Résumé
Let
S
be a compact, orientable surface of hyperbolic type. Let
(
k
+
,
k
-
)
be a pair of negative numbers and let
(
g
+
,
g
-
)
be a pair of marked metrics over
S
of constant curvature equal to
k
+
and
k
-
respectively. Using a functional introduced by Bonsante, Mondello and Schlenker, we show that there exists a unique affine deformation
Γ
:
=
(
ρ
,
τ
)
of a Fuchsian group such that
(
S
,
g
+
)
and
(
S
,
g
-
)
embed isometrically as locally strictly convex Cauchy surfaces in the future and past complete components respectively of the quotient by
Γ
of an open subset
Ω
of Minkowski space. Such quotients are known as Globally Hyperbolic, Maximal, Cauchy compact Minkowski spacetimes and are naturally dual to the half-pipe spaces introduced by Danciger. When translated into this latter framework, our result states that there exists a unique, marked, quasi-Fuchsian half-pipe space in which
(
S
,
g
+
)
and
(
S
,
g
-
)
are realised as the third fundamental forms of future- and past-oriented, locally strictly convex graphs.