## Weighted Hsiung-Minkowski formulas and rigidity of umbilic hypersurfaces

1. Motivation and Main Results

A. D. Alexandrov [Ale1956], [Ale1962] proved that the only closed hypersurfaces of constant (higher order) mean curvature embedded in ${{\mathbb{R}}^{n \geq 3}}$ are round hyperspheres. The embeddedness assumption is essential. For instance, ${{\mathbb{R}}^{3}}$ admits immersed tori with constant mean curvature, constructed by U. Abresch [Abr1987] and H. Wente [Wen1986] . R. C. Reilly [Rei1977] and A. Ros [Ros1987], [Ros1988] presented alternative proofs, employing the Hsiung-Minkowski formula. See also Osserman’s wonderful survey [Oss1990] .

In 1999, S. Montiel [Mon1999] established various general rigidity results in a class of warped product manifolds, including the Schwarzschild manifolds and Gaussian spaces. Some of his results require the additional assumption that the closed hypersurfaces are star-shaped with respect to the conformal vector field induced from the ambient warped product structure. As a corollary [Mon1999][Example 5] , he also recovers Huisken’s theorem [Hui1990] that the closed, star-shaped, self-shrinking hypersurfaces to the mean curvature flow in ${{\mathbb{R}}^{n \geq 3}}$ are round hyperspheres. In 2016, S. Brendle [B2016] solved the open problem that, in ${{\mathbb{R}}^{3}}$, closed, embedded, self-shrinking topological spheres to the mean curvature flow should be round. The embeddedness assumption is essential. Indeed, in 2015, G. Drugan [Dru2015] employed the shooting method to prove the existence of a self-shrinking sphere with self-intersections in ${{\mathbb{R}}^{3}}$.

In 2001, H. Bray and F. Morgan [BM2002] proved a general isoperimetric comparison theorem in a class of warped product spaces, including Schwarzschild manifolds. In 2013, S. Brendle [B2013] showed that Alexandrov Theorem holds in a class of sub-static warped product spaces, including Schwarzschild and Reissner-Nordstrom manifolds. S. Brendle and M. Eichmair [BE2013] extended Brendle’s result to the closed, convex, star-shaped hypersurfaces with constant higher order mean curvature. See also [Gim2015] by V. Gimeno, [LX2016] by J. Li and C. Xia, and [WW2016] by X. Wang and Y.-K. Wang.

In this post, we provide new rigidity results (Theorem 1, 2 and 3). First, we associate the manifold ${M^{n \geq 3} = \left( {N}^{n-1} \times [0,\bar{r}), \bar{g} = dr^2 + h(r)^2 \, g_ {N} \right)}$, where ${(N^{n-1}, g_N)}$ is a compact manifold with constant curvature ${K}$. As in [B2013], [BE2013] , we consider four conditions on the warping function ${h: [0,\bar{r}) \rightarrow [0, \infty)}$:

• (H1) ${h'(0) = 0}$ and ${h''(0) > 0}$.
• (H2) ${h'(r) > 0}$ for all ${r \in (0,\bar{r})}$.
• (H3) ${2 \, \frac{h''(r)}{h(r)} - (n-2) \, \frac{K - h'(r)^2}{h(r)^2}}$ is monotone increasing for ${r \in (0,\bar{r})}$.
• (H4) For all ${r \in (0,\bar{r})}$, we have ${\frac{h''(r)}{h(r)} + \frac{K-h'(r)^2}{h(r)^2} > 0}$.

Examples of ambient spaces satisfying all the conditions include the classical Schwarzschild and Reissner-Nordstrom manifolds [B2013] [Section 5].

 Theorem 1 Let ${\Sigma}$ be a closed hypersurface embedded in ${{ M }^{n \geq 3}}$ with the ${k}$-th normalized mean curvature function ${H_{k}=\eta(r)>0}$ on ${\Sigma}$ for some smooth radially symmetric function ${\eta(r)}$. Assume that ${\eta(r)}$ is monotone decreasing in ${r}$. ${k=1:}$ Assume (H1), (H2), (H3). Then ${\Sigma}$ is umbilic. ${k \in \{2, \cdots, n-1\}:}$ Assume (H1), (H2), (H3), (H4). If ${\Sigma}$ is star-shaped (Section 2), then it is a slice ${{N}^{n-1} \times \left\{ r_{0} \right\}}$ for some constant ${r_{0}}$.

