Harmonic Mappings of Spheres

Introduction and statement of results. This announcement describes an elementary method of constructing harmonic maps in some cases not covered by the general existence theory. Recall that given smooth Riemannian manifolds N and M, with N compact, then the energy functional E:H(N,M) -> R is defined on a suitable manifold of maps H(iV, M ) and is given by E{ f ) = \ \n \df | . A map ƒ is said to be harmonic iff is a critical point of E; equivalently, if the tension field x(f) vanishes, where T is the Euler-Lagrange operator associated to E. The study of harmonic maps was initiated by Eells and Sampson, and the basic problem they consider is the following: given a homotopy class of maps between JV and M, is there a harmonic representative of that class? The case when M is compact is of particular interest, and under the assumption that all sectional curvatures of M are nonpositive, they succeeded in giving an affirmative answer to this question [2]. Their method was to find solutions of the heat equation T( ƒ ) — df/dt = 0 and obtain harmonic maps in the limit. More recently, the global theory of the calculus of variations has been successfully employed to recapture the existence theory [3], [4]. One may take the following as a starting point for the direct construction approach given here: Observation. Let/:R -» R be a map each of whose coordinates is given by a homogeneous harmonic polynomial of degree k. Suppose f(S) s S. Then ƒ = f/S:S -* S is harmonic. The examples of interest arise from putting two such maps together: THEOREM. Let f:R -* R and g:R-+R be homogeneous harmonic polynomial maps of degree I and k respectively which send sphere to sphere. Assume (1) k>((y/Tl)/2)(r 2), (2) I > ((VTl)/2)(p 2). Further assume r, p ^ 2. Then there is harmonic map F:S J -• S 1 which is given in Euclidean coordinates by (3) F(u, o) = ((g(«)/|«|*) cos ait), (f(v)/\v\) sin oit)) where t = log(|w|/|i;|) and a is a smooth monotone function on the reals which