Harmonic tori in spheres and complex projective spaces

Introduction A map : M ! N of Riemannian manifolds is harmonic if it extremises the energy functional: Z jdj 2 dvol on every compact subdomain of M. Harmonic maps arise in many diierent contexts in Geometry and Physics (for an overview, see 15,16]) but the setting of concern to us is the following: take M to be 2-dimensional and N to be a Riemannian symmetric space of compact type. In this case, the energy is conformally invariant so that we may take the domain to be a Riemann surface and the methods of complex analysis may be brought to bear. Moreover, the symmetric nature of the target allows us to reformulate the harmonic map equations in a gauge-theoretic way so that harmonic maps may be viewed as simple analogues of Yang{Mills elds. This paper treats harmonic maps of a 2-torus into a sphere S n or a complex projective space C P n and makes use of the ideas and methods of two separate developments in the theory of harmonic maps. The rst, and more recent, of these is the soliton-theoretic approach which has its roots in the Pinkall{Sterling classiication 23] of constant mean curvature 2-tori in R 3 (which are equivalent, via the Gauss map, to non-conformal harmonic 2-tori in S 2). Pinkall{Sterling showed that all such maps could be constructed from solutions to a family of nite-dimensional completely integrable Hamiltonian ordinary diierential equations. There followed a rapid development and extension of these ideas 5,19] which culminated in a rather general theory of harmonic maps into symmetric spaces due to Burstall{Ferus{Pedit{Pinkall 9]. This theory distinguishes special harmonic maps of R 2 into a symmetric space called harmonic maps of nite type which are constructed from commuting Hamiltonian ows on nite-dimensional subspaces of a loop algebra. Viewing maps of 2-tori as doubly periodic maps of R 2 , these authors prove: Theorem A non-conformal harmonic map of a 2-torus into a rank one symmetric space is of nite type. In particular, this result accounts for all non-conformal harmonic 2-tori in S n and C P n but excludes the conformal harmonic (i.e., branched, minimal) tori. The second development of importance to us is the well-established twistor theory of harmonic maps which goes back to Calabi's study 11,12] of minimal surfaces and, especially, minimal 2-spheres in S n. Recall 11,17] that a harmonic map of a Riemann surface into 1 …