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Jacques Smulevici: On the initial value problem for the Einstein equations in the maximal gauge

Time: Thu 2019-10-10 11.00 - 12.00

Location: Seminar Hall Kuskvillan, Institut Mittag-Leffler

Participating: Jacques Smulevici, Sorbonne University

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Work in progress in collaboration with Grigorios Fournodavlos.
We address the initial value problem for the vacuum Einstein equations in the maximal gauge, or more generally, in a gauge where the mean curvature of the level sets of a time function is prescribed. Assuming that the induced metric on the boundary is given in the geodesic gauge, or more generally that the lapse is prescribed, the main boundary data is given by the one-parameter family of conformal metrics on each 2d section. These conformal metrics can be written in terms of 2 scalar functions and can be seen as incoding the usual gravitational degrees of freedom. The main analytical tools are wave equations for the second-fundamental form of the maximal foliation together with well-chosen boundary conditions (on top of the geometrical ones already mentioned) that allow both the propagation of constraints and energy type estimates for the main unknown.