We also prove the following rather general rigidity result for linear combinations of higher order mean cuvatures, with less stringent assumptions on the ambient space.

 Theorem 2 Suppose ${(M^{n \geq 3}, \bar g)}$ satisfies (H2) and (H4). Let ${\Sigma}$ be a closed star-shaped ${k}$-convex (${H_k>0}$) hypersurface immersed in ${M^n}$, ${\{a_i(r)\}_{i=1}^{l-1}}$ and ${\{b_j(r)\}_{j=l}^k}$ (${2\le l) be a family of monotone decreasing, smooth, non-negative functions and a family of monotone increasing, smooth, non-negative functions respectively (where at least one ${ a_i(r) }$ and one ${ b_j(r) }$ are positive). Suppose $\displaystyle \sum_{i=1}^{l-1}a_i(r)H_i= \sum_{j=l}^{k} b_j(r) H_j.$ Then ${\Sigma}$ is totally umbilic.

Theorem 2 contains the case where ${\frac{H_k}{H_l}=\eta(r)}$ for some monotone decreasing function ${\eta}$ and ${k>l}$. We notice that the same result also applies to the space forms ${\mathbb R^n}$, ${\mathbb H^n}$ and ${\mathbb S^n_+}$ (open hemisphere) without the star-shapedness assumption (Theorem 11). Our result extends [Koh2000][Theorem B] by S.-E. Koh, [Kwo2016][Corollary 3.11] by Kwong and [WX2014] [Theorem 11] by J. Wu and C. Xia.

We next prove, in Section 4, a rigidity theorem for self-expanding soliton to the inverse curvature flow. Let us first recall the well-known inverse curvature flow of hypersurfaces

$\displaystyle \frac{d}{dt} \mathcal{F} =\frac{ {\sigma}_{k-1} }{ {\sigma}_{k} } \nu, \ \ \ \ \ (1)$

where ${\nu}$ denotes the outward pointing unit normal vector field and ${{\sigma}_{k}}$ the ${k}$-th symmetric function of the principal curvature functions. We point out that the inverse curvature flow has been used to prove various geometric inequalities and rigidities: Huisken-Ilmanen [HI2001] , Ge-Wang-Wu [GWW2014] , Li-Wei-Xiong [LWX2014], Kwong-Miao [KM2014] , Brendle-Hung-Wang [BHW2016] , Guo-Li-Wu GLW2016 , and Lambert-Scheuer [LS2016] .

In the Euclidean space, the long time existence of smooth solutions to (1) was proved by Gerhardt in [G1990] and by Urbas in [U1990] , when the initial closed hypersurface is star-shaped and ${k}$-convex ${({\sigma}_{k}>0)}$. Furthermore, they showed that the rescaled hypersurfaces converge to a round hypersphere as ${ t \rightarrow \infty}$.

 Theorem 3 Let ${\Sigma}$ be a closed hypersurface immersed in ${{\mathbb{R}}^{n \geq 3}}$. If it becomes a self-expanding soliton to the inverse curvature flow, it must be round.

In the proof of our main results, we shall use several integral formulas and inequalities. Theorem 1 requires the embeddedness assumption as in the classical Alexandrov Theorem and is proved for the space forms in [Kwo2016] . Theorem 2 and 3 require no embeddedness assumption and Theorem 3 is proved in [DLW2015] for the inverse mean curvature flow.

2. Preliminaries

Let ${(N^{n-1}, g_N)}$ be an ${(n-1)}$-dimensional compact manifold with constant curvature ${K}$. Our ambient space is the warped product manifold ${M^{n \geq 3} = {N}^{n-1} \times [0,\bar{r})}$ equipped with the metric ${\bar{g} = dr^2 + h(r)^2 \, g_ {N} }$. The precise conditions on the warping function ${h}$ will be stated separately for each result.

