On the Size of Minimal Surfaces in ${\mathbb{R}}^4$

Abstract

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

19 pages, 3 figures