Scattering and Complete Integrability in Conformally Invariant Nonlinear Theories

We study conformally invariant nonlinear wave equations in four dimensions corresponding to multicomponent massless scalar elds with a quartic interaction. We prove that the scattering operator S on the space H of nite-Einstein-energy Cauchy data has innnitely many xed points, as well as periodic points of all orders. There are also 2 H such that S n is almost periodic but not periodic, and 2 H such that S n is not almost periodic. We also prove that H admits no conformally invariant KK ahler metrics but innnitely many distinct KK ahler metrics invariant under the Poincar e group and scale transformations. Moreover, we prove that time evolution for these nonlinear wave equations is completely integrable on the space H.