On the cauchy problem for harmonic maps defined on two‐dimensional Minkowski space

Let R 1+1 be two‐dimensional Minkowski space and M a complete Riemannian manifold of dimension n . It is proved that the solution of the Cauchy problem for the harmonic map ϕ : R 1+1 → M exists globally. As an application to physics we conclude that the field function in a two‐dimensional chiral field theory is regular for all time, if it is regular initially.