Attaching handles to Delaunay nodo\"ıds

Abstract

For all $m\in \mathbb{N}-\{0\}$, we prove the existence of a one dimensional family of genus $m$, constant mean curvature (equal to 1) surfaces which are complete, immersed in ${\mathbb{R}}^3$ and have two Delaunay ends asymptotic to nodo\"ıdal ends. Moreover, these surfaces are invariant under the group of isometries of ${\mathbb{R}}^3$ leaving a horizontal regular polygon with $m+1$ sides fixed.