Attaching handles to Delaunay nodo\"{\i}ds

TL;DR

This paper constructs a family of complete, constant mean curvature surfaces in three-dimensional space with specific symmetry and asymptotic properties, expanding the understanding of Delaunay nodo"{}ds.

Contribution

It introduces a new family of genus m CMC surfaces with Delaunay ends, invariant under symmetries of a regular polygon, for all natural numbers m.

Findings

Existence of a one-parameter family of such surfaces for each genus m

Surfaces have two Delaunay ends asymptotic to nodo"{}dal ends

Surfaces are invariant under symmetries of a regular polygon

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\"{\i}dal 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.