Title:

Harmonic mapping of spheres

This thesis is addressed to the following fundamental problem: given a homotopy class of maps between compact Riemannian manifolds N and M, is there a harmonic representative of that class? Eells and Sampson have given a general existence theorem for the case that M has no positive sectional curvatures [ES]. Otherwise, very little is known. Certainly no counterexample has ever been established. The most important contributions of this dissertation are two: firstly, we have a direct construction technique for producing some essential harmonic maps between Euclidean spheres. Topologically, this consists simply of joining two harmonic polynomial mappings (e.g., the Hopf fibrations). Analytically, however, this method has a novel physical motivation: we study the equation of motion of an exotic pendulum driven by a gravity which chances sign. If this system has an exceptional trajectory of the right sort, it defines a harmonic map of spheres. One consequence or our theorem is that πn(Sn) is represented by harmonic maps for n= 1,...,7. Finally, the rudiments of an equivariant theory of harmonic maps having been set out earlier, we find that our examples can also be put in this framework. The second significant result which arose from this study is a strong candidate for a counterexample: suppose Sn is stretched to a length b in one direction to make an ellipsoid En(b). Then if n > 3 and b is large enough, there is no harmonic stretching (of degree one) of Sn onto En(b). However, if b=1 the identity is such a harmonic map, so it certainly appears that the existence of a harmonic representative in a homotopy class can depend upon the metric. We also examine here a large collection of examples of harmonic maps of spheres which are defined by harmonic polynomials and orthogonal multiplications. The last chapter takes up the study of the Morse theory of a harmonic map: amongst several pleasing results, we have an example of a simple map whose index and degeneracy can be made arbitrarily large by equally simple changes in the metrics.
