ^2f)_i^j +(k+1)H_{k+1}Ff+ H_k \langle \overline \nabla f , F\nu\rangle\right)d\mu. \ \ \ \ \ (7)$

For any function ${f}$, we have

$\displaystyle \nabla ^2 _{ij}f= \overline \nabla ^2_{ij}f- A_{ij}\overline \nabla f\cdot \nu.$

Now we fix ${f= \frac{r^2 }{2}}$, then ${\overline \nabla f=X}$ and ${\overline \nabla ^2 f = g}$ (the Euclidean metric). i.e.

$\displaystyle \nabla ^2_{ij}f=g_{ij}-A_{ij}u$

where ${u =X\cdot \nu\ge 0}$ as ${\Sigma_t}$ is star-shaped. Putting this into (7), and using Proposition 1, for this ${f}$, we have

$\displaystyle \begin{array}{rl} \left(\int_\Sigma H_k fd\mu\right)' =&\int_\Sigma \left(-(T_{k-1})_j^iF (\nabla ^2f)_i^j +(k+1)H_{k+1}Ff+ H_k \langle \overline \nabla f , F\nu\rangle\right)d\mu\\ =&\int_\Sigma \left(F \left((T_{k-1})_i^jA_j^iu- \textrm{tr}\left(T_{k-1}\right)\right) +(k+1)H_{k+1}Ff+ FH_k u\right)d\mu\\ =&\int_\Sigma \left(F \left(k H_k u- (n-k)H_{k-1}\right) +(k+1)H_{k+1}Ff+ FH_k u\right)d\mu\\ =&\int_\Sigma \left((k+1)FH_k u -(n-k)H_{k-1}F+(k+1)H_{k+1}Ff\right)d\mu. \end{array}$

This is equivalent to

$\displaystyle \begin{array}{rl} \left(\int_\Sigma \sigma_k fd\mu\right)' =&\int_\Sigma \left((k+1)F\sigma_k u -k \sigma_{k-1}F+(m-k)\sigma_{k+1}Ff\right)d\mu \end{array}$

where ${m=n-1}$. Let ${F=\frac{\sigma_{k-2}}{\sigma_{k-1}}}$, then the above equation becomes

$\displaystyle \begin{array}{rl} \left(\int_\Sigma \sigma_k fd\mu\right)' =&\int_\Sigma \left((k+1)\frac{\sigma_k \sigma_{k-2}}{\sigma_{k-1}} u -k \sigma_{k-2}+(m-k)\frac{\sigma_{k+1}\sigma_{k-2}}{\sigma_{k-1}}f\right)d\mu. \end{array}$

By the fact that ${\sigma_k>0}$ implies ${\sigma_l>0}$ for ${l\le k}$ (cf. e.g. [BC] Proposition 3.2), the Hsiung-Minkowski formula (Theorem 7), and by the Maclaurin’s inequality ${\sigma_{m-1}\sigma_{m+1}\le \sigma_m^2}$, which implies ${\frac{\sigma_{k+1}\sigma_{k-2}}{\sigma_{k-1}}\le \sigma_k}$, we have

$\displaystyle \begin{array}{rl} \left(\int_\Sigma \sigma_k fd\mu\right)' \le&\int_\Sigma \left((k+1)\sigma_{k-1}u -k \sigma_{k-2}+(m-k)\sigma_k f\right)d\mu\\ =&\int_\Sigma \left((k+1)\sigma_{k-2} -k \sigma_{k-2}+(m-k)\sigma_k f\right)d\mu\\ =&\int_\Sigma \left(\sigma_{k-2} +(m-k)\sigma_k f\right)d\mu. \end{array}$

i.e.

$\displaystyle \left(\int_\Sigma \sigma_k r^2 d\mu\right)'\le \int_\Sigma \left(2\sigma_{k-2} +(m-k)\sigma_k r^2\right)d\mu. \ \ \ \ \ (8)$

On the other hand, by (7), we have

$\displaystyle \left(\int _\Sigma\sigma_{k-2}d\mu\right)'= (n-k+1)\int_{\Sigma}\sigma_{k-1}Fd\mu=(n-k+1)\int_{\Sigma}\sigma_{k-2}d\mu. \ \ \ \ \ (9)$

