  We introduce a new method for analyzing nonlinear wave-Klein-Gordon systems
and establishing global-in-time existence results for the Cauchy problem when
the initial data need not have compact support. This method, which we call the
Euclidian-Hyperboidal Foliation Method (EHFM), relies on the construction of a
spacetime foliation obtained by glueing together asymptotically Euclidian and
asymptotically hyperboloidal hypersurfaces. Well-chosen frames of vector fields
(null-semi-hyperboloidal, Euclidian-hyperboloidal) allow us to exhibit the
structure of the equations under consideration and analyze the decay of
solutions in timelike and in spacelike directions. New Sobolev inequalities for
Euclidian-hyperboloidal foliations involving the Killing fields of Minkowski
spacetime (but not the scaling field), as well as pointwise bounds for wave and
Klein-Gordon equations on curved spacetimes are established. Our bootstrap
argument involves a hierarchy of (almost sharp) energy and pointwise bounds and
distinguishes between low- and high-order derivatives of the solutions. We
apply this method to the Einstein equations when the matter model is a massive
field and the methods by Christodolou and Klainerman and by Lindblad and
Rodnianski do not apply.
