  The studies in general relativity of rotating finite objects in equilibrium
have usually focused on the case when they are truly isolated, this is, the
models to describe finite objects are embedded in an asymptotically flat
exterior vacuum. Known results ensure the uniqueness of the vacuum exterior
field by using the boundary data for the exterior field given at the surface of
the object plus the decay of the exterior field at infinity. The final aim of
the present work is to study the consequences on the interior models by
changing the boundary condition at infinity to one accounting for the embedding
of the object in a cosmological background. Considering first the FLRW standard
cosmological backgrounds, we are studying the general matching of FLRW with
stationary axisymmetric spacetimes in order to find the new boundary condition
for the vacuum region. Here we present the first results.
