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

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.