New examples of constant mean curvature surfaces in ${\mathbb{S}}^2\times \mathbb{R}$ and ${\mathbb{H}}^2\times \mathbb{R}$

Abstract

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.

22 pages, 5 figures

References in corpus (1)

• Estimates for constant mean curvature graphs in MxR

Cited by in corpus (3)

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• Invariant constant mean curvature tubes around a horizontal geodesic in $\mathbb{E}(\kappa ,\tau )$-spaces
• Genus one minimal k-noids and saddle towers in ${\mathbb{H}}^2\times \mathbb{R}$