The other result on global existence for small data is that of
Christodoulou and Klainerman on the stability of Minkowski space[28]. The formulation of the result is close to that given in
Section (4.1) but now de Sitter space is replaced by Minkowski space. Suppose
then that initial data are given which are asymptotically flat
and sufficiently close to those induced by Minkowski space on a
hyperplane. Then Christodoulou and Klainerman prove that the
maximal Cauchy development of these data is geodesically
complete. They also provide a wealth of detail on the asymptotic
behaviour of the solutions. The proof is very long and technical.
The central tool is the Bel-Robinson tensor which plays an
analogous role for the gravitational field to that played by the
energy-momentum tensor for matter fields. Apart from the book of
Christodoulou and Klainerman itself some introductory material on
geometric and analytic aspects of the proof can be found in [8] and [27] respectively.