In this post, all hypersurfaces we consider are assumed to be connected, closed, and orientable. On a given hypersurface ${\Sigma}$ in ${M}$, we define the normalized ${k}$-th mean curvature function

$\displaystyle \begin{array}{rl} H_k:=H_k(\Lambda)=\frac{1}{\binom{n-1}{k}}\sigma_k(\Lambda), \end{array}$

where ${\Lambda=(\l_1,\cdots,\l_{n-1})}$ are the principal curvature functions on ${\Sigma}$ and the homogenous polynomial ${\sigma_k}$ of degree ${k}$ is the ${k}$-th elementary symmetric function

$\displaystyle \sigma_k(\Lambda)=\sum_{i_1<\cdots

We adopt the usual convention ${\sigma_0=H_{0}=1}$.

 Definition 4 (Potential function and conformal vector fieldi) In our ambient warped product manifold ${M}$, we define the potential function ${f(r) = h'(r)>0.}$ We define the vector field ${X = h(r) \, \frac{\partial}{\partial r}=\overline \nabla \psi}$, where ${\psi'(r)=h(r)}$ and ${\overline \nabla}$ is the connection on ${M}$. We note that it is conformal: ${\mathcal L_X \overline g = 2 f\overline g}$ [B2013] [Lemma 2.2].
 Definition 5 ( Star-shapeness ) For a hypersurface ${\Sigma}$ oriented by the outward pointing unit normal vector field ${\nu}$, we say that it is star-shaped when ${\langle X, \nu \rangle \ge 0}$.

A useful tool in studying higher order mean curvatures is the ${k}$-th Newton transformation ${T_k: T\Sigma \rightarrow T \Sigma}$ cf. [Rei1973], [Rei1977]. If we write

$\displaystyle T_k ( e_j ) =\sum_{i=1}^{n-1} ( T_k )_j^i e_{i},$

then ${ (T_k) _j^i }$ are given by

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

where ${(A_i^j)}$ is the second fundamental form of ${\Sigma}$. If ${ \{ e_i \}_{i=1}^{n-1} }$ consist of eigenvectors of ${A}$ with

$\displaystyle A (e_j) = \lambda_j e_{j},$

then we have

$\displaystyle T_k (e_j) = \Lambda_j e_{j},$

where

$\displaystyle \Lambda_j = \sum_{ \substack{ 1 \le i_1 < \cdots < i_k \le n-1, \\ j \notin \{ i_1, \cdots, i_k \} } } \lambda_{i_1} \cdots \lambda_{i_k} =\sigma_k(\lambda_1, \cdots, \lambda_{j-1}, \lambda_{j+1}, \cdots, \lambda_{n-1}). \ \ \ \ \ (2)$

One also defines ${T_0 = \mathrm{Id}}$, the identity map. We have the following basic facts:

 Lemma 6 Let ${\Sigma}$ be a closed hypersurface in a warped product manifold ${M}$ satisfying the condition (H2). On ${\Sigma}$, there is an elliptic point, where all principal curvatures are positive. Assume that ${\Sigma}$ is ${p}$-convex ${(H_{p}>0)}$. Then the following assertions hold
1. For all ${k \in \{1, \cdots, p-1\}}$, we have ${T_k>0}$ and ${H_k>0}$. For any ${j \in \{1,\cdots, n-1\}}$, we have ${H_{k;j}:=H_k(\lambda_1, \cdots, \lambda_{j-1}, \lambda_{j+1}, \cdots, \lambda_{n-1})>0}$.
2. The inequality ${\frac{iH_{p-1}}{H_p} \geq \frac{1}{H_{1}}>0}$ holds.
3. For ${1\le i and for any ${l=\{1, \cdots, n-1\}}$,

$\displaystyle \begin{array}{rl} j H_i H_{j-1;l}> i H_j H_{i-1;l}. \end{array}$

Proof: The first assertion is proved in [LWX] [Lemma 4]. As in the proof of [BC1997] [Proposition 3.2], ${T_k>0}$ when ${k \in \{1, \cdots, p-1\}}$, which implies

$\displaystyle H_k(\lambda_1, \cdots, \lambda_{j-1}, \lambda_{j+1}, \cdots, \lambda_{n-1})>0$

by (2). Also, ${H_k=\frac{1}{(n-1-k){{n-1}\choose k}}\mathrm{tr}_\Sigma (T_k)>0}$. The classical Newton-Maclaurin inequality ${H_{1} H_{p-1} \geq H_{p}}$ then gives ${\frac{H_{p-1}}{H_p} \geq \frac{1}{H_{1}}>0}$.

