  Cauchy-compact flat spacetimes with extreme BTZ are Lorentzian analogue of
complete hyperbolic surfaces of finite volume. Indeed, the latter are
2-manifolds locally modeled on the hyperbolic plane, with group of isometries
$\mathrm{PSL}_2(\R)$, admitting finitely many cuspidal ends while the regular
part of the former are 3-manifolds locally models on 3 dimensionnal Minkowski
space, with group of isometries $\mathrm{PSL}_2(\R)\ltimes \RR^3$, admitting
finitely many ends whose neighborhoods are foliated by cusps. We prove a
Theorem akin to the classical parametrization result for finite volume complete
hyperbolic surfaces: the tangent bundle of the Teichm\"uller space of a
punctured surface parametrizes globally hyperbolic Cauchy-maximal and
Cauchy-compact locally Minkowski manifolds with extreme BTZ. Previous results
of Mess, Bonsante and Barbot provide already a satisfactory parametrization of
regular parts of such manifolds, the particularity of the present work reside
in the consideration of manifolds with a singular geometrical structure with a
singularities modeled on extreme BTZ. We present a BTZ-extension procedure akin
to the procedure compactifying finite volume complete hyperbolic surface by
adding cusp points at infinity
