CMC surfaces of revolution, Elliptic curves, and Weierstrass- functions

This paper establishes an interesting connection between the family of CMC surfaces of revolution in E₁³ 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 E₁³, only spacelike cylinders and standard hyperboloids are algebraic. We also show that a similar connection exists between CMC surfaces of revolution in E³ and elliptic curves. Further, we use this to reestablish the fact that the only CMC algebraic surfaces of revolution in E³ are spheres and right circular cylinders.