We now show (6). Let ${\lambda=\lambda_l}$, ${m=n-1}$, and ${\sigma_{i;l}={{m-1}\choose i} H_{i;l}}$. Note that ${\sigma_{i}= \lambda \sigma_{i-1;l}+\sigma_{i;l}}$, which implies

$\displaystyle H_{i}=\frac{i}{m}\lambda H_{i-1;l}+\frac{m-i}{m}H_{i;l}.$

Using this identity, (6), and the Newton-Maclaurin inequality, we have

$\displaystyle \begin{array}{rl} &j H_i H_{j-1;l}- i H_j H_{i-1;l}\\ =& j \left(\frac{i}{m}\lambda H_{i-1;l}+\frac{m-i}{m}H_{i;l}\right)H_{j-1;l} - i \left(\frac{j}{m}\lambda H_{j-1;l}+\frac{m-j}{m}H_{j;l}\right) H_{i-1;l}\\ =& \frac{j(m-i)}{m}H_{i;l}H_{j-1;l}- \frac{i(m-j)}{m}H_{j;l}H_{i-1;l}\\ =& (j-i)H_{j-1;l}H_{i-1;l}+ \frac{i(m-j)}{m}(H_{i;l} H_{j-1;l}-H_{j;l}H_{i-1;l})\\ >&0. \end{array}$

$\Box$

For the reader’s convenience, let us also record the following Heintze-Karcher-type inequality due to Brendle [B2013] [Theorem 3.5 and 3.11], which is crucial in our proof of Theorem 1.

 Proposition 7 ( Brendle’s Inequality ) Suppose the warped product manifold ${(M,\bar g)}$ satisfies (H1), (H2), and (H3). Let ${\Sigma}$ be a closed embedded hypersurface in ${(M,\bar g)}$ with positive mean curvature. Then $\displaystyle \int_\Sigma \frac{f}{H_1}\ge \int_\Sigma \langle X, \nu\rangle.$ The equality holds if and only if ${\Sigma}$ is umbilic. If, futhermore, (H4) is satisfied, then ${\Sigma}$ is a slice ${N\times \{r_0\}}$.

3. Proof of Theorem 1 and 2

The following formulas will play an essential role in our proof.

 Proposition 8 Let ${\phi}$ be a smooth function on a closed hypersurface ${\Sigma}$ in a Riemannian manifold ${M^n}$. (Weighted Hsiung-Minkowski formulas) For ${k \in \{1, \cdots, n-1\}}$, we have $\displaystyle \int_\Sigma \phi \left( f H_{k-1} - H_{k} \langle X, \nu\rangle \right) +\frac{1}{k{{n-1}\choose k }}\int_\Sigma \phi \left(\mathrm{div}_\Sigma T_{k-1}\right)(\xi) =-\frac{1}{k{{n-1}\choose k}}\int_\Sigma \langle T_{k-1}(\xi), \nabla _\Sigma \phi\rangle. \ \ \ \ \ (3)$ Here, ${\xi=X^T}$ is the tangential projection of the conformal vector field ${X}$ onto ${T\Sigma}$. (Note that ${\mathrm{div}(T_0)=0}$.) Suppose ${(M^n, \bar g)}$ is the warped product manifold in Section 2. Then, for ${k \in \{2, \cdots, n-1\}}$, $\displaystyle (\mathrm{div}_\Sigma T_{k-1}) (\xi) =-{{n-3}\choose {k-2}}\sum_{j=1}^{n-1} H_{k-2;j} \xi^j \mathrm{Ric}(e_j, \nu). \ \ \ \ \ (4)$ Here, ${\{e_j\}_{j=1}^{n-1}}$ and ${\{\lambda_j\}_{j=1}^{n-1}}$ are the principal directions and principal curvatures of ${\Sigma}$, respectively, and ${H_{k-2;j} =H_{k-2} (\lambda_1, \cdots, \lambda_{j-1}, \lambda_{j+1}, \cdots, \lambda_{n-1})}$. If ${\Sigma}$ is star-shaped and (H4) is satisfied, then, for each ${j\in \{1, \cdots, n-1\}}$, $\displaystyle -\xi^j\mathrm{Ric}(e_j, \nu)\ge 0 . \ \ \ \ \ (5)$

