On the size of stable 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 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 the question of Barbosa and Do Carmo (Math Z 173:13–28, 1980) in \({\mathbb {R}}^4\).