Minimal surfaces with symmetries

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