De l'\'equation de prescription de courbure scalaire aux \'equations de contrainte en relativit\'e g\'en\'erale sur une vari\'et\'e asymptotiquement hyperbolique

TL;DR

This paper investigates scalar curvature prescription and boundary value problems on asymptotically hyperbolic manifolds, applying results to Einstein vacuum equations and constructing initial data with constant mean curvature.

Contribution

It introduces new methods for scalar curvature and boundary condition problems on asymptotically hyperbolic manifolds, with applications to Einstein's equations.

Findings

Solutions for scalar curvature prescription with boundary conditions

Construction of Cauchy data with constant mean curvature

Application to Einstein vacuum equations

Abstract

Two problems concerning asymptotically hyperbolic manifolds with an inner boundary are studied. First, we study scalar curvature presciption with either Dirichlet or mean curvature prescription interior boundary condition. Then we apply those results to the Lichnerowicz equation with (future or past) apparent horizon interior boundary condition. In the last part we show how to construct TT-tensors. Thus we obtain Cauchy data with constant mean curvature for Einstein vacuum equations.