Global Regularity of Wave Maps Vii. Control of Delocalised or Dispersed Solutions

This is the final paper in the series [18], [19], [20], [21] that establishes global regularity for two-dimensional wave maps into hyperbolic targets. In this paper we establish the remaining claims required for this statement, namely a divisible perturbation theory, and a means of synthesising solutions for frequencydelocalised, spatially-dispersed, or spatially-delocalised data out of solutions of strictly smaller energy. As a consequence of the perturbation theory here and the results obtained earlier in the series, we also establish spacetime bounds and scattering properties of wave maps into hyperbolic space. Note: This is a first draft only, and is NOT publication quality! I hope to reorganise this series of papers in the future, hopefully with a somewhat more simplified and streamlined approach to the subject.