Solutions of the constraints with controlled decay to Kerr, including Schwartz decay

TL;DR

This paper constructs solutions to Einstein's constraint equations near Minkowski spacetime with controlled decay, incorporating Kerr black hole data and Schwartz decay, enabling regular conformal compactification of the resulting spacetime.

Contribution

It introduces a method to correct linearized solutions with small Kerr data and Schwartz decay to full solutions, using a new inverse operator with optimal decay properties.

Findings

Solutions admit regular conformal compactification

Construction uses a right inverse with optimal decay properties

Constraint equations derived via homotopy transfer theorem

Abstract

We show that to every small and decaying solution of the linearized constraint equations about Minkowski spacetime, one can add a quadratically small correction to obtain a solution of the full constraint equations. Near spacelike infinity, the correction is given by Kerr black hole initial data, up to a term that decays faster than the linearized solution, and that has Schwartz decay if the linearized solution has Schwartz decay. Using a recent result, we obtain that the solutions of the Einstein equations with these initial data admit a regular conformal compactification along null and timelike infinity. The construction is based on a right inverse (up to necessary integrability conditions) for the linearized constraint operator about Minkowski initial data obtained previously, that has optimal mapping properties relative to weighted b-Sobolev spaces, where the weights measure decay towards infinity. On an algebraic level, we show that the constraint equations can be derived using the homotopy transfer theorem, rather than using the geometric Gauss and Codazzi equations.