On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications

TL;DR

This paper investigates the surjectivity of the general relativistic constraint map in weighted Sobolev spaces, leading to new results on spacetime constructions, black hole initial data, and gluing techniques.

Contribution

It extends analysis of the constraint map's properties, enabling new perturbation, gluing, and extension results in general relativity.

Findings

Existence of singularity-free, vacuum spacetimes stationary near infinity.

Construction of initial data for black holes matching Kerr or Schwarzschild solutions.

Development of localized gluing techniques for initial data modifications.

Abstract

Generalising an analysis of Corvino and Schoen, we study surjectivity properties of the constraint map in general relativity in a large class of weighted Sobolev spaces. As a corollary we prove several perturbation, gluing, and extension results: we show existence of non-trivial, singularity-free, vacuum space-times which are stationary in a neighborhood of $i^0$; for small perturbations of parity-covariant initial data sufficiently close to those for Minkowski space-time this leads to space-times with a smooth global Scri; we prove existence of initial data for many black holes which are exactly Kerr -- or exactly Schwarzschild -- both near infinity and near each of the connected components of the apparent horizon; under appropriate conditions we obtain existence of vacuum extensions of vacuum initial data across compact boundaries; we show that for generic metrics the deformations in the Isenberg-Mazzeo-Pollack gluings can be localised, so that the initial data on the connected sum manifold coincide with the original ones except for a small neighborhood of the gluing region; we prove existence of asymptotically flat solutions which are static or stationary up to $r^{-m}$ terms, for any fixed $m$, and with multipole moments freely prescribable within certain ranges.