Constant mean curvature surfaces in warped product manifolds

By S. Brendle. Publications mathématiques (Bures-sur-Yvette), 2011

234 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/23bddd6894f387910a093d66c342d4369ce99609

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In Applied Mathematics

In Radiology, Nuclear Medicine and Imaging

Citations in the corpus

A Generalization of Reilly's Formula and its Applications to a New Heintze–Karcher Type Inequality (2014, methodology): “Quite recently, in order to study the Alexandrov type rigidity problem in general relativity, Brendle [2] gave a version of Theorem A in Sn+ and H n, more generally, in a large class of warped product spaces, including the Schwarzschild manifold.”
A volume preserving flow and the isoperimetric problem in warped product spaces (2016, background): “ω2φ ′. (4.14) Plugging the deﬁnition of uinto (4.14), we ﬁnished the proof. Remark 4.2. By direct computations, if the ambient space is substatic, namely satisfying conditions H1-H4 in Brendle’s work [2], the last two lines in (4.10) are nonnegative, which immediately implies a uniform upper bound f”
Isoperimetric type problems and Alexandrov-Fenchel type inequalities in the hyperbolic space (2013, unknown): “y. For the case l = 0, we have Hk = c. As explained in [34], this implies that c is a positive constant and Hj is positive for 1 ≤ j ≤ k. A Heintze-Karcher type inequality, recently proved by Brendle [4], says that Z Σ h∇¯ρ,νidµ ≤ Z Σ 1 H1 (3.15) ρdµ, and equality holds if and only if Σ is a geodesi”
On the 1/H-flow by p-Laplace approximation: New estimates via fake distances under Ricci lower bounds (2019, background): “To our knowledge, the problem of deciding which curvature conditions guarantee that balls Br ⊂ Mh are isoperimetric sets is still partly open, and in this respect see [12].”
On static three-manifolds with positive scalar curvature (2015, result): “We show that the flow (30) starting at a minimal surface typically decreases the area (compare with [16], Proposition 3.”
Minimal surfaces in $$S^3$$: a survey of recent results (2013, methodology): “Finally, in a recent paper [8], we obtained a uniqueness theorem for embedded constant mean curvature surfaces in certain rotationally symmetric spaces.”
An integral formula and its applications on sub-static manifolds (2016, methodology): “5, by adding RicN = (n − 2)ρgN , we remove conditions (H4) and (H4) from [2].”
Alexandrov’s theorem revisited (2017, background): “∂vΩ = ∂Ω and Hv Ω(x) = HΩ(x) for any x∈ ∂Ω. Hence, the viscosity notion generalizes the mean convexity in the classical sense. This said, following Brendle’s point of view on the Montiel-Ros argument [Bre13], we have the following generalized form of the Heintze-Karcher inequality, see (4.4) below. ”
A sharp quantitative version of Alexandrov's theorem via the method of moving planes (2015, methodology): “Alexandrov Theorem has been studied also for warped product manifolds by Montiel [36], Brendle [9] and Brendle and Eichmair [10].”