Combining this equation and (8), we have

$\displaystyle \left(\int _\Sigma\sigma_{k-2}d\mu- \int_\Sigma \sigma_k r^2 d\mu\right)'\ge (n-1-k) \left(\int _\Sigma\sigma_{k-2}d\mu- \int_\Sigma \sigma_k r^2 d\mu\right).$

Equivalently,

$\displaystyle \left[e^{-(n-1-k)t}\left(\int _\Sigma\sigma_{k-2}d\mu- \int_\Sigma \sigma_k r^2 d\mu\right)\right]'\ge 0.$

In view of (9), this can also be written as

$\displaystyle \left[\left(\int_\Sigma\sigma_{k-2}d\mu\right)^{-\frac{n-k-1}{n-k+1}}\left(\int _\Sigma\sigma_{k-2}d\mu- \int_\Sigma \sigma_k r^2 d\mu\right)\right]'\ge 0.$

Suppose the equality holds, then by the equality case of the Newton’s inequality, ${\Sigma_t}$ must be umbilical, and therefore ${\Sigma}$ is a sphere. $\Box$

Proof of Theorem 3: By Theorem 6, there exists a smooth solution ${\{ \Sigma_t \}}$ to the inverse mean curvature flow with initial condition ${ \Sigma}$. Moreover, the rescaled hypersurface ${ \widetilde \Sigma_t = \{ e^{-\frac t{n-1}}x \ | \ x \in \Sigma_t \} }$ converges exponentially fast in the ${C^\infty}$ topology to a sphere. In particular, ${\widetilde \Sigma_t}$ and hence ${\Sigma_t}$, must be convex for large ${t}$.

Let ${ t_1 }$ be a time when ${ \Sigma_{t_1}}$ becomes convex. By using Theorem 7 twice, first with ${f=1}$ and then with ${f=u=X\cdot \nu}$, we have

$\displaystyle \begin{array}{rl} \int_{\Sigma_{t_1}} \sigma_{k-2} d\mu =&\int_{\Sigma_{t_1}} \sigma_{k-1}ud\mu\\ =&\int_{\Sigma_{t_1}} \sigma_k u^2 d\mu - \frac{1}{(m-k+1){m\choose {k-1}}}\int_{\Sigma_{t_1}} \langle T_{k-1}A(X^T), X^T\rangle d\mu\\ \le &\int_{\Sigma_{t_1}} \sigma_k r^2 d\mu - \frac{1}{(m-k+1){m\choose {k-1}}}\int_{\Sigma_{t_1}} \langle T_{k-1}A(X^T), X^T\rangle d\mu \end{array}$

Here we have used the fact that ${\nabla u=A(X^T)}$, which can be easily verified. As ${\Sigma_{t_1}}$ is convex, we have ${A>0}$ and ${T_{k-1}>0}$, and hence ${T_{k-1}A>0}$ (note that ${T_{k-1}}$ and ${A}$ are simultaneously diagonalizable), therefore

$\displaystyle \int_{\Sigma_{t_1}} \sigma_{k-2} d\mu\le\int_{\Sigma_{t_1}} \sigma_k r^2d\mu.$

In other words, ${Q(t_1)\le 0}$ for ${Q}$ defined in Theorem 1.

By Theorem 1, we know ${Q(t)}$ is monotone increasing, hence

$\displaystyle Q(0) \le Q(t_1) \le 0$

which proves (4).

If the equality in (4) holds, then ${ Q(0) = 0 }$. It follows from the monotonicity of ${ Q(t)}$ and the fact ${ Q(t) \le 0 }$ for large ${t}$ that

$\displaystyle Q(t) = 0, \ \forall \ t .$

By Theorem 1, this implies that ${ \Sigma_t }$ is a sphere for each ${t}$. By the above argument, we have ${X\cdot \nu= r}$ on ${\Sigma_{t_1}}$, from which we can deduce that ${\Sigma_{t_1}}$ is a sphere centered at ${O}$. Therefore, we conclude that the initial hypersurface ${ \Sigma}$ is a sphere centered at ${O}$. $\Box$

 Remark 1 We remark that the inequality in Theorem 3 also holds when ${k=1}$ by [KM] (using the convention that ${\sigma_{-1}=X\cdot \nu}$). When ${k=2}$, the assumption that ${\Sigma}$ is star-shaped can be dropped, by [K] Corollary 3.3.