The adiabatic limit of wave map flow on a two torus

TL;DR

This paper rigorously proves that wave maps from a two-torus to a sphere, with small initial velocity, converge to geodesics in the moduli space of holomorphic maps, confirming a long-standing conjecture.

Contribution

It establishes the adiabatic limit of wave map flow on a two torus, connecting wave maps to geodesics in the moduli space of holomorphic maps.

Findings

Wave maps converge to geodesics in the moduli space as initial velocity tends to zero.

The convergence is in a precise mathematical sense.

Confirms Ward's conjecture on the adiabatic limit of wave maps.

Abstract

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.