Constant mean curvature $k$-noids in homogeneous manifolds

Abstract

For each $k\ge 2$, we construct two families of surfaces with constant mean curvature $H$ for $H\in [0,1/2]$ in $\Sigma (\kappa )\times \mathbb{R}$ where $\kappa +4H^2\le 0$. The surfaces are invariant under $2\pi /k$-rotations about a vertical fiber of $\Sigma (\kappa )\times \mathbb{R}$, have genus zero, and a finite number of ends. The first family generalizes the notion of $k$-noids: It has $k$ ends, one horizontal and $k$ vertical symmetry planes. The second family is less symmetric and has two types of ends. Each surface arises as the conjugate (sister) surface of a minimal graph in a homogeneous 3-manifold. The domain of the graph is non-convex in the second family. For $\kappa =-1$ the surfaces with constant mean curvature $H$ arise from a minimal surface in $\widetilde{\PSL}_2(\R)$ for $H\in (0,1/2)$ and in $\Nil$ for H=1/2. For H=0, the conjugate surfaces are both minimal in a product space.