Compact embedded surfaces with constant mean curvature in ${\mathbb{S}}^2\times \mathbb{R}$

Abstract

We obtain compact orientable embedded surfaces with constant mean curvature $0<H<\frac{1}{2}$ and arbitrary genus in ${\mathbb{S}}^2\times \mathbb{R}$. These surfaces have dihedral symmetry and desingularize a pair of spheres with mean curvature $\frac{1}{2}$ tangent along an equator. This is a particular case of a conjugate Plateau construction of doubly periodic surfaces with constant mean curvature in ${\mathbb{S}}^2\times \mathbb{R}$, ${\mathbb{H}}^2\times \mathbb{R}$, and ${\mathbb{R}}^3$ with bounded height and enjoying the symmetries of certain tessellations of ${\mathbb{S}}^2$, ${\mathbb{H}}^2$, and ${\mathbb{R}}^2$ by regular polygons.

arXiv comment12 pages, 3 figures, 1 table

References in corpus (3)

• Associate and conjugate minimal immersions in MxR
• Compact stable surfaces with constant mean curvature in Killing submersions
• Estimates for constant mean curvature graphs in MxR

Cited by in corpus (3)

• Horizontal Delaunay surfaces with constant mean curvature in ${\mathbb{S}}^2\times \mathbb{R}$ and ${\mathbb{H}}^2\times \mathbb{R}$
• Conjugate Plateau constructions in product spaces
• Genus one minimal k-noids and saddle towers in ${\mathbb{H}}^2\times \mathbb{R}$