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 $n\geq 3$ spatial dimensions. We glue an $\epsilon$-rescaling of an asymptotically flat data set $(\hat\gamma,\hat k)$ into the neighborhood of a point $\mathfrak{p}\in X$ inside of another initial data set $(X,\gamma,k)$, under a local genericity condition (non-existence of KIDs) near $\mathfrak{p}$. As the scaling parameter $\epsilon$ tends to $0$, the rescalings $\frac{x}{\epsilon}$ of normal coordinates $x$ on $X$ around $\mathfrak{p}$ become asymptotically flat coordinates on the asymptotically flat data set; outside of any neighborhood of $\mathfrak{p}$ on the other hand, the glued initial data converge back to $(\gamma,k)$. The initial data we construct enjoy polyhomogeneous regularity jointly in $\epsilon$ and the (rescaled) spatial coordinates. Applying our construction to unit mass black hole data sets $(X,\gamma,k)$ and appropriate boosted Kerr initial data sets $(\hat\gamma,\hat k)$ produces initial data which conjecturally evolve into the extreme mass ratio inspiral of a unit mass and a mass $\epsilon$ 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.