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

TL;DR

This paper introduces novel fixed point methods for solving vacuum Einstein constraint equations with non-constant mean curvature, unifying and simplifying existing results while relaxing assumptions.

Contribution

It presents two new approaches based on Schaefer's fixed point theorem and half-continuity, expanding the toolkit for Einstein constraint problem solutions.

Findings

Unified recent existence results

Simplified proofs of key theorems

Weakened assumptions in Einstein constraint solutions

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.