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

• Published in: Mathematische annalen Vol. 389; no. 4; pp. 3865 - 3884
• Main Authors: Gálvez, José A., Mira, Pablo, Tassi, Marcos P.
• Format: Journal Article
• Language: English
• Published: 01-08-2024
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-023-02741-4

Abstract A theorem by Almgren establishes that any minimal 2-sphere immersed in $$\mathbb {S}^3$$ 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 $$\mathbb {S}^3$$ S 3 that is saddle, i.e., of non-positive extrinsic curvature, must be an equator of $$\mathbb {S}^3$$ S 3 . We remark that, contrary to Almgren’s theorem, no geometric PDE is imposed on the surface. The result is not true for $$C^{\infty }$$ C ∞ spheres.