The L^2 geometry of spaces of harmonic maps S^2 -> S^2 and RP^2 -> RP^2

Abstract

Harmonic maps from S^2 to S^2 are all weakly conformal, and so are represented by rational maps. This paper presents a study of the L^2 metric gamma on M_n, the space of degree n harmonic maps S^2 -> S^2, or equivalently, the space of rational maps of degree n. It is proved that gamma is Kaehler with respect to a certain natural complex structure on M_n. The case n=1 is considered in detail: explicit formulae for gamma and its holomorphic sectional, Ricci and scalar curvatures are obtained, it is shown that the space has finite volume and diameter and codimension 2 boundary at infinity, and a certain class of Hamiltonian flows on M_1 is analyzed. It is proved that \tilde{M}_n, the space of absolute degree n (an odd positive integer) harmonic maps RP^2 -> RP^2, is a totally geodesic Lagrangian submanifold of M_n, and that for all n>1, \tilde{M}_n is geodesically incomplete. Possible generalizations and the relevance of these results to theoretical physics are briefly discussed.