CMC surfaces of revolution, Elliptic curves, Weierstrass-$\wp$ functions, and Algebraicity

TL;DR

This paper explores the relationship between constant mean curvature (CMC) surfaces of revolution and elliptic curves, revealing algebraic properties and classifying such surfaces in Euclidean and Lorentzian spaces.

Contribution

It establishes a novel connection between CMC surfaces of revolution and elliptic curves, leading to classification results for algebraic CMC surfaces in different geometries.

Findings

Spacelike cylinders and hyperboloids are the only algebraic spacelike CMC surfaces of revolution in _1^3.

In _1^3, only spheres and right circular cylinders are algebraic CMC surfaces of revolution.

A similar elliptic curve connection exists for CMC surfaces of revolution in E^3.

Abstract

This paper establishes an interesting connection between the family of CMC surfaces of revolution in $\mathbb E_1^3$ and some specific families of elliptic curves. As a consequence of this connection, we show in the class of spacelike CMC surfaces of revolution in the $\mathbb E_1^3$, only spacelike cylinders and standard hyperboloids are algebraic. We also show that a similar connection exists between CMC surfaces of revolution in $\mathbb E^3$ and elliptic curves. Further, we use this to reestablish the fact that the only CMC algebraic surfaces of revolution in $\mathbb E^3$ are spheres and right circular cylinders.