Einstein Type Systems on Complete Manifolds

Abstract

In this paper, we study the coupled Einstein constraint equations on complete manifolds through the conformal method, focusing on non-compact manifolds with flexible asymptotics. This is physically well-motivated by standard cosmological space-times with non-compact Cauchy hypersurfaces, which favour general bounded geometry manifolds rather than a specific model for infinity. First, we prove an existence criterion on complete manifolds with appropriate barrier functions for physically well-motivated coupled systems. Then, in the bounded geometry case, we build barrier functions and thus show existence. We also prove an existence result on compact manifolds with boundary for a wider family of coupled systems.

Changes for the new version : many typos were corrected, the appendixes were expanded, to improve readability former sections 2.3 and 3.2 were moved away to a separate note Changes v3: implemented referee's suggestions and corrections for publication