Algebra Seminar

Harmonic maps from a Riemann surface to a compact Lie group or symmetric space are of both physical and geometric interest. The harmonic map equations are then a reduction of the self-dual Yang-Mills equations, and thus physicists study harmonic maps of R² and R^1,1. A surface has constant mean curvature precisely when its Gauss map is harmonic, and Willmore surfaces and surfaces of constant negative Gauss curvature also possess characterisations in terms of such harmonic maps. We focus on a simple case of particular interest, harmonic maps f from a 2-torus (with any conformal structure tau) to the 3-sphere (with standard metric). In 1990 Hitchin showed that the data (f,tau) is in one-to-one correspondence with certain algebro-geometric data, namely a hyperelliptic curve (called the spectral curve), a pair of meromorphic differentials on this curve, and a line bundle, all satisfying certain conditions. He proved that for g\leq 3, there are curves of genus g that support the required data, and hence describe harmonic maps f:(T², tau)-> S³. One is particularly interested in conformal harmonic maps as their images are minimal surfaces. We show that for each g>= 0, there is a conformal harmonic map f:(T², tau)-> S³ whose spectral curve has genus g. We also show that one obtains a 1-dimensional family of non-conformal harmonic maps for each g.

Please note that this seminar will be held in Carslaw 709.