2017
DOI: 10.48550/arxiv.1706.09394
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Constant mean curvature spheres in homogeneous three-manifolds

Abstract: We prove that two spheres of the same constant mean curvature in an arbitrary homogeneous three-manifold only differ by an ambient isometry, and we determine the values of the mean curvature for which such spheres exist. This gives a complete classification of immersed constant mean curvature spheres in three-dimensional homogeneous manifolds.

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Cited by 5 publications
(7 citation statements)
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“…Here we use the results in section (4) and we consider the unitary normal vector field ω to π directed as the geodesic γ in M n + satisfying γ(0) = v. Lemma 5.2. Assume (37) g p (N p , ω p ) ≤ µ for every p on the boundary of Σ, for some µ ≤ 1/2, and let t 0 = ρ 1 − µ 2 . Then Σ t is connected for any 0 < t < t 0 .…”
Section: Approximate Symmetry In One Directionmentioning
confidence: 99%
See 1 more Smart Citation
“…Here we use the results in section (4) and we consider the unitary normal vector field ω to π directed as the geodesic γ in M n + satisfying γ(0) = v. Lemma 5.2. Assume (37) g p (N p , ω p ) ≤ µ for every p on the boundary of Σ, for some µ ≤ 1/2, and let t 0 = ρ 1 − µ 2 . Then Σ t is connected for any 0 < t < t 0 .…”
Section: Approximate Symmetry In One Directionmentioning
confidence: 99%
“…[27] and [46] for classical counterexamples in higher dimension and in R 3 , respectively). However, for immersed hypersurfaces, one can add some condition in order to guarantee that S is a sphere: in particular Hopf proved in [25] that every constant mean curvature C 2 -regular sphere immersed in the 3-dimensional Euclidean space is necessary a round sphere (see also [1,37,38] for a very recent generalization of Hopf's theorem to simply-connected homogeneous 3-manifolds), and Barbosa and DoCarmo [6] proved that every compact, orientable and stable hypersurface immersed in R n is a round sphere (see also [7] for generalizations of this result). Moreover there exists non-closed constant mean curvature hypersufaces embedded in R 3 which are not diffeomorphic to a sphere, like for instance the unduloids (see [17] and [28] for the generalization to higher dimensions).…”
mentioning
confidence: 99%
“…Hopf's theorem was generalized in three-dimensional space forms by Chern in [8] and, more recently, in 3-dimensional homogeneous spaces (see [13] and the references therein). The study of constant mean curvature spacial Riemannian manifolds is a central subject in differential geometry and there are many interesting results on this topic (see e.g.…”
Section: Introductionmentioning
confidence: 99%
“…The value of H, if any, such that 4H 2 + κ = 0 is usually called the critical mean curvature. The Hopf problem has been solved very recently in the rest of homogeneous 3-manifolds by Meeks, Mira, Pérez, and Ros [29].…”
Section: Introductionmentioning
confidence: 99%