Reading instructions, the initial value problem, Vienna summer
school
The introductory material (putting the Cauchy problem into
a historical context), as well as a sketch of the proof
of local existence is to be found in
The Cauchy problem in general relativity. Acta Phys. Polon. B44 (2013), no. 12, 2621--2641.
A preprint version is to be found here.
A sketch of a proof of future global non-linear stability in a cosmological
setting can be found here.
A more extensive sketch of a proof of local existence can be
found in Chapter 2 of the book
Ringström, Hans On the topology and future stability of the universe.
Oxford Mathematical Monographs. Oxford University Press, Oxford, 2013.
xiv+718 pp. ISBN: 978-0-19-968029-0
Moreover, that book contains a proof of local existence of solutions in
the Einstein-Vlasov setting.
For a proof of local existence of solutions in the vacuum setting, one reference
is
Ringström, Hans The Cauchy problem in general relativity. ESI Lectures
in Mathematics and Physics. European Mathematical Society (EMS), Zürich,
2009. xiv+294 pp. ISBN: 978-3-03719-053-1
Concerning the existence of a maximal globally hyperbolic development, one
proof can be found here.