Einstein Constraint Equations for Kaluza-Klein Spacetimes

The aim of this article is to construct initial data for the Einstein equations on manifolds of the form R n+1 x T m , which are asymptotically flat at infinity, without assuming any symmetry condition in the compact direction. We use the conformal method to reduce the constraint equations to a system of elliptic equation and work in the near CMC (constant mean curvature) regime. The main new feature of our paper is the introduction of new weighted Sobolev spaces, adapted to the inversion of the Laplacian on product manifolds.