Holomorphicity of stable minimal surfaces of low genus

Abstract

We prove that a (branched) minimal immersion from $\mathbb{C}$ to $\mathbb{R}^n$ is stable if and only if it lives in an even dimensional affine subspace and is holomorphic for some orthogonal complex structure on the subspace. More generally, we prove that the same result holds for a class of genus $0$ surfaces that can have infinite total curvature. This contributes to an inquiry initiated by Micallef, who previously proved the equivalence in genus $0$ assuming completeness and finite total curvature. As a corollary, we prove a holomorphicity result for covering stable minimal surfaces of genus $0$ and $1$, recovering a theorem of Fraser and Schoen as a particular case. Our approach is new, based on a method of constructing variations developed by the first named author and Markovi\'c. For unstable surfaces, we get explicit destabilizations and destabilization radii that can be read from the Weierstrass-Enneper data.