Constant mean curvature $k$-noids in homogeneous manifolds

Abstract

For each $k\geq2$, we construct a family of surfaces in $\Sigma(\kappa)\times\mathbb{R}$ with constant mean curvature $H\in[0,1/2]$, where $\kappa+4H^{2}\leq0$ and $\Sigma(\kappa)$ is a two-dimensional space form. The surfaces are invariant under $2\pi/k$-rotations about a vertical fiber of $\Sigma(\kappa)\times\mathbb{R}$, have genus zero, and $2k$ ends. Each surface arises as the sister surface of a minimal graph in a homogeneous $3$-manifold. The domain of the graph is non-convex. We use the sisters of a generalization of Jorge–Meeks-$k$-noids in homogeneous $3$-manifolds as barriers in the conjugate Plateau construction.