New examples of constant mean curvature surfaces in S^2xR and H^2xR

Details New examples of constant mean curvature surfaces in S^2xR and H^2xR New examples of constant mean curvature surfaces in S^2xR and H^2xR

2011-04-07

arxiv.org/abs/1104.1259v3

Mathematics

Differential Geometry

22 pages, 5 figures; 1 figure has been added and the paper has been revised

Scientific paper

We construct non-zero constant mean curvature H surfaces in the product spaces $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2\times \mathbb{R}$ by using suitable conjugate Plateau constructions. The resulting surfaces are complete, have bounded height and are invariant under a discrete group of horizontal translations. In $\mathbb{S}^2\times\mathbb{R}$ (for any H > 0) or $\mathbb{H}^2\times\mathbb{R}$ (for H > 1/2), a 1-parameter family of unduloid-type surfaces is obtained, some of which are shown to be compact in $\mathbb{S}^2\times\mathbb{R}$. Finally, in the case of H = 1/2 in $\mathbb{H}^2 \times \mathbb{R}$, the constructed examples have the symmetries of a tessellation of $\mathbb{H}^2$ by regular polygons.

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