  We consider the global evolution problem for Einstein's field equations in
the near-Minkowski regime and study the long-time dynamics of a massive scalar
field evolving under its own gravitational field. We establish the existence of
a globally hyperbolic Cauchy development associated with any initial data set
that is sufficiently close to a data set in Minkowski spacetime. In addition to
applying to massive fields, our theory allows us to cover metrics with slow
decay in space. The strategy of proof, proposed here and referred to as the
Euclidean-Hyperboloidal Foliation Method, applies, more generally, to nonlinear
systems of coupled wave and Klein-Gordon equations. It is based on a spacetime
foliation defined by merging together asymptotically Euclidean hypersurfaces
(covering spacelike infinity) and asymptotically hyperboloidal hypersurfaces
(covering timelike infinity). A transition domain (reaching null infinity)
limited by two asymptotic light cones is introduced in order to realize this
merging. On the one hand, we exhibit a boost-rotation hierarchy property (as we
call it) which is associated with Minkowski's Killing fields and is enjoyed by
commutators of curved wave operators and, on the other hand, we exhibit a
metric hierarchy property (as we call it) enjoyed by components of Einstein's
field equations in frames associated with our Euclidean-hyperboloidal
foliation. The core of the argument is, on the one hand, the derivation of
novel integral and pointwise estimates which lead us to almost sharp decay
properties (at timelike, null, and spacelike infinity) and, on the other hand,
the control of the (quasi-linear and semi-linear) coupling between the
geometric and matter parts of the Einstein equations.