Proof: Let ${\xi=X^T=X-\langle X, \nu\rangle \nu}$, and recall that ${X}$ is conformal: ${\mathcal L_X \overline g = 2 f\overline g}$. By [Kwo2016] [Proposition 3.1], we have

$\displaystyle \mathrm{div}_\Sigma (\phi T_{k-1}(\xi)) = (n-k) f \sigma_{k-1}\phi- k \sigma_k \phi\langle X, \nu\rangle+ \phi (\mathrm{div }_\Sigma T_{k-1})(\xi)+\langle T_{k-1}(\xi), \nabla _\Sigma \phi\rangle.$

Integrating this equation, we get (3).

We now show (8). Take a local orthonormal frame ${\nu}$, ${e_{1}}$, ${\cdots}$, ${e_{n-1}}$, so that ${e_{1}}$, ${\cdots}$, ${e_{n-1}}$ are the principal directions of ${\Sigma}$. By the proof of [BE2013] [Proposition 8] (note that ${T^{(k)}}$ in [BE2013] is the ${(k-1)}$-th Newton transformation), we have

$\displaystyle (\mathrm{div}_\Sigma T_{k-1}) \xi =-\frac{n-k }{n-2}\sum_{j=1}^{n-1} \sigma_{k-2;j} \xi^j \mathrm{Ric}(e_j, \nu),$

where ${\sigma_{k-2;j} =\sigma_{k-2} (\lambda_1, \cdots, \lambda_{j-1}, \lambda_{j+1}, \cdots, \lambda_{n-1})}$, which is equivalent to (4).

It remains to show (5). As in [B2013] [Equation (2)], we compute

$\displaystyle \mathrm{Ric} =-\left(\frac{h''(r)}{h(r)} -(n-2)\frac{K-h'(r)^2}{h(r)^2}\right)\overline g -(n-2) \left(\frac{h''(r)}{h(r)} +\frac{ K-h'(r)^2}{h(r)^2}\right) {dr}^{2}.$

By the assumption (H4) and star-shaped condition ${ \langle \frac{\partial}{\partial r}, \nu\rangle>0}$, we have

$\displaystyle \begin{array}{rl} -\xi^j\mathrm{Ric}( e_j, \nu) =(n-2)\left(\frac{h''(r)}{h(r)}+\frac{K-h'(r)^2}{h(r)^2}\right) \frac{(\xi^j)^2}{h(r)} \langle \frac{\partial}{\partial r}, \nu \rangle \geq 0. \end{array}$

$\Box$

 Theorem 9 (Theorem 1) } Suppose ${(M^{n\geq 3}, \bar g)}$ is the warped product manifold in Section 2. Let ${\Sigma}$ be a closed hypersurface embedded in ${M^n}$ with the ${p}$-th normalized mean curvature ${H_{p}=\eta(r)>0}$ for some smooth radially symmetric function ${\eta(r)}$. Assume that ${\eta(r)}$ is monotone decreasing. ${p=1:}$ Assume (H1), (H2), (H3). Then ${\Sigma}$ is umbilic. ${p \in \{2, \cdots, n-1\}:}$ Assume (H1), (H2), (H3), (H4). If ${\Sigma}$ is star-shaped, then it is a slice ${{N}^{n-1} \times \left\{ r_{0} \right\}}$ for some constant ${r_{0}}$.

Proof: Taking the radial weight ${{\phi}(r)=\frac{1}{\eta(r)}>0}$ on ${\Sigma}$, we have ${{\phi}'(r) \geq 0}$ and

$\displaystyle f \frac{H_{p-1}}{H_{p}} - \langle X, \nu\rangle = \phi \left( f H_{p-1} - H_{p} \langle X, \nu\rangle \right). \ \ \ \ \ (6)$

Assume first ${p\in \{2, \cdots, n-1\}}$. By Proposition 8 (8) and Lemma 6, ${(\mathrm{div}_\Sigma T_{p-1}) \xi\ge 0}$. So by Proposition 8 (8), we have

