Main Theorem

This chapter presents the main theorem, its main conclusions, as well as a full strategy of its proof, divided in nine supporting intermediate results, Theorems M0–M8. The chapter specifies the closeness to Schwarzschild of the initial data in the context of the Characteristic Cauchy problem. The conclusions of the main theorem can be immediately extended to the case where the data are specified to be close to Schwarzschild on a spacelike hypersurface Σ‎. The chapter then outlines the main bootstrap assumptions. It also provides a short description of the results concerning the General Covariant Modulation procedure, which is at the heart of the proof.

CONFORMAL SCATTERING FOR A NONLINEAR WAVE EQUATION

We establish a geometric scattering theory for a conformally invariant nonlinear wave equation on an asymptotically simple space-time. The scattering operator is defined via some trace operators at null infinity, and the proof is decomposed into three steps. A priori linear estimates are obtained via an adaptation of the Morawetz vector field to the Schwarzschild space-time and a method introduced by Hörmander for the Goursat problem. A well-posedness theorem for the characteristic Cauchy problem on a light cone at infinity is then obtained. Its proof requires a control of the nonlinearity that is uniform in time and follows from, both, an estimate of the Sobolev constant and a decay assumption on the nonlinearity of the equation. Finally, the trace operators on conformal infinity are introduced and allow us to define the conformal scattering operator of interest.