Cambridge Journal of Mathematics

Volume 10 (2022)

Number 3

Horizontal Delaunay surfaces with constant mean curvature in $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$

Pages: 657 – 688

DOI: https://dx.doi.org/10.4310/CJM.2022.v10.n3.a2

Authors

José M. Manzano (Departamento de Matemáticas, Universidad de Jaén, Spain)

Francisco Torralbo (Departamento de Geometría y Topología, Universidad de Granada, Spain)

Abstract

We obtain a $1$-parameter family of horizontal Delaunay surfaces with positive constant mean curvature in $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$, being the mean curvature larger than $1$ $2$ in the latter case. These surfaces are not equivariant but singly periodic, and they lie at bounded distance from a horizontal geodesic. We study in detail the geometry of the whole family and show that horizontal unduloids are properly embedded in $\mathbb{H}^2 \times \mathbb{R}$. We also find (among unduloids) families of embedded constant mean curvature tori in $\mathbb{S}^2 \times \mathbb{R}$ which are continuous deformations from a stack of tangent spheres to a horizontal invariant cylinder. These are the first non-equivariant examples of embedded tori in $\mathbb{S}^2 \times \mathbb{R}$, and have constant mean curvature $H \gt \frac{1}{2}$ . Finally, we prove that there are no properly immersed surfaces with constant mean curvature $H \leq \frac{1}{2}$ at bounded distance from a horizontal geodesic in $\mathbb{H}^2 \times \mathbb{R}$.

Keywords

constant mean curvature surfaces, product spaces, conjugate constructions

2010 Mathematics Subject Classification

Primary 53A10. Secondary 53C30.

Received 5 November 2020

Published 22 July 2022