Harmonic maps of spheres and the Hopf construction
Given a bi-eigenmap F: \(S^ p\times S^ q\to S^ r\) with eigenvalues \(\lambda_ 1\) and \(\lambda_ 2\) (i.e. F is harmonic in each variable separately with (constant) energy density \(\lambda_ 1/2\) in the first variable and \(\lambda_ 2/2\) in the second), the Hopf construction associates to F a map H(F): \(S^{p+q+1}\to S^{r+1}\). The main result of the paper is that H(F) can be harmonically represented provided that the damping conditions \((p-1)^ 2>4\lambda_ 1\) and \((q-1)^ 2>4\lambda_ 2\) or \(p=q\) and \(\lambda_ 1=\lambda_ 2\) hold. Taking F to be an orthogonal multiplication yields many interesting corollaries, e.g., for \(p\geq 6\), every element of the stable group \(\pi_ p\) in the image of the stable J-homomorphism can be harmonically represented. The author also shows that the damping conditions can be removed provided that one allows to choose a suitable Riemannian metric on the domain. Using bi-eigenmaps provides many new harmonic maps between spheres that were not attainable by the join construction developed previously. (Note that a new classification of bi-eigenmaps can be found in the reviewer's book: Harmonic maps and minimal immersions through representation theory (Academic Press, 1990).)
scientific article; zbMATH DE number 4133116