  The Hyperboloidal Foliation Method presented in this monograph is based on a
(3+1)-foliation of Minkowski spacetime by hyperboloidal hypersurfaces. It
allows us to establish global-in-time existence results for systems of
nonlinear wave equations posed on a curved spacetime and to derive uniform
energy bounds and optimal rates of decay in time. We are also able to encompass
the wave equation and the Klein-Gordon equation in a unified framework and to
establish a well-posedness theory for nonlinear wave-Klein-Gordon systems and a
large class of nonlinear interactions. The hyperboloidal foliation of Minkowski
spacetime we rely upon in this book has the advantage of being geometric in
nature and, especially, invariant under Lorentz transformations. As stated, our
theory applies to many systems arising in mathematical physics and involving a
massive scalar field, such as the Dirac-Klein-Gordon system. As it provides
uniform energy bounds and optimal rates of decay in time, our method appears to
be very robust and should extend to even more general systems.
