Minimal surfaces with symmetries

Abstract

Let $G$ be a finite group acting on a connected open Riemann surface $X$ by holomorphic automorphisms and acting on a Euclidean space ${\mathbb{R}}^n$ $(n\ge 3)$ by orthogonal transformations. We identify a necessary and sufficient condition for the existence of a $G$-equivariant conformal minimal immersion $F:X\to {\mathbb{R}}^n$. 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 ${\mathbb{R}}^n$ by rigid transformations.

arXiv commentTo appear in Proc. London Math. Soc

References in corpus (5)

• Null curves and directed immersions of open Riemann surfaces
• Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves
• The Calabi-Yau property of superminimal surfaces in self-dual Einstein four-manifolds
• Approximation theory for non-orientable minimal surfaces and applications
• Holomorphic Legendrian curves in ${\mathbb{CP}}^3$ and superminimal surfaces in ${\mathbb{S}}^4$