Gluing Small Black Holes into Initial Data Sets

Abstract We prove a strong localized gluing result for the general relativistic constraint equations (with or without cosmological constant) in n 3 n ≥ 3 spatial dimensions. We glue an ϵ -rescaling of an asymptotically flat data set (, k) (γ ^, k ^) into the neighborhood of a point p X p ∈ X inside of another initial data set (X, , k) (X, γ, k), under a local genericity condition (non-existence of KIDs) near p p. As the scaling parameter ϵ tends to 0, the rescalings x x ϵ of normal coordinates x on X around p p become asymptotically flat coordinates on the asymptotically flat data set; outside of any neighborhood of p p on the other hand, the glued initial data converge back to (, k) (γ, k). The initial data we construct enjoy polyhomogeneous regularity jointly in ϵ and the (rescaled) spatial coordinates. Applying our construction to unit mass black hole data sets (X, , k) (X, γ, k) and appropriate boosted Kerr initial data sets (, k) (γ ^, k ^) produces initial data which conjecturally evolve into the extreme mass ratio inspiral of a unit mass and a mass ϵ black hole. The proof combines a variant of the gluing method introduced by Corvino and Schoen with geometric singular analysis techniques originating in Melrose’s work. On a technical level, we present a fully geometric microlocal treatment of the solvability theory for the linearized constraints map.