  In this paper, we sketch the proof of the extension of the stability theorem
of the Minkowski space in General Relativity done explicitly in previous work
by the present author. We discuss solutions of the Einstein vacuum (EV)
equations. We solve the Cauchy problem for more general, asymptotically flat
initial data than in the pioneering work `The Global Nonlinear Stability of the
Minkowski Space' of D. Christodoulou and S. Klainerman or than in any other
work. Moreover, we describe precisely the asymptotic behaviour. Our relaxed
assumptions on the initial data yield a spacetime curvature which is not
bounded in $L^{\infty}(M)$. 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. 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 `The
Global Nonlinear Stability of the Minkowski Space' of D. Christodoulou and S.
Klainerman.
