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arXiv · 2405.19742

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

Rukmini Dey·Anantadulal Paul·Rahul Kumar Singh
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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.

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Rukmini Dey, Anantadulal Paul, Rahul Kumar Singh. 2024-05-30. CMC surfaces of revolution, Elliptic curves, Weierstrass-$\wp$ functions, and Algebraicity. https://arxiv.org/abs/2405.19742

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