  The two-sphere valued wave map flow on a Lorentzian domain R x Sigma, where
Sigma is any flat two-torus, is studied. The Cauchy problem with initial data
tangent to the moduli space of holomorphic maps Sigma -> S^2 is considered, in
the limit of small initial velocity. It is proved that wave maps, in this
limit, converge in a precise sense to geodesics in the moduli space of
holomorphic maps, with respect to the L^2 metric. This establishes, in a
rigorous setting, a long-standing informal conjecture of Ward.
