An immersed minimal surface in \(\mathbb{R}^n\), \(n\geq3\), is the image of a conformal harmonic immersion \(F:X\to\mathbb{R}^n\) from an open conformal surface \(X\). There has been much work on symmetries of specific minimal surfaces, but the author is interested in general existence results. Let \(X\) be a connected open Riemann surface and \(G\) be a finite group of holomorphic automorphisms of \(X\) and acting on \(\mathbb{R}^n\) by orthogonal transformations. The author identifies a necessary and sufficient condition for the existence of a \(G\)-equivariant conformal minimal immersion \(F:X\to\mathbb{R}^n\).
Let \(G_x=\{g\in G \, : \, gx=x\}\) be the stabiliser of a point \(x\in X\). The group \(G_x\) is a cyclic subgroup of \(G\), which is trivial for points in the complement of a closed discrete subset of \(X\). The author proves in Theorem 1.1 that if for every non-trivial \(G_x\) there is a 2-plane \(\Lambda_x\subset\mathbb{R}^n\) on which \(G_x\) acts effectively by rotations, then there exists a minimal immersion \(F:X\to\mathbb{R}^n\) such that \[ \forall\,x\in X,\,g\in G: \quad F(gx) = gF(x) \] and the image \(F(X)\) is not contained in any affine hyperplane of \(\mathbb{R}^n\).
The author shows (Remark 1.2) that the conditions on stabilisers in Theorem 1.1 are necessary. The proof of Theorem 1.1 gives several additions concerning approximation, interpolation, and the control of the flux. (Theorem 5.1 should be compared with the results in the non-equivariant case, e.g., Theorems 3.6.1 and 3.6.2 of [A. Alarcón et al., Minimal surfaces from a complex analytic viewpoint. Cham: Springer (2021; Zbl 1520.53001)].) In particular, the map \(F\) in Theorem 1.1 can be chosen to be the real part of a \(G\)-equivariant null holomorphic immersion \(X\to\mathbb{C}^n\) (Theorem 5.3).
If \(H\) is the normal subgroup of \(G\) consisting of all elements \(g\in G\) which act trivially on \(\mathbb{R}^n\), then \(G/H\) is a symmetry group of the minimal surface \(F(X)\) in Theorem 1.1. The author derives some corollaries and states many problems concerning, among other things, the construction of complete \(G\)-equivariant minimal surfaces of finite total curvature and finding examples of minimal immersions with given groups of symmetries.
Finally, the author also provides an analogue of Theorem 1.1 for an infinite discrete group \(G\) acting on \(\mathbb{R}^n\) by rigid transformations and acting on a Riemann surface \(X\) properly discontinuously by holomorphic automorphisms such that the quotient surface \(X/G\) is non-compact. This case is only relevant if \(X\) has genus at most one.