After a broad contextualisation, I will describe the recent construction
(joint with Richard Schoen) of a new class of solutions to the Einstein
constraint equations that exhibit highly anomalous properties both at a
geometric and at a physical level. In the purely Riemannian setting our
methods produce asymptotically flat manifolds that have positive ADM mass
but are exactly flat outside a cone of arbitrarily small, pre-assigned
opening angle. In particular, using basic facts about Huisken's
isoperimetric mass one can see that these data contain arbitrarily large
stable CMC spheres that are not isoperimetric for they volume they
enclose. Furthermore, the gluing scheme that we develop allows to produce
novel classes of N-body solutions for the Einstein equation, which
patently exhibit the phenomenon of gravitational shielding: for any large
T we can engineer solutions where any two massive bodies do not interact
at all for any time up to T, in striking contrast with the Newtonian
gravity scenario.http://cvgmt.sns.it/seminar/539/
