Gluing small black holes along timelike geodesics III: construction of true solutions and extreme mass ratio mergers

Given a smooth globally hyperbolic $(3+1)$-dimensional spacetime $(M,g)$ satisfying the Einstein vacuum equations (possibly with cosmological constant) and an inextendible timelike geodesic $\mathcal{C}$, we construct, on any compact subset of $M$, solutions $g_\epsilon$ of the Einstein equations which describe a mass $\epsilon$ Kerr black hole traveling along $\mathcal{C}$. More precisely, away from $\mathcal{C}$ one has $g_\epsilon\to g$ as $\epsilon\to 0$, while the $\epsilon^{-1}$-rescaling of $g_\epsilon$ around every point of $\mathcal{C}$ tends to a fixed subextremal Kerr metric. Our result applies on all spacetimes with noncompact Cauchy hypersurfaces, and also on spacetimes which do not admit nontrivial Killing vector fields in a neighborhood of a point on the geodesic. As an application, we construct spacetimes which model the merger of a very light subextremal Kerr black hole with a slowly rotating unit mass Kerr(-de Sitter) black hole, followed by the relaxation of the resulting black hole to its final Kerr(-de Sitter) state. In Part I, we constructed approximate solutions $g_{0,\epsilon}$ of the gluing problem which satisfy the Einstein equations only modulo $\mathcal{O}(\epsilon^\infty)$ errors. Part II introduces a framework for obtaining uniform control of solutions of linear wave equations on $\epsilon$-independent precompact subsets of the original spacetime $(M,g)$. In this final part, we show how to correct $g_{0,\epsilon}$ to a true solution $g_\epsilon$ by adding a metric perturbation of size $\mathcal{O}(\epsilon^\infty)$ which solves a carefully chosen gauge-fixed version of the Einstein vacuum equations. The main novel ingredient is the proof of suitable mapping properties for the linearized gauge-fixed Einstein equations on subextremal Kerr spacetimes.