Minimal surfaces with symmetries

Let G$G$ be a finite group acting on a connected open Riemann surface X$X$ by holomorphic automorphisms and acting on a Euclidean space Rn$\mathbb {R}^n$ (n⩾3)$(n\geqslant 3)$ by orthogonal transformations. We identify a necessary and sufficient condition for the existence of a G$G$‐equivariant conformal minimal immersion F:X→Rn$F:X\rightarrow \mathbb {R}^n$. We show in particular that such a map F$F$ always exists if G$G$ acts without fixed points on X$X$. Furthermore, every finite group G$G$ arises in this way for some open Riemann surface and n=2|G|$n=2|G|$. We obtain an analogous result for minimal surfaces having complete ends with finite total Gaussian curvature, and for discrete groups acting on X$X$ properly discontinuously and acting on Rn$\mathbb {R}^n$ by rigid transformations.