26 research outputs found
Linearization stability of the Einstein constraint equations on an asymptotically hyperbolic manifold
We study the linearization stability of the Einstein constraint equations on
an asymptotically hyperbolic manifold. In particular we prove that these
equations are linearization stable in the neighborhood of vacuum solutions for
a non-positive cosmological constant and of
Friedman--Lema\^itre--Robertson--Walker spaces in a certain range of decays. We
also prove that this result is no longer true for faster decays. The
construction of the counterexamples is based on a new construction of
TT-tensors on the Euclidean space and on positive energy theorems.Comment: 19 pages, no figur
Limit equation for vacuum Einstein constraints with a translational Killing vector field in the compact hyperbolic case
We construct solutions to the constraint equations in general relativity
using the limit equation criterion introduced by Dahl, Humbert and the first
author. We focus on solutions over compact 3-manifolds admitting a
\bS^1-symmetry group. When the quotient manifold has genus greater than 2, we
obtain strong far from CMC results.Comment: 14 page
Bifurcating solutions of the Lichnerowicz equation
We give an exhaustive description of bifurcations and of the number of
solutions of the vacuum Lichnerowicz equation with positive cosmological
constant on with -invariant seed data. The
resulting CMC slicings of Schwarzschild-de Sitter and Nariai are described.Comment: 33 pages, 25 figure
On the asymptotic behavior of Einstein manifolds with an integral bound on the Weyl curvature
In this paper we consider the geometric behavior near infinity of some
Einstein manifolds with Weyl curvature belonging to a certain
space. Namely, we show that if , , admits an essential set
and has its Weyl curvature in for some , then must be asymptotically locally hyperbolic. One interesting application of
this theorem is to show a rigidity result for the hyperbolic space under an
integral condition for the curvature.Comment: 25 page
A non-existence result for a generalization of the equations of the conformal method in general relativity
The constraint equations of general relativity can in many cases be solved by
the conformal method. We show that a slight modification of the equations of
the conformal method admits no solution for a broad range of parameters. This
suggests that the question of existence or non-existence of solutions to the
original equations is more subtle than could perhaps be expected.Comment: minor changes from previous versio
Asymptotically hyperbolic manifolds with small mass
For asymptotically hyperbolic manifolds of dimension with scalar
curvature at least equal to the conjectured positive mass theorem
states that the mass is non-negative, and vanishes only if the manifold is
isometric to hyperbolic space. In this paper we study asymptotically hyperbolic
manifolds which are also conformally hyperbolic outside a ball of fixed radius,
and for which the positive mass theorem holds. For such manifolds we show that
the conformal factor tends to one as the mass tends to zero