Constant Mean Curvature surfaces with prescribed finite topologies

TL;DR

This paper presents a method to construct complete embedded constant mean curvature surfaces in three-dimensional space with arbitrary genus and at least four ends, by resolving tangencies between spheres.

Contribution

It introduces a new construction technique for constant mean curvature surfaces with prescribed finite topologies, extending previous methods to more general configurations.

Findings

Successfully constructs surfaces with arbitrary genus and multiple ends.

Provides a new approach based on resolving tangencies between spheres.

Builds on previous work with catenoidal necks for surface construction.

Abstract

In this article, we construct complete embedded constant mean curvature surfaces in $\mb{R}^3$ with freely prescribed genus and any number of ends greater than or equal to four. Heuristically, the surfaces are obtained by resolving finitely many points of tangency between collections of spheres. The construction relies a family of constant mean curvature surfaces constructed in \cite{Kleene}, constructed as graphs over catenoidal necks of small scale.