Constant mean curvature surfaces in warped product manifolds

Abstract

Abstract. We consider surfaces with constant mean curvature in certain warped product manifolds. We show that any such surface is umbilic, provided that the warping factor satisfies certain structure conditions. This theorem can be viewed as a generalization of the classical Alexandrov theorem in Euclidean space. In particular, our results apply to the deSitter-Schwarzschild and Reissner-Nordstrom manifolds.

A Minkowski Inequality for Hypersurfaces in the Anti‐de Sitter‐Schwarzschild Manifold

“…We have jh have shown above that jM j is uniformly bounded, and H D n 1 C O.t e /. Hence, it follows from(3…”

“…This inequality was used in [3] to prove a generalization of Alexandrov's theorem (see also [4]). Finally, we study the limit of Q(t) as t → ∞.…”

“…This inequality generalizes the classical Minkowski inequality for surfaces in the three dimensional Euclidean space, and has a natural interpretation in terms of the Penrose inequality for collapsing null shells of dust. The proof relies on a new monotonicity formula for inverse mean curvature flow, and uses a geometric inequality established by the first author in [3]. …”

The Area Preserving Willmore Flow and Local Maximizers of the Hawking Mass in Asymptotically Schwarzschild Manifolds

“…On the other hand, studying the uniqueness of stable constant mean curvature spheres which are not necessarily isoperimetric turned out to be a more difficult problem. As a first step in this direction, Brendle showed a Heintze-Karcher type inequality and used a conformal flow in an elegant way to show that the centred spheres are the only constant mean curvature surfaces contained in one half of the Schwarzschild manifold (see [Bre13]). Finally, in a series of crucial results, Chodosh and Eichmair obtained the unconditional characterization of stable constant mean curvature surfaces in asymptotically flat manifolds which are C 6 −close to Schwarzschild and whose scalar curvature is non-negative and satisfies a certain decay condition.…”

Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds I: minimization

“…But then is a Euclidean round sphere which contradicts that was not spherical. More precisely, by a maximum principle argument using comparison with CMC slices, it is possible to show that the only embedded closed minimal hypersurface in is the horizon , which in fact is totally geodesic. Regarding CMC surfaces in , it was proved by Brendle that the only embedded closed CMC surfaces in the outer Schwarzschild are the spherical slices (let us mention that the results of Brendle include a larger class of warped products metrics). The embeddedness assumption is crucial for this classification result, in view of possible phenomena analogous to the Wente tori (which are immersed and CMC) in .…”