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

Abstract

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 main theorem in arXiv:1012.2188) and weaken the assumptions of many recent results.

In this version, I change from 3-dimensional case to n-dimensional case

References in corpus (4)

• Rough Solutions of the Einstein Constraint Equations on Compact Manifolds
• Rough solutions of the Einstein constraints on closed manifolds without near-CMC conditions
• A limit equation associated to the solvability of the vacuum Einstein constraint equations using the conformal method
• A new point of view on the solutions to the Einstein constraint equations with arbitrary mean curvature and small TT-tensor

Cited by in corpus (3)

• Einstein-type elliptic systems
• Construction of vacuum initial data by the conformally covariant split system
• What Uniqueness for the Holst-Nagy-Tsogtgerel--Maxwell Solutions to the Einstein Conformal Constraint Equations?