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

TL;DR

This paper constructs new constant mean curvature surfaces in the product spaces  and , using conjugate Plateau methods, revealing families of complete, bounded height surfaces with symmetries and tessellation properties.

Contribution

It introduces novel constant mean curvature surfaces in  and , including unduloid-type and tessellated symmetric examples, expanding the known classes of such surfaces.

Findings

Constructed complete, bounded height CMC surfaces in  and .

Generated a 1-parameter family of unduloid-type surfaces with invariance under discrete translations.

Produced examples with tessellation symmetries in  for H=1/2.

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.