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The Einstein-Straus model consists of a Schwarzschild spherical vacuole in aFriedman-Lema\^{\i}tre-Robertson-Walker (FLRW) dust spacetime (with or without\Lambda). It constitutes the most widely accepted model to answer the questionof the influence of large scale (cosmological) dynamics on local systems. Theconclusion drawn by the model is that there is no influence from the cosmicbackground, since the spherical vacuole is static. Spherical generalizations toother interior matter models are commonly used in the construction of lumpyinhomogeneous cosmological models. On the other hand, the model has proven tobe reluctant to admit non-spherical generalizations. In this review, wesummarize the known uniqueness results for this model. These seem to indicatethat the only reasonable and realistic non-spherical deformations of theEinstein-Straus model require perturbing the FLRW background. We review resultsabout linear perturbations of the Einstein-Straus model, where theperturbations in the vacuole are assumed to be stationary and axially symmetricso as to describe regions (voids in particular) in which the matter has reachedan equilibrium regime.

Review on exact and perturbative deformations of the Einstein-Straus model: uniqueness and rigidity results