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}
}
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
Abstract We classify constant mean curvature surfaces invariant by a 1-parameter group of isometries in the Berger spheres and in the special linear group Sl ( 2 , R ) . In particular, all constantExpand
Compact stable constant mean curvature surfaces in homogeneous 3-manifolds
We classify the stable constant mean curvature spheres in the homogeneous Riemannian 3-manifolds: the Berger spheres, the special linear group and the Heisenberg group. We show that all of them areExpand
The Gauss map of surfaces in ~PSL_2(R)
We define a Gauss map for surfaces in the universal cover of the Lie group PSL_2(R) endowed with a left-invariant Riemannian metric having a 4-dimensional isometry group. This Gauss map is notExpand
Constant mean curvature hypersurfaces on symmetric spaces, minimal graphs on semidirect products and properly embedded surfaces in hyperbolic 3-manifolds
We prove results concerning the geometry of hypersurfaces on different ambient spaces. First, we define a generalized Gauss map for a hypersurface Mn−1 ⊆ N, where N is a symmetric space of dimensionExpand
Constant mean curvature surfaces in 3-dimensional Thurston geometries
This is a survey on the global theory of constant mean curvature surfaces in Riemannian homogeneous 3-manifolds. These ambient 3-manifolds include the eight canonical Thurston 3-dimensionalExpand
Harmonic maps and constant mean curvature surfaces in H 2 x R
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
The Bonnet problem for surfaces in homogeneous $3$-manifolds
We solve the Bonnet problem for surfaces in the homogeneous 3-manifolds with a 4-dimensional isometry group. More specifically, we show that a simply connected real analytic surface in H2×R or S2×RExpand
PARABOLIC AND HYPERBOLIC SCREW MOTION SURFACES IN ℍ2×ℝ
  • R. Earp
  • Mathematics
  • Journal of the Australian Mathematical Society
  • 2008
Abstract In this paper we find many families in the product space ℍ2×ℝ of complete embedded, simply connected, minimal and surfaces with constant mean curvature H such that |H|≤1/2. We study completeExpand
Harmonic maps and constant mean curvature surfaces in $\H^2 \times \R$
We introduce a hyperbolic Gauss map into the Poincare disk for any surface in H^2xR with regular vertical projection, and prove that if the surface has constant mean curvature H=1/2, this hyperbolicExpand
Compact stable constant mean curvature surfaces in the Berger spheres
In the 1-parameter family of Berger spheres S^3(a), a > 0 (S^3(1) is the round 3-sphere of radius 1) we classify the stable constant mean curvature spheres, showing that in some Berger spheres (aExpand
...
1
2
3
4
5
...

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
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
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
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
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
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
Universidad de Granada, E-18071 Granada, Spain E-mail address: isafer@ugr
  • Universidad de Granada, E-18071 Granada, Spain E-mail address: isafer@ugr