Horizontal Delaunay surfaces with constant mean curvature in product spaces

For each $0< H \leq 1/2$ and for each integer $k \geq 2$, we show the existence of a $2$-parameter family of properly Alexandrov-embedded surfaces with constant mean curvature $H$ in $\mathbb{H}^2\times \mathbb{R}$ which are symmetric with respect to a horizontal slice and a $k$ vertical planes disposed symmetrically. These surfaces are obtained from solutions to Jenkins-Serrin problems in $\tilde{SL}_2(\mathbb{R})$ or $\mathrm{Nil}_3$ via the Daniel sister correspondence. We recover Plehnert’s $(H,k)$-noids and generalize Rodríguez and Morabito’s minimal saddle towers. We also obtain new complete examples that we call $(H,k)$-nodoids, whose $k$ ends are asymptotic to vertical cylinders from the convex side. These $(H,k)$-nodoids show that the so-called Krust property for minimal graphs obtained by Hauswirth, Sa Earp and Toubiana, does not extend to the case of positive constant mean curvature, i.e., there are minimal graphs over convex domains of $\tilde{SL}_2(\mathbb{R})$ and $\mathrm{Nil}_3$ whose conjugate surfaces are not embedded.