Classification of harmonic mappings of constant energy density into spheres
type: Item
instance of: scholarly article
title: Classification of harmonic mappings of constant energy density into spheres
zbMATH Open document ID: 0731.58019
DOI: 10.1007/BF00147415
author: Joon-Sik Park
author: Hajime Urakawa
published in: Geometriae Dedicata
publication date: 1991-01-01T00:00:00Z
review text: Eigenmaps, i.e. harmonic maps of constant energy density, of a flat torus \({\mathbb{R}}^ m/\Lambda\), \(\Lambda\) is a lattice, into spheres can be parametrized by a compact convex body in a finite dimensional vector space in much the same way as in the work of Do Carmo and Wallach for minimal immersions of spheres and as extended by Li and the reviewer for irreducible homogeneous domains and eigenmaps, respectively. The main contribution here is the determination of the dimension of the parameter space in terms of the dual lattice of \(\Lambda\). This is then used to describe the eigenmaps and minimal immersions of the flat 2-torus into spheres very explicitly. In particular, unicity of Clifford tori in \(S^ 3\) due to Lawson and Sasaki is recovered. Further examples are given corresponding to \(m\geq 3\) and for the first two eigenvalues of \({\mathbb{R}}^ m/\Lambda\).
reviewed by: Gábor Tóth
Mathematics Subject Classification ID: 58E20
zbMATH DE Number: 4209127
zbMATH Keywords: harmonic maps of constant energy density
zbMATH Keywords: torus
zbMATH Keywords: eigenmaps
zbMATH Keywords: minimal immersions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00147415
OpenAlex ID: W2058615783
description: scientific article; zbMATH DE number 4209127