Analytic saddle spheres in $\mathbb{S}^3$ are equatorial
| Autor: | Galvez, Jose A., Mira, Pablo, Tassi, Marcos P. |
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| Rok vydání: | 2023 |
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| Druh dokumentu: | Working Paper |
| Popis: | A theorem by Almgren establishes that any minimal $2$-sphere immersed in $\mathbb{S}^3$ is a totally geodesic equator. In this paper we give a purely geometric extension of Almgren's result, by showing that any immersed, real analytic $2$-sphere in $\mathbb{S}^3$ that is saddle, i.e., of non-positive extrinsic curvature, must be an equator of $\mathbb{S}^3$. We remark that, contrary to Almgren's theorem, no geometric PDE is imposed on the surface. The result is not true for $C^{\infty}$ spheres. Comment: 17 pages, 9 figures |
| Databáze: | arXiv |
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