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



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

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


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 $\mathbb{H}^2 \times \mathbb{R}$.


constant mean curvature surfaces, product spaces, conjugate constructions

2010 Mathematics Subject Classification

Primary 53A10. Secondary 53C30.

The authors are supported by the project PID2019.111531GA.I00 and by the IMAG–María de Maeztu grant CEX2020-001105-M, both of them funded by MCIN/AEI/10.13039/501100011033.

The first-named author is also supported by the Ramón y Cajal programme of MCIN/AEI and by a FEDER-UJA project (ref. 1380860).

The second-named author is also supported by the Programa Operativo FEDER Andalucía 2014-2020, grant no. E-FQM-309-UGR18.

Received 5 November 2020

Published 22 July 2022