### 5.1 Stability of de Sitter space

In [141], 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. An alternative proof of this result which extends to all
higher even dimensions was given in [6]. For some comments on the case of odd dimensions
see [304].
There are results obtained using the regular conformal field equations for negative or vanishing
cosmological constant [143, 146], but a detailed discussion of their nature would be out of place here
(cf. however Section 9.1).