Calculus of Variations and Geometric Measure Theory
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Localizing solutions of the Einstein constraint equations

Alessandro Carlotto

created by malchiodi on 06 Feb 2015

13 feb 2015 -- 14:30   [open in google calendar]

Scuola Normale, Aula Bianchi

Abstract.

A well-known corollary of the positive mass theorem by Schoen-Yau is that if an asymptotically flat manifold (of non-negative scalar curvature) is exactly flat outside of a compact set, then it has to be globally flat: in other terms any such metric can never be localized inside a compact set. So what is the "optimal" localization of those metrics? For instance, can one produce scalar non-negative metrics that have positive ADM mass and still are trivial in a half-space?

In recent joint work with Schoen, we answer these questions by giving a systematic method for constructing solutions to the Einstein constraint equations that are localized inside a cone of arbitrarily small aperture. This sharply contrasts with various recent scalar curvature rigidity phenomena both in the closed and in the free-boundary setting. Moreover, the gluing scheme that we adopt allows to produce a new class of exotic 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 t\in(0,T), in striking contrast with the Newtonian gravity scenario.

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