49 research outputs found
Recent results in mathematical relativity
We review selected recent results concerning the global structure of
solutions of the vacuum Einstein equations. The topics covered include
quasi-local mass, strong cosmic censorship, non-linear stability, new
constructions of solutions of the constraint equations, improved understanding
of asymptotic properties of the solutions, existence of solutions with low
regularity, and construction of initial data with trapped surfaces or apparent
horizons.
This is an expanded version of a plenary lecture, sponsored by Classical and
Quantum Gravity, held at the GR17 conference in Dublin in July 2004.Comment: 16 pages, to appear in the proceedings of GRG17, Dubli
On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications
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.Comment: latex2e, now 87 pages, several style files; various typos corrected,
treatment of weighted Hoelder spaces improved, to appear in Memoires de la
Societe Mathematique de Franc
The mass of asymptotically hyperbolic Riemannian manifolds
We present a set of global invariants, called "mass integrals", which can be
defined for a large class of asymptotically hyperbolic Riemannian manifolds.
When the "boundary at infinity" has spherical topology one single invariant is
obtained, called the mass; we show positivity thereof. We apply the definition
to conformally compactifiable manifolds, and show that the mass is
completion-independent. We also prove the result, closely related to the
problem at hand, that conformal completions of conformally compactifiable
manifolds are unique.Comment: 27 pages, Latex2e with several style files; various misprints
corrected, positivity theorem for black holes considerably strengthened, to
appear in Pacific Jour. of Mathematic