$\displaystyle \begin{array}{rcl} \int_\Sigma \phi \left( f H_{p-1} - H_{p} \langle X, \nu\rangle \right) &\le& -\frac{1}{(n-p){{n-1}\choose {p-1}}}\int_\Sigma \langle T_{p-1}(X^T), \nabla _\Sigma \phi\rangle\\ &=& -\frac{1}{(n-p){{n-1}\choose {p-1}}}\int_\Sigma h(r) \phi'(r)\langle T_{p-1}(\nabla _\Sigma r), \nabla _\Sigma r\rangle\\ &\le&0 \end{array} \ \ \ \ \ (7)$

as ${T_{p-1}>0}$ by Lemma 6. We note that this inequality also holds for ${p=1}$ by Proposition 8 (without assuming (H4) and the star-shapedness). Combining (6) and (7), we deduce

$\displaystyle \int_\Sigma \, \left( f \frac{H_{p-1}}{H_{p}} - \langle X, \nu\rangle \right) = \int_\Sigma \phi \left( f H_{p-1} - H_{p} \langle X, \nu\rangle \right) \leq 0.$

From Lemma 6, we have ${\frac{H_{p-1}}{H_p} \geq \frac{1}{H_{1}}>0}$. Then, we obtain the inequality

$\displaystyle \int_\Sigma \, \left( \frac{f}{H_{1}} - \langle X, \nu\rangle \right) \leq 0.$

However, Brendle’s inequality (Proposition 7) is the reverse inequality

$\displaystyle \int_\Sigma \, \left( \frac{f}{H_{1}} - \langle X, \nu\rangle \right) \geq 0.$

These two inequalities imply the equality in Brendle’s inequality. We conclude that ${\Sigma}$ is umbilic and that, in the case when the condition (H4) holds, it is a slice. $\Box$

 Remark 1 Recently, Brendle’s inequality is extended in several ways, for instance, see [LX2016], [WW2016], [WWZ2014] . We observe that the proof of (1) in Theorem 1 works on more general warped product manifold ${M^n = {N}^{n-1} \times [0,\bar{r})}$, which admits the property that Brendle’s inequality holds. For instance, the fiber ${{N}^{n-1}}$ can be a compact Einstein manifold, as in Brendle’s paper [B2013] .

We now give another rigidity result which contains as a special case where the ratio of two distinct higher order mean curvatures is a radial function.

 Theorem 10 (Theorem 1) } Suppose ${(M^{n \geq 3}, \bar g)}$ is the warped product manifold in Section 2 satisfying (H2) and (H4). Let ${\Sigma}$ be a closed star-shaped ${k}$-convex (${H_k>0}$) hypersurface immersed in ${M^n}$, ${\{a_i(r)\}_{i=1}^{l-1}}$ and ${\{b_j(r)\}_{j=l}^k}$ (${2\le l) be a family of monotone decreasing, smooth, non-negative functions and a family of monotone increasing, smooth, non-negative functions respectively (where at least one ${ a_i(r) }$ and one ${ b_j(r) }$ are positive). Suppose $\displaystyle \sum_{i=1}^{l-1}a_i(r)H_i= \sum_{j=l}^{k} b_j(r) H_j.$ Then ${\Sigma}$ is totally umbilic.

Proof: Let ${\xi=X^T}$ and ${A_p=-\frac{1}{(n-1)(n-2)} \xi^p \mathrm{Ric}(e_p, \nu)}$. Since we are assuming (H2) and (H4), we can apply Lemma 6 (6) and Proposition 8 to have, for each ${i}$ and ${j}$,

$\displaystyle \begin{array}{rcl} &&\int_\Sigma a_i(r) \left( f H_{i-1} - H_i \langle X, \nu\rangle \right)+(i-1)\int_\Sigma a_i(r)\sum_{p=1}^{n-1}A_p H_{i-2;p}\\ &=&-\frac{1}{i{{n-1}\choose i}}\int_\Sigma \langle T_{i-1}(\xi), \nabla _\Sigma a_i\rangle =-\frac{1}{i{{n-1}\choose i}}\int_\Sigma h(r)a_i'(r)\langle T_{i-1}(\nabla _\Sigma r), \nabla _\Sigma r\rangle \ge 0 \end{array} \ \ \ \ \ (8)$

and

$\displaystyle \begin{array}{rcl} &&\int_\Sigma b_j(r) \left( f H_{j-1} - H_j \langle X, \nu\rangle \right)+(j-1)\int_\Sigma b_j(r)\sum_{p=1}^{n-1}A_p H_{j-2;p}\\ &=&-\frac{1}{j{{n-1}\choose j}}\int_\Sigma \langle T_{j-1}(\xi), \nabla _\Sigma b_j\rangle =-\frac{1}{j{{n-1}\choose j}}\int_\Sigma h(r)b_j'(r)\langle T_{j-1}(\nabla _\Sigma r), \nabla _\Sigma r\rangle \le 0. \end{array} \ \ \ \ \ (9)$

