Seminar

# On the initial boundary value problem for the Einstein equations in the maximal gauge

#### General Relativity, Geometry and Analysis: beyond the first 100 years after Einstein

#### 10 October 11:00 - 12:00

#### Jacques Smulevici - Sorbonne University

Work in progress in collaboration with Grigorios Fournodavlos. We address the initial boundary 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.

Organizers

Lars Andersson

Max Planck Institute for Gravitational Physics (Albert Einstein Institute)

Mattias Dahl

KTH Royal Institute of Technology

Philippe G. LeFloch

Sorbonne University

Richard Schoen

University of California, Irvine