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}.

  1. {k=1:} Assume (H1), (H2), (H3). Then {\Sigma} is umbilic.
  2. {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<k\le n-1}) 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<i_{k}}\lambda_{i_1}\cdots\lambda_{i_k}.

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).

  1. On {\Sigma}, there is an elliptic point, where all principal curvatures are positive.
  2. 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 <j\le k} 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}.

  1. (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}.)

  2. 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.

  1. {p=1:} Assume (H1), (H2), (H3). Then {\Sigma} is umbilic.
  2. {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<k\le n-1}) 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<j\le k}, 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<k\le n-1}) 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

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This entry was posted in Calculus, General Relativity, Geometry, Inequalities. Bookmark the permalink.

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