In this paper, we sketch the proof of the extension of the stability
theorem of the Minkowski space in General Relativity done
explicitly in [6] and [7]. We discuss solutions of the Einstein
vacuum (EV) equations (obtained in the author’s Ph.D. thesis
([6]) in 2007). We solve the Cauchy problem for more general,
asymptotically flat initial data than in the pioneering work of D.
Christodoulou and S. Klainerman [21] or than in any other work.
Moreover, we describe precisely the asymptotic behavior. Our relaxed
assumptions on the initial data yield a spacetime curvature
which is not bounded in L1(M). As a major result, we encounter
in our work borderline cases, which we discuss in this paper as
well. The fact that certain of our estimates are borderline in view
of decay indicates that the conditions in our main theorem are
sharp in so far as the assumptions on the decay at infinity on the
initial data are concerned. Thus, the borderline cases are a consequence
of our relaxed assumptions on the data, [6, 7]. They are
not present in the other works, as all of them place stronger assumptions
on their data. We work with an invariant formulation
of the EV equations. Our main proof is based on a bootstrap argument.
To close the argument, we have to show that the spacetime
curvature and the corresponding geometrical quantities have the
required decay. In order to do so, the Einstein equations are decomposed
with respect to specific foliations of the spacetime. This
result generalizes the work of D. Christodoulou and S. Klainerman
[21].