
School of Mathematics and Statistics
Emma Carberry
Princeton University
Harmonic Tori: an Algebraic perspective.
Friday 22nd December, 121pm,
Carslaw 709.
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 selfdual
YangMills equations, and thus physicists study harmonic maps of
R^{2} 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 2torus (with any
conformal structure tau) to the 3sphere (with standard
metric). In 1990 Hitchin showed that the data (f,tau) is
in onetoone correspondence with certain algebrogeometric 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:(T^{2}, tau)> S^{3}. 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:(T^{2}, tau)> S^{3} whose spectral
curve has genus g. We also show that one obtains a
1dimensional family of nonconformal harmonic maps for each
g.
Please note that this seminar will be held in Carslaw 709.
