CMC surfaces of revolution, Elliptic curves, Weierstrass-$\wp$ functions, and Algebraicity
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913408841940992 |
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| author | Dey, Rukmini Paul, Anantadulal Singh, Rahul Kumar |
| author_facet | Dey, Rukmini Paul, Anantadulal Singh, Rahul Kumar |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_19742 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | CMC surfaces of revolution, Elliptic curves, Weierstrass-$\wp$ functions, and Algebraicity Dey, Rukmini Paul, Anantadulal Singh, Rahul Kumar Differential Geometry Number Theory 53A10, 11G07 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. |
| title | CMC surfaces of revolution, Elliptic curves, Weierstrass-$\wp$ functions, and Algebraicity |
| topic | Differential Geometry Number Theory 53A10, 11G07 |
| url | https://arxiv.org/abs/2405.19742 |