Review on exact and perturbative deformations of the Einstein-Straus
  model: uniqueness and rigidity results
  The Einstein-Straus model consists of a Schwarzschild spherical vacuole in a
Friedman-Lema\^{\i}tre-Robertson-Walker (FLRW) dust spacetime (with or without
\Lambda). It constitutes the most widely accepted model to answer the question
of the influence of large scale (cosmological) dynamics on local systems. The
conclusion drawn by the model is that there is no influence from the cosmic
background, since the spherical vacuole is static. Spherical generalizations to
other interior matter models are commonly used in the construction of lumpy
inhomogeneous cosmological models. On the other hand, the model has proven to
be reluctant to admit non-spherical generalizations. In this review, we
summarize the known uniqueness results for this model. These seem to indicate
that the only reasonable and realistic non-spherical deformations of the
Einstein-Straus model require perturbing the FLRW background. We review results
about linear perturbations of the Einstein-Straus model, where the
perturbations in the vacuole are assumed to be stationary and axially symmetric
so as to describe regions (voids in particular) in which the matter has reached
an equilibrium regime.
