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

TL;DR

This paper establishes geometric criteria for the stability of minimal surfaces in four-dimensional space based on the area of their Gauss map, answering a question posed by Barbosa and Do Carmo.

Contribution

It provides new stability criteria for minimal surfaces in $\

Findings

If the spherical area of the Gauss map is less than 2π, the surface is stable under boundary-fixing deformations.

The paper answers a specific question of Barbosa and Do Carmo regarding stability in $\

It links the stability of minimal surfaces to the geometric properties of their Gauss map in $\

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 $\mathbb{R}^4$.