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 :
- (H1) and .
- (H2) for all .
- (H3) is monotone increasing for .
- (H4) For all , we have .
Examples of ambient spaces satisfying all the conditions include the classical Schwarzschild and Reissner-Nordstrom manifolds [B2013] [Section 5].
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. Continue reading