A characteristic property of Delaunay surfaces

TL;DR

This paper characterizes Delaunay surfaces, showing they are uniquely defined by a specific implicit equation form among constant mean curvature surfaces in Euclidean space, excluding the plane and catenoid.

Contribution

It proves a new characterization of Delaunay surfaces based on their implicit equation form, distinguishing them from other constant mean curvature surfaces.

Findings

Delaunay surfaces are uniquely characterized by the implicit form $f(x)+g(y)+h(z)=0$.

The plane and catenoid are exceptions to this characterization.

This result narrows the classification of constant mean curvature surfaces in Euclidean space.

Abstract

We prove that Delaunay surfaces, except the plane and the catenoid, are the only surfaces in Euclidean space with nonzero constant mean curvature that can be expressed as an implicit equation of type $f(x)+g(y)+h(z)=0$, where $f$, $g$ and $h$ are smooth real functions of one variable.