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

TL;DR

This paper constructs compact, embedded surfaces with constant mean curvature in , exhibiting dihedral symmetry and desingularizing sphere pairs, expanding the family of known CMC surfaces in product spaces.

Contribution

It introduces a new family of compact, embedded CMC surfaces with arbitrary genus and dihedral symmetry in , using a conjugate Plateau method.

Findings

Constructed CMC surfaces with 0<H<1/2 in .

Surfaces desingularize pairs of tangent spheres with mean curvature 1/2.

Surfaces exhibit symmetries of regular tessellations.

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.