Applications of Fixed Point Theorems to the Vacuum Einstein Constraint Equations with Non-Constant Mean Curvature

In this paper, we introduce new methods for solving the vacuum Einstein constraints equations: the first one is based on Schaefer’s fixed point theorem (known methods use Schauder’s fixed point theorem), while the second one uses the concept of half-continuity coupled with the introduction of local supersolutions. These methods allow to: unify some recent existence results, simplify many proofs (for instance, the one of the main theorems in Dahl et al., Duke Math J 161(14):2669–2697, 2012) and weaken the assumptions of many recent results.