Classification of quadratic harmonic maps of \(S^7\) into \(S^7\)
type: Item
instance of: scholarly article
title: Classification of quadratic harmonic maps of \(S^7\) into \(S^7\)
zbMATH Open document ID: 1325.58008
author: Xinnuan Zhao
author: Fa En Wu
author: Yue-Shan Xiong
published in: The Journal of Geometric Analysis
publication date: 2015-08-28T00:00:00Z
review text: Harmonic polynomial maps between spheres is one of interesting cases in studying harmonic maps between Riemannian manifolds. Recall that a harmonic polynomial map between spheres is a harmonic map between spheres whose components are (restriction of) homogeneous harmonic polynomial of degree \(k\). These maps are of constant energy density, and the components are just spherical harmonics of order \(k\) with eigenvalue \(\lambda_k = k(k+m-1)\) of spherical Laplacian on \(S^m\). In view of this, such maps are also called \(\lambda_k\)-eigenmaps. In particular, a quadratic harmonic polynomial map between Euclidean spheres is called a \(\lambda_2\)-eigenmap as well. In this paper, the authors give a complete classification of full quadratic harmonic maps of the seven sphere \(S^7\) into itself. A map between spheres is said to be full if the image of the map is not contained in any proper greater sphere of the range sphere. It is well known that the gradient map of a Cartan cubic isoparametric polynomial gives a full \(\lambda_2\)-eigenmap \(h: S^{3l+1} \to S^{3l+1}\) for \(l=0,1,2,4\) and \(8\). For the case \(l=2\), the authors prove that any full quadratic harmonic map of \(S^7\) into itself is equivalent to either the gradient map of the cubic isoparametric polynomial of Cartan or a map depending only on one parameter. They also provide the eight components of the map in the latter case concretely. Two full harmonic maps between spheres are called equivalent if they differ only by orthogonal transformations on the domain and the range. The main result in this paper shows that the rigidity property is not true for \(l=2\) in above. If a certain class of harmonic maps forms a single equivalence class, then the respective maps are called rigid. In [Bull. Lond. Math. Soc. 35, No. 3, 344--354 (2003; Zbl 1033.58014)] \textit{H. He} et al. proved a rigidity result that any non-constant \(\lambda_2\)-eigenmap between \(S^4\) is equivalent to the gradient map of the cubic is…
reviewed by: Gabjin Yun
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 53C43
zbMATH DE Number: 6476838
zbMATH Keywords: eigenmap
zbMATH Keywords: harmonic map
zbMATH Keywords: quadratic harmonic map
zbMATH Keywords: orthogonal multiplication
zbMATH Keywords: moduli space
zbMATH Keywords: Euclidean sphere
zbMATH Keywords: classification
MaRDI profile type: Publication
full work available at URL: s12220-014-9501-6
OpenAlex ID: W150672020
cites work: On Minimal Immersions of S 2 Into S 2m
cites work: Minimal immersions of surfaces in Euclidean spheres
cites work: Symmetric harmonic maps between spheres
cites work: Q3662316
cites work: Q4848739
cites work: Harmonic Maps and Minimal Immersions with Symmetries (AM-130)
cites work: Harmonic Mappings of Riemannian Manifolds
cites work: Harmonic maps from surfaces to complex projective spaces
cites work: Constructions of harmonic polynomial maps between spheres
cites work: Normed bilinear pairings for semi-Euclidean spaces near the Hurwitz-Radon range
cites work: A note on free differentiable actions of \(S^ 1\) and \(S^ 3\) on homotopy spheres
cites work: ON EIGENMAPS BETWEEN SPHERES
cites work: On the first Pontrjagin class of homotopy complex projective spaces
cites work: Characteristic Classes. (AM-76)
cites work: Orthogonal Multiplications in small Dimensions
cites work: Harmonic maps from spheres to spheres
cites work: Q5749997
cites work: The existence of minimal immersions of 2-spheres
cites work: Compositions of quadratic forms
cites work: Harmonic Mappings of Spheres
cites work: Q3764913
cites work: Q3996395
cites work: Mappings of moduli spaces for harmonic eigenmaps and minimal immersions between spheres
cites work: Infinitesimal rotations of isometric minimal immersions between spheres
cites work: Finite Möbius groups, minimal immersions of spheres, and moduli
cites work: NEW CONSTRUCTIONS OF EIGENMAPS BETWEEN SPHERES
cites work: Polynomial maps from spheres to spheres
cites work: Non-existence of quadratic harmonic maps of $S^{4}$ into $S^{5}$ or $S^{6}$
cites work: Quadratic forms between spheres and the non-existence of sums of squares formulae
DOI: 10.1007/S12220-014-9501-6
description: scientific article; zbMATH DE number 6476838