Summing (8) over ${i}$ and (9) over ${j}$, and then taking the difference gives

$\displaystyle \begin{array}{rcl} 0&=&\int \left(\sum_{j=l}^k b_j(r) H_j-\sum_{i=1}^{l-1} a_i(r) H_i\right)\langle X, \nu\rangle\\ &\ge& \int_\Sigma f \left(\sum_{j=l}^k b_j(r) H_{j-1}-\sum_{i=1}^{l-1} a_i(r) H_{i-1}\right)\\ &&+\int_\Sigma \sum_{p=1}^{n-1}A_p\left(\sum_{j=l}^k(j-1)b_j(r) H_{j-2;p}-\sum_{i=1}^{l-1} (i-1)a_i(r) H_{i-2;p}\right). \end{array} \ \ \ \ \ (10)$

Note that ${H_j>0}$ for ${j\le k}$ by Lemma 6. Let ${1\le i, then multiplying the Newton’s inequality ${H_i H_{j-1}\ge H_{i-1}H_j}$ by ${a_i(r) b_j(r)}$ and summing over ${i, j}$ gives

$\displaystyle \sum_{i=1}^{l-1} a_i(r) H_i \sum_{j=1}^k b_j(r) H_{j-1}\ge \sum_{i=1}^{l-1} a_i(r) H_{i-1}\sum_{j=1}^k b_j(r) H_j.$

Since ${\sum_{i=1}^{l-1}a_i(r)H_i=\sum_{j=l}^{k}b_j(r)H_j>0}$, we deduce

$\displaystyle \sum_{j=l}^k b_j(r)H_{j-1}\ge \sum_{i=1}^{l-1}a_i(r)H_{i-1}. \ \ \ \ \ (11)$

Similar to (11), we can obtain from Lemma 6 (6) the inequality

$\displaystyle \sum_{j=l}^k(j-1)b_j(r) H_{j-2;p}-\sum_{i=1}^{l-1} (i-1)a_i(r) H_{i-2;p} > 0. \ \ \ \ \ (12)$

On the other hand, ${A_p\ge 0}$ by Proposition 8. Combining this with (12) and (11), we conclude that all the integrands in (10) are zero. This implies (11) is an equality and hence ${\Sigma}$ is totally umbilic by the Newton-Maclaurin inequality. $\Box$

Due to the analogous, but simpler, weighted Hsiung-Minkowski integral formulas in the space forms (cf. [Kwo2016] ), without assuming the star-shapedness condition, following the proof of Theorem 10, we can deduce

 Theorem 11 Let ${\Sigma}$ be a closed ${k}$-convex hypersurface immersed in ${M^{n \geq 3}=\mathbb R^n}$, ${\mathbb H^n}$ or ${\mathbb S^n_+}$ (open hemisphere). Let ${r}$ be the distance in ${M^n}$ from a fixed point ${p_0\in M}$ (chosen to be the center if ${M=\mathbb S^n_+}$). Let ${\{a_i(r)\}_{i=1}^{l-1}}$ and ${\{b_j(r)\}_{j=l}^k}$ (${2\le l) be a family of monotone decreasing, smooth, non-negative functions and a family of monotone increasing, smooth, non-negative functions respectively (where at least one ${ a_i(r) }$ and one ${ b_j(r) }$ are positive). Suppose $\displaystyle \sum_{i=1}^{l-1}a_i(r)H_i= \sum_{j=l}^{k} b_j(r) H_j.$ Then it is a geodesic hypersphere.

