Minimal surfaces with symmetries

TL;DR

This paper characterizes when finite groups acting on Riemann surfaces can be realized through equivariant minimal immersions into Euclidean spaces, establishing existence conditions and constructing examples for various group actions.

Contribution

It provides necessary and sufficient conditions for the existence of G-equivariant conformal minimal immersions, including explicit constructions for all finite groups and extensions to infinite groups.

Findings

Existence of G-equivariant minimal immersions under fixed point free actions

Construction of minimal surfaces for all finite groups with n=2|G|

Extension of results to surfaces with finite total Gaussian curvature

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.