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 1 Let 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 3 Let 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.
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 .
then are given by
where is the second fundamental form of . If consist of eigenvectors of with
then we have
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
The following formulas will play an essential role in our proof.
Proposition 8 Let be a smooth function on a closed hypersurface in a Riemannian manifold .
- (Weighted Hsiung-Minkowski formulas) For , we have
Here, 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).
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 .
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 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 , 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.
Note that for by Lemma 6. Let , then multiplying the Newton’s inequality by and summing over gives
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
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 12 We 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
Since on , we have . The soliton equation (15) reduces to
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.