Classification of quadratic harmonic maps of \(S^7\) into \(S^7\)
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 isoparametric polynomial of É. Cartan. Thus the known results show that the map \(h\) has the rigidity property for \(l=0,1\).
scientific article; zbMATH DE number 6476838
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