Harmonic maps of spheres and equivariant theory
In Chapter I we produce many new harmonic maps of spheres by the qualitative study of the pendulum equations for the join and the Hopf construction. In particular, we obtain Corollary 1.7.1. Let Φ1 : Sp -> Sr be any harmonic homogeneous polynomial of degree greater or equal than two, and let Φ2 be the identity map id : Sq -> Sq. Then the (q+1)-suspension of Φ1 is harmonically representable by an equivariant map of the form Φ1 * Φ2 if and only if q=0 ....5. Corollary 1.11.1. Let [f] E ΠSp be a stable class in the image of the stable J-ho momorphism Jp :Πp (0) -> ΠSp, p >= 6. Then there exists q > p such that [f] can be represented by a harmonic map Φ : Sp+q+1 -> Sq+1. In Chapter II we illustrate equivariant theory and study the rendering problems: in particular, we show that the restriction q=o ...5 in Corollary 1.7.1. can be removed provided that the domain is given a suitable riemannian metric; then, for istance, the groups Πn(Sn) = Z can be rendered harmonic for every n. In Chapter III we describe applications of equivariant theory to the study of Dirichlet problems and warped products; and extensions of the theory to spaces with conical singularities.