Examples of constant mean curvature surfaces obtained from harmonic maps to the two sphere
type: Item
instance of: scholarly article
title: Examples of constant mean curvature surfaces obtained from harmonic maps to the two sphere
zbMATH Open document ID: 0882.53010
DOI: 10.1007/PL00004326
author: Manuel Ritoré
published in: Mathematische Zeitschrift
publication date: 1997-12-18T00:00:00Z
review text: In this paper families of surfaces with constant mean curvature in quotients of the Euclidean space are obtained from their Gauss maps. We start by considering a compact Riemann surface \(M\) of positive genus \(g\leq 3\) with three commuting anticonformal involutions \(u_1, u_2, u_3\). Following \textit{L. Lemaire} [J. Differ. Geom. 13, 51-78 (1978; Zbl 0388.58003)], we construct a smooth harmonic map \(\varphi: M\to \mathbb{S}^2\) minimizing the energy functional in the class of equivariant maps with respect to \(\{u_1, u_2, u_3\}\) and with respect to the symmetries around the three coordinate planes in \(\mathbb{R}^3\). From this harmonic map we obtain an unbranched constant mean curvature immersion \(\psi:M \to \mathbb{R}^3/ \Gamma\) whose Gauss map is \(\varphi\). Here \(\Gamma\) is a discrete subgroup of translations of rank \(g\) generated by orthogonal vectors. We also prove that the immersions are embeddings and exhibit a geometrical characterization of the examples.
reviewed by: Manuel Ritoré
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 1099266
zbMATH Keywords: families of constant mean curvature surfaces
zbMATH Keywords: harmonic maps
zbMATH Keywords: Gauss maps
MaRDI profile type: Publication
Recommended article: Q4397529
Recommended article: Harmonic maps and constant mean curvature surfaces in H 2 x R
Recommended article: Two examples of harmonic maps into spheres
Recommended article: New examples of constant mean curvature surfaces in \(\mathbb S^2 \times \mathbb R\) and \(\mathbb H^2 \times \mathbb R\)
Recommended article: Examples of surfaces of constant mean curvature
Recommended article: Constant mean curvature surfaces in 𝑀²×𝐑
Recommended article: Constant mean curvature hypersurfaces with two principal curvatures in a sphere
Recommended article: Constant mean curvature surface, harmonic maps, and universal Teichmüller space
Recommended article: Q2723196
Recommended article: Hypersurfaces with Constant Mean Curvature in Spheres
description: scientific article; zbMATH DE number 1099266