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arXiv · 2303.17445

Analytic saddle spheres in $\mathbb{S}^3$ are equatorial

Jose A. Galvez·Pablo Mira·Marcos P. Tassi
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Abstract

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.

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Jose A. Galvez, Pablo Mira, Marcos P. Tassi. 2023-03-30. Analytic saddle spheres in $\mathbb{S}^3$ are equatorial. https://arxiv.org/abs/2303.17445

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