Decay Estimates (Theorems M4, M5)

This chapter evaluates the proof for Theorems M4 and M5. It relies on the decay of q, α‎ and α‎ to prove the decay estimates for all the other quantities. More precisely, the chapter relies on the results of Theorems M1, M2, and M3 to prove Theorems M4 and M5. The detailed proof of Theorem M4 provides the main decay estimates in (ext)M. The proof depends in a fundamental way on the geometric properties of the GCM hypersurface Σ‎∗, the spacelike future boundary of (ext)M introduced in section 3.1.2. The chapter then reformulates the main bootstrap assumptions in the form needed in the proof of Theorem M4.