# A characterization of constant mean curvature surfaces in homogeneous 3-manifolds

@article{Fernndez2005ACO,
title={A characterization of constant mean curvature surfaces in homogeneous 3-manifolds},
author={Isabel Fern{\'a}ndez and Pablo Mira},
journal={Differential Geometry and Its Applications},
year={2005},
volume={25},
pages={281-289}
}
• Published 2005
• Mathematics
• Differential Geometry and Its Applications
Abstract It has been recently shown by Abresch and Rosenberg that a certain Hopf differential is holomorphic on every constant mean curvature surface in a Riemannian homogeneous 3-manifold with isometry group of dimension 4. In this paper we describe all the surfaces with holomorphic Hopf differential in the homogeneous 3-manifolds isometric to H 2 × R or having isometry group isomorphic either to the one of the universal cover of PSL ( 2 , R ) , or to the one of a certain class of Berger… Expand
Rotationally invariant constant mean curvature surfaces in homogeneous 3-manifolds
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• Mathematics
• Journal of the Australian Mathematical Society
• 2008
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#### References

SHOWING 1-10 OF 10 REFERENCES
Isometric immersions into 3-dimensional homogeneous manifolds
We give a necessary and sufficient condition for a 2-dimensional Riemannian manifold to be locally isometrically immersed into a 3-dimensional homogeneous Riemannian manifold with a 4-dimensionalExpand
Harmonic maps and constant mean curvature surfaces in H 2 x R
• Mathematics
• 2007
We introduce a hyperbolic Gauss map into the Poincar´e disk for any surface in H2×R with regular vertical projection, and prove that if the surface has constant mean curvature H = 1/2, thisExpand
Examples and structure of CMC surfaces in some Riemannian and Lorentzian homogeneous spaces
• Mathematics
• 2005
It is proved that the holomorphic quadratic differential associated to CMC surfaces in Riemannian products S × R and H × R discovered by U. Abresch and H. Rosenberg could be obtained as a linearExpand
A Hopf differential for constant mean curvature surfaces inS2×R andH2×R
• Mathematics
• 2004
A basic tool in the theory of constant mean curvature (cmc) surfaces Σ in space forms is the holomorphic quadratic differential discovered by H. Hopf. In this paper we generalize this differential toExpand
Surfaces in Three-Dimensional Lie Groups
• Mathematics
• 2005
AbstractWe derive the Weierstrass (or spinor) representation for surfaces in the three-dimensional Lie groups Nil, $$\widetilde{SL}_2$$ , and Sol with Thurston's geometries and establish theExpand
Complete surfaces of constant curvature in H2 × R and S2 × R
• Mathematics
• 2005
We study isometric immersions of surfaces of constant curvature into the homogeneous spaces $${\mathbb{H}^2\times\mathbb{R}}$$ and $${\mathbb{S}^2\times\mathbb{R}}$$ . In particular, we prove thatExpand
Taimanov, Surfaces in three-dimensional Lie groups
• Siberian Math. J
• 2005
A Hopf differential for constant mean curvature surfaces in S 2 × R and H 2 × R, Acta Math
• A Hopf differential for constant mean curvature surfaces in S 2 × R and H 2 × R, Acta Math
• 2004
Generalized Hopf differentials, preprint
• Generalized Hopf differentials, preprint