4. Proof of Theorem 3

We consider the inverse curvature flow in Euclidean space ${{\mathbb{R}}^{n \geq 3}}$:

$\displaystyle \frac{d}{dt} \mathcal{F} =\frac{ {\sigma}_{k-1} }{ {\sigma}_{k} } \nu. \ \ \ \ \ (13)$

Here, ${\nu}$ denotes the outward pointing unit normal vector field and ${{\sigma}_{k}}$ the ${k}$-th symmetric function of the principal curvatures. When ${k=1}$, the evolution (13) is equivalent to the inverse mean curvature flow.

 Definition 12 We say that a hypersurface ${\Sigma}$ with ${{\sigma}_{k}>0}$ is a self-expander to the inverse curvature flow when we have $\displaystyle \frac{ {\sigma}_{k-1} }{ {\sigma}_{k} } = C \; \langle X, \nu \rangle \quad \text{on} \; \Sigma \ \ \ \ \ (14)$   for some constant ${C>0}$.
 Theorem 13 (Theorem 3) Let ${\Sigma}$ be a closed hypersurface immersed in ${{\mathbb{R}}^{n \geq 3}}$. If it becomes a self-expanding soliton to the inverse curvature flow, it must be round.

Proof: Let ${\mathbf{p}= \langle X, \nu \rangle}$ denote the support function. We shall repeatedly use Hsiung–Minkowski integral formula [Hs1954], [Hs1956] for the normalized higher order mean curvature functions ${H_i=\frac{1}{\binom{n-1}{i}}\sigma_i}$. Then, the soliton equation (14) is equivalent to

$\displaystyle \frac{ {H}_{k-1} }{ {H}_{k} } = \mu \; \mathbf{p} \ \ \ \ \ (15)$

for some constant ${\mu>0}$. By the ${k}$-convexity assumption ${{\sigma}_{k}>0}$, the item (6) in Lemma 6 implies that ${{\sigma}_{i}>0}$ and ${H_{i}>0}$ for ${i \in \{0, \cdots, k \}}$. We claim ${\mu=1}$. Indeed, the Hsiung–Minkowski integral formula shows that

$\displaystyle 0 = \int_{\Sigma} \; \left({H}_{k-1} - {H}_{k} \mathbf{p}\right) \; = \left( 1 - \frac{1}{\mu} \right) \int_{\Sigma} {H}_{k-1}.$

Since ${{H}_{k-1}>0}$ on ${\Sigma}$, we have ${\mu=1}$. The soliton equation (15) reduces to

$\displaystyle {H}_{k-1} = {H}_{k} \mathbf{p}.$

Notice that ${\mathbf{p}>0}$. We distinguish two cases: ${k=1}$ and ${k \in \{2, \cdots, n-1\}}$. When ${k=1}$, by the Newton-Maclaurin inequality, we have

$\displaystyle H_2 \mathbf{p}=\frac{H_2}{H_1}\le \frac{H_1}{H_0}= H_1. \ \ \ \ \ (16)$

Integrating this inequality gives

$\displaystyle \int_\Sigma H _2 \mathbf{p} \le \int_\Sigma H_1. \ \ \ \ \ (17)$

However, by the Hsiung-Minkowski formula, we have the equality

$\displaystyle \int_\Sigma H_1 =\int_\Sigma H_2 \mathbf{p}.$

When ${k \in \{2, \cdots, n-1\}}$, since ${H_i>0}$ for ${i\le k}$, by the Newton-Maclaurin inequality,

$\displaystyle H_0=1=\frac{H_k }{H_{k-1}} \mathbf{p} \le \frac{H_{k-1}}{H_{k-2}} \mathbf{p} \le\cdots\le\frac{H_1}{H_0} \mathbf{p} =H_1 \mathbf{p}. \ \ \ \ \ (18)$

Integrating this gives

$\displaystyle \int_\Sigma H_0 \le \int_\Sigma H_1 \mathbf{p}. \ \ \ \ \ (19)$

However, by the Hsiung-Minkowski inequality, we have the equality

$\displaystyle \int_\Sigma H_1 \mathbf{p}=\int_\Sigma H_{0}.$

In both cases, the Hsiung-Minkowski formula assures that the equality in (17) or (19) actually holds, and so the equality in (16) or (18) holds. Therefore, the closed hypersurface ${\Sigma}$ must be umbilic and so is round. $\Box$