Abstract
The conformal approach introduced by Lichnerowicz aims to construct solutions to the Einstein constraint equations by conformally deforming a given metric, giving rise to a system of coupled equations satisfied by the conformal factor and a 1-form. When the starting metric is rotationally symmetrical, we can reduce this system of equations and search for radial solutions.In a previous work, the second author described these radial solutions in the Euclidean space. In this paper we extend this approach to the general framework of harmonic manifolds (for which the Laplacian of a radial function is still a radial function). When the starting metric has constant curvature, we get an explicit relation between the conformal factor and the equation data. In the case of the round sphere we get a non-existence result of solutions with zero cosmological constant and we prove that the Einstein constraint equation is unstable with respect to the data. In the case of the hyperbolic space, we obtain an existence result for asymptotically hyperbolic solutions and compute their mass.