  We provide a significant extension of the Hyperboloidal Foliation Method
introduced by the authors in 2014 in order to establish global existence
results for systems of quasilinear wave equations posed on a curved space, when
wave equations and Klein-Gordon equations are coupled. This method is based on
a (3+1) foliation (of the interior of a future light cone in Minkowski
spacetime) by spacelike and asymptotically hyperboloidal hypersurfaces. In the
new formulation of the method, we succeed to cover wave-Klein-Gordon systems
containing "strong interaction" terms at the level of the metric, and then
generalize our method in order to establish a new existence theory for the
Einstein equations of general relativity. Following pioneering work by Lindblad
and Rodnianski on the Einstein equations in wave coordinates, we establish the
nonlinear stability of Minkowski spacetime for self-gravitating massive scalar
fields.
