Caterina Vâlcu
We study the constraint equations for Einstein equations on manifolds of the form $\mathbb{R}^{n+1}\times T^m$, where $T^m$ is a flat m-dimensional torus. Spacetimes with compact directions were introduced almost a century ago by Theodor Kaluza and Oskar Klein as an early attempt of unifying electromagnetism and general relativity in a simple, elegant way. The aim of this article is to construct initial data for the Einstein equations on manifolds of the form $\mathbb{R}^{n+1}\times T^m$, 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 the proof is the introduction of new weighted Sobolev spaces, adapted to the inversion of the Laplacean on product manifolds. Classical linear elliptic results need to rigorously proved in this new setting. This is joint work with Cécile Huneau.
