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

Let R¹⁺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 ? : R¹⁺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.