**1. Motivation and Main Results **

A. D. Alexandrov [Ale1956], [Ale1962] proved that the only closed hypersurfaces of constant (higher order) mean curvature embedded in are round hyperspheres. The embeddedness assumption is essential. For instance, 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 are round hyperspheres. In 2016, S. Brendle [B2016] solved the open problem that, in , 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 .

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 , where is a compact manifold with constant curvature . As in [B2013], [BE2013] , we consider four conditions on the warping function :

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

Theorem 1Let be a closed hypersurface embedded in with the -th normalized mean curvature function on for some smooth radially symmetric function . Assume that is monotone decreasing in .

- Assume (H1), (H2), (H3). Then is umbilic.
- Assume (H1), (H2), (H3), (H4). If is star-shaped (Section 2), then it is a slice for some constant .

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 contains the case where for some monotone decreasing function and . We notice that the same result also applies to the space forms , and (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

where denotes the outward pointing unit normal vector field and the -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 -convex . Furthermore, they showed that the rescaled hypersurfaces converge to a round hypersphere as .

Theorem 3Let be a closed hypersurface immersed in . 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 be an -dimensional compact manifold with constant curvature . Our ambient space is the warped product manifold equipped with the metric . The precise conditions on the warping function 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 in , we define the normalized -th mean curvature function

where are the principal curvature functions on and the homogenous polynomial of degree is the -th elementary symmetric function

We adopt the usual convention .

Definition 4 (Potential function and conformal vector fieldi)In our ambient warped product manifold , we define the potential function We define the vector field , where and is the connection on . We note that it is conformal: [B2013] [Lemma 2.2].

Definition 5 ( Star-shapeness )For a hypersurface oriented by the outward pointing unit normal vector field , we say that it is star-shaped when .

A useful tool in studying higher order mean curvatures is the -th Newton transformation cf. [Rei1973], [Rei1977]. If we write

then are given by

where is the second fundamental form of . If consist of eigenvectors of with

then we have

One also defines , the identity map. We have the following basic facts:

Lemma 6Let be a closed hypersurface in a warped product manifold satisfying the condition (H2).

**Proof**: The first assertion is proved in [LWX] [Lemma 4]. As in the proof of [BC1997] [Proposition 3.2], when , which implies

by (2). Also, . The classical Newton-Maclaurin inequality then gives .

We now show (6). Let , , and . Note that , which implies

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

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.

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

Proposition 8Let be a smooth function on a closed hypersurface in a Riemannian manifold .

(Weighted Hsiung-Minkowski formulas)For , we haveHere, is the tangential projection of the conformal vector field onto . (Note that .)

- Suppose is the warped product manifold in Section 2. Then, for ,
Here, and are the principal directions and principal curvatures of , respectively, and .

**Proof**: Let , and recall that is conformal: . By [Kwo2016] [Proposition 3.1], we have

Integrating this equation, we get (3).

We now show (8). Take a local orthonormal frame , , , , so that , , are the principal directions of . By the proof of [BE2013] [Proposition 8] (note that in [BE2013] is the -th Newton transformation), we have

where , which is equivalent to (4).

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

By the assumption (H4) and star-shaped condition , we have

Theorem 9 (Theorem 1)} Suppose is the warped product manifold in Section 2. Let be a closed hypersurface embedded in with the -th normalized mean curvature for some smooth radially symmetric function . Assume that is monotone decreasing.

- Assume (H1), (H2), (H3). Then is umbilic.
- Assume (H1), (H2), (H3), (H4). If is star-shaped, then it is a slice for some constant .

**Proof**: Taking the radial weight on , we have and

* Assume first . By Proposition 8 (8) and Lemma 6, . So by Proposition 8 (8), we have
*

* as by Lemma 6. We note that this inequality also holds for by Proposition 8 (without assuming (H4) and the star-shapedness). Combining (6) and (7), we deduce
*

From Lemma 6, we have . Then, we obtain the inequality

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

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

Remark 1Recently, 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 , which admits the property that Brendle’s inequality holds. For instance, the fiber 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 is the warped product manifold in Section 2 satisfying (H2) and (H4). Let be a closed star-shaped -convex () hypersurface immersed in , and () 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 and one are positive). Suppose

Then is totally umbilic.

**Proof**: Let and . Since we are assuming (H2) and (H4), we can apply Lemma 6 (6) and Proposition 8 to have, for each and ,

* Summing (8) over and (9) over , and then taking the difference gives
*

* Note that for by Lemma 6. Let , then multiplying the Newton’s inequality by and summing over gives
*

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

On the other hand, 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 is totally umbilic by the Newton-Maclaurin inequality.

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

**4. Proof of Theorem 3**

We consider the *inverse curvature flow* in Euclidean space :

Here, denotes the outward pointing unit normal vector field and the -th symmetric function of the principal curvatures. When , the evolution (13) is equivalent to the inverse mean curvature flow.

Definition 12We say that a hypersurface with is a self-expander to the inverse curvature flow when we have

Theorem 13 (Theorem 3)Let be a closed hypersurface immersed in . If it becomes a self-expanding soliton to the inverse curvature flow, it must be round.

**Proof**: Let denote the support function. We shall repeatedly use Hsiung–Minkowski integral formula [Hs1954], [Hs1956] for the normalized higher order mean curvature functions . Then, the soliton equation (14) is equivalent to

* for some constant . By the -convexity assumption , the item (6) in Lemma 6 implies that and for . We claim . Indeed, the Hsiung–Minkowski integral formula shows that
*

Since on , we have . The soliton equation (15) reduces to

Notice that . We distinguish two cases: and . When , by the Newton-Maclaurin inequality, we have

Integrating this inequality gives

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

When , since for , by the Newton-Maclaurin inequality,

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

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 must be umbilic and so is round.