Minimal surfaces with symmetries

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