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

Abstract

The S 2 S^2 valued wave map flow on a Lorentzian domain R × Σ \mathbb {R}\times \Sigma , where Σ \Sigma is any flat two-torus, is studied. The Cauchy problem with initial data tangent to the moduli space of holomorphic maps Σ → S 2 \Sigma \rightarrow 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 L^2 metric. This establishes, in a rigorous setting, a long-standing informal conjecture of Ward.