In  Friedrich proved a result on the stability of de Sitter space.
This concerns the Einstein vacuum equations with positive
cosmological constant. His result 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 which are geodesically
complete. In fact 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 [42,
39] but a detailed discussion of their nature would be out of place
here. (Cf. however Section (5.2).)
||Local and global existence theorems for the Einstein
Alan D. Rendall
© Max-Planck-Gesellschaft. ISSN 1433-8351