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

Piotr T. Chrusciel, Erwann Delay

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.

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

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 ; 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 terms, for any fixed , and with multipole moments freely prescribable within certain ranges.

Paper Structure

This paper contains 33 sections, 438 equations.

Table of Contents

  1. Introduction
  2. The constraints map
  3. Isomorphism theorems
  4. An asymptotic inequality
  5. Compact boundaries
  6. Conformally compactifiable initial data
  7. Asymptotically flat initial data
  8. Applications
  9. Space-times that are Kerrian near $i^0$
  10. Gluing asymptotically flat initial data sets
  11. Initial data which are stationary to high asymptotic order
  12. Space-times that are Kerrian near ${\mycal I}^{+}$
  13. Bondi-type asymptotic expansions at ${\mycal I}^{+}$.
  14. Local and global extensions of initial data sets
  15. Localised Isenberg-Mazzeo-Pollack gluings
  16. ...and 18 more sections

Theorems & Definitions (22)

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  • ...and 12 more