Einstein Type Systems on Complete Manifolds

TL;DR

This paper investigates the existence of solutions to the Einstein constraint equations on complete, non-compact manifolds with various asymptotic conditions, using the conformal method and barrier functions.

Contribution

It provides new existence criteria for coupled Einstein systems on complete manifolds, including non-compact and bounded geometry cases, expanding the understanding of initial data in general relativity.

Findings

Established existence criteria with barrier functions for non-compact manifolds.

Proved existence results for bounded geometry manifolds.

Extended existence results to compact manifolds with boundary.

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.