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In the present paper, we study the coupled Einstein Constraint Equations(ECE) on complete manifolds through the conformal method, focusing onnon-compact manifolds with flexible asymptotics. In particular, we do notimpose any specific model for infinity. First, we prove an existence criteriaon compact manifolds with boundary which applies to more general systems andcan be seen as a natural extension of known existence theory for the coupledECE. Building on this, we prove an L^p existence based on existence ofappropriate barrier functions for a family of physically well-motivated coupledsystems on complete manifolds. We prove existence results for these systems bybuilding barrier functions in the bounded geometry case. We conclude bytranslating our result to the H^s formulation, making contact with classicworks. To this end, we prove several intermediary L^2 regularity results forthe coupled systems in dimensions n \le 12 which fills certain gaps in currentinitial data analysis.

Einstein Type Systems on Complete Manifolds