Constant mean curvature $k$-noids in homogeneous manifolds

TL;DR

This paper constructs new families of constant mean curvature surfaces with specific symmetries in homogeneous 3-manifolds, generalizing classical $k$-noids and exploring their geometric properties.

Contribution

It introduces two novel families of constant mean curvature $k$-noids in homogeneous manifolds, extending classical minimal surface theory to new symmetric and less symmetric configurations.

Findings

Surfaces have genus zero and finitely many ends.

Surfaces are invariant under $2 extpi/k$-rotations.

Construction links to minimal surfaces in homogeneous 3-manifolds.

Abstract

For each $k\geq2$, we construct two families of surfaces with constant mean curvature $H$ for $H\in[0,1/2]$ in $\Sigma(\kappa)\times\R$ where $\kappa+4H^2\leq0$. The surfaces are invariant under $2\pi/k$-rotations about a vertical fiber of $\Sigma(\kappa)\times\R$, 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.