Horizontal Delaunay surfaces with constant mean curvature in product spaces

In this talk, we will describe the 1-parameter family of horizontal Delaunay surfaces in $\mathbb{S}^2\times \mathbb{R}$ and $\mathbb{H}^2\times \mathbb{R}$ with supercritical constant mean curvature. These surfaces are not equivariant but singly periodic, and they lie at bounded distance from a horizontal geodesic. We will show that horizontal unduloids are properly embedded surfaces in $\mathbb{H}^2\times \mathbb{R}$. We also describe the first non-trivial examples 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. They have constant mean curvature H>12. Finally, we prove that there are no properly immersed surface with critical or subcritical constant mean curvature at bounded distance from a horizontal geodesic in $\mathbb{H}^2\times \mathbb{R}$.