2023/03/30 by José A. Gálvez, Pablo Mira, Galvez, Jose A. +3 · 1 citation
Mathematics · Physics and Astronomy · #53A10 #53C42 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2303.17445
openalex publication_date 2023/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A theorem by Almgren establishes that any minimal 2-sphere immersed in \mathbbS3 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 \mathbbS3 that is saddle, i.e., of non-positive extrinsic curvature, must be an equator of \mathbbS3. We remark that, contrary to Almgren's theorem, no geometric PDE is imposed on the surface. The result is not true for C∞ spheres.