On the Size of Minimal Surfaces in ℝ⁴

2021/06/11 by Arì Aiolfi, Aiolfi, Ari, Marc Soret +3
Mathematics · Physics and Astronomy · #53A10 #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.2106.06318

openalex publication_date 2021/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Gauss map g of a surface Σ in ℝ⁴ takes its values in the Grassmannian of oriented 2-planes of ℝ⁴: G⁺(2,4). We give geometric criteria of stability for minimal surfaces in ℝ⁴ in terms of g. We show in particular that if the spherical area of the Gauss map |g(Σ)| of a minimal surface is smaller than 2π then the surface is stable by deformations which fix the boundary of the surface.This answers a question of Barbosa and Do Carmo in ℝ⁴.

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