The adiabatic limit of wave-map flow on the two-torus

The S2 valued wave map flow on a Lorentzian domain R × Σ, where Σ is any flat two-torus, is studied. The Cauchy problem with initial data tangent to the moduli space of holomorphic maps Σ → S2 is onsidered, 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 L2 metric. This establishes, in a rigorous setting, a long-standing informal conjecture of Ward.