We present a local gluing construction for general relativistic initial data sets. The method applies togenericinitial data, in a sense which is made precise. In particular the trace of the extrinsic curvature is not assumed to be constant near the gluing points, which was the case for previous such constructions. No global conditions on the initial data sets such as compactness, completeness, or asymptotic conditions are imposed. As an application, we prove existence of spatially compact, maximal globally hyperbolic, vacuum space-times without any closed constant mean curvature spacelike hypersurface.

We establish an optimal gluing construction for general relativistic initial dats sets. The construction is optimal in two distinct ways. First, it applies togenericinitial data sets and the required (generically satisfied) hypotheses are geometrically and physically natural. Secondly, the construction is completelylocalin the sense that the initial data is left unaltered on the complement of arbitrarily small neighborhoods of the points about which the gluing takes place. Using this construction we establish the existence of cosmological, maximal globally hyperbolic, vacuum space-times with no constant mean curvature spacelike Cauchy surfaces.

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 space-times which arestationaryin a neighborhood of $i^0$; for small perturbations ofparity-covariantinitial 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, with arbitrary $m$, and with multipole moments freely prescribable within certain ranges.

We review recent results by the author, in collaboration with Erwann Delay, Olivier Lengard, and Rafe Mazzeo (the papers below or here), on existence and properties of space-times with controlled asymptotic behavior at null infinity.

**Authors:** Piotr
T. Chruściel, Erwann
Delay

**Comments:** gr-qc/0203053, 11
pages, Latex2e, various style files

We construct non-trivial vacuum space-times with a global Scri. The construction proceeds by proving extension results across compact boundaries for initial data sets, adapting the gluing arguments of Corvino and Schoen. Another application of the extension results is existence of initial data which are exactly Schwarzschild both near infinity and near each of the connected component of the apparent horizon.

Version published in Classical and Quantum Gravity
**19**, L71-L79 (2002) [
from the CQG web site] [local file]

[Erratum to the CQG published version]

We construct manifold structures on various sets of solutions of the general relativistic initial data sets.

**Authors:** Robert Beig, Piotr
T. Chruściel and Richard Schoen

**Comments:**[gr-qc/0403042], 37 pages

We prove that generic solutions of the vacuum constraint Einstein equations do not possess any global or local space-time Killing vectors, on an asymptotically flat Cauchy surface, or on a compact Cauchy surface with mean curvature close to a constant, or for CMC asymptotically hyperbolic initial data sets. More generally, we show that non-existence of global symmetries implies, generically, non-existence of local ones. As part of the argument, we prove that generic metrics do not possess any local or global conformal Killing vectors.

**Authors:** Piotr
T. Chruściel, Rafe
Mazzeo

**Comments:**[gr-qc/0210103], 29
pages, Latex2e, various style files, several figures

We analyze the horizon structure of families of space times obtained by evolving initial data sets containing apparent horizons with several connected components. We show that under certain smallness conditions the outermost apparent horizons will also have several connected components. We further show that, again under a smallness condition, the maximal globally hyperbolic development of the many black hole initial data constructed in the letter with Erwann Delay above, or of glued hyperboloidal data of Isenberg, Mazzeo and Pollack, will have an event horizon, the intersection of which with the initial data hypersurface is not connected. This justifies the "many black hole" character of those space-times.

Here you can watch my talk in Cargèse about this work.