Minimal surfaces with symmetries

Forstnerič, Franc (author)

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.

• Language: English
• Keywords: Riemann surfaces, minimal surfaces, G-equivariant conformal minimal immersion
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2024
• Number of pages: 32 str.
• Numbering: Vol. 128, iss. 3, art. e12590
• PID: 20.500.12556/RUL-154999
• UDC: 517.5
• ISSN on article: 0024-6115
• DOI: 10.1112/plms.12590
• COBISS.SI-ID: 188644867
• Publication date in RUL: 13.03.2024

Record is a part of a journal

• Title: Proceedings of the London Mathematical Society
• Shortened title: Proc. Lond. Math. Soc.
• Publisher: Wiley, London Mathematical Society
• ISSN: 0024-6115
• COBISS.SI-ID: 26179584

Licences

• License: CC BY-NC-ND 4.0, Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
• Link: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Description: The most restrictive Creative Commons license. This only allows people to download and share the work for no commercial gain and for no other purposes.

Secondary language

• Language: Slovenian
• Keywords: Riemannove ploskve, minimalne ploskve, G-ekvivariantna konformna minimalna imerzija