In [101], Friedrich proved a result on the stability of de Sitter space.
He gives data at infinity but the same type of argument can be
applied starting from a Cauchy surface in spacetime to give an
analogous result. This concerns the Einstein vacuum equations
with positive cosmological constant and is as follows. Consider
initial data induced by de Sitter space on a regular Cauchy
hypersurface. Then all initial data (vacuum with positive
cosmological constant) near enough to these data in a suitable
(Sobolev) topology have maximal Cauchy developments that are
geodesically complete. The result gives much more detail on the
asymptotic behaviour than just this and may be thought of as
proving a form of the cosmic no hair conjecture in the vacuum
case. (This conjecture says roughly that the de Sitter solution
is an attractor for expanding cosmological models with positive
cosmological constant.) This result is proved using conformal
techniques and, in particular, the regular conformal field
equations developed by Friedrich.
There are results obtained using the regular conformal field
equations for negative or vanishing cosmological constant [103,
106], but a detailed discussion of their nature would be out of
place here (cf. however Section
9.1).

Theorems on Existence and Global Dynamics for the
Einstein Equations
Alan D. Rendall
http://www.livingreviews.org/lrr20026
© MaxPlanckGesellschaft. ISSN 14338351
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