Einstein Constraint Equations for Kaluza-Klein Spacetimes

TL;DR

This paper develops a method to construct initial data for Einstein equations on Kaluza-Klein manifolds without symmetry assumptions, introducing new weighted Sobolev spaces to handle elliptic equations.

Contribution

It introduces novel weighted Sobolev spaces tailored for product manifolds, enabling the analysis of Einstein constraint equations in Kaluza-Klein spacetimes without symmetry.

Findings

Successfully constructs initial data on R^{n+1} x T^{m}

Extends the conformal method to product manifolds

Introduces new weighted Sobolev spaces for elliptic PDEs

Abstract

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.