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

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.

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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