72 research outputs found
Localized gluing of Riemannian metrics in interpolating their scalar curvature
We show that two smooth nearby Riemannian metrics can be glued interpolating
their scalar curvature. The resulting smooth metric is the same as the starting
ones outside the gluing region and has scalar curvature interpolating between
the original ones. One can then glue metrics while maintaining inequalities
satisfied by the scalar curvature. We also glue asymptotically Euclidean
metrics to Schwarzschild ones and the same for asymptotically Delaunay metrics,
keeping bounds on the scalar curvature, if any. This extend the Corvino gluing
near infinity to non-constant scalar curvature metrics
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
Studies of Some Curvature Operators in a Neighborhood of an Asymptotically Hyperbolic Einstein Manifold
AbstractOn an asymptotically hyperbolic Einstein manifold (M,g0) for which the Yamabe invariant of the conformal structure on the boundary at infinity is nonnegative, we show that the operators of Ricci curvature, and of Einstein curvature, are locally invertible in a neighborhood of the metric g0. We deduce in the C∞ case that the image of the Riemann–Christoffel curvature operator is a submanifold in a neighborhood of g0
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