  The notion of maximal extension of a globally hyperbolic space-time arises
from the notion of maximal solutions of the Cauchy problem associated to the
Einstein's equations of general relativity. In 1969 Choquet-Bruhat and Geroch
proved that if the Cauchy problem has a local solution, this solution has a
unique maximal extension. Since the causal structure of a space-time is
invariant under conformal changes of metrics we may generalize this notion of
maximality to the conformal setting. In this article we focus on conformally
flat space-times of dimension greater or equal than 3. In this case, by a
Lorentzian version of Liouville theorem, these space-times are locally modeled
on the Einstein space-time. In the first part we use this fact to prove the
existence and uniqueness of the maximum extension for globally hyperbolic
conformally flat space-times. In the second part we find a causal
characterization of globally hyperbolic conformally flat maximal space-times
whose developing map is a global diffeomorphism.
