Stability and instability of the Einstein-Lichnerowicz constraint system

TL;DR

This paper studies the stability of the Einstein-Lichnerowicz conformal constraint system in scalar-field settings, proving stability under certain conditions and demonstrating sharpness through explicit instability examples.

Contribution

It establishes the stability of the system on closed conformally flat manifolds and constructs explicit instability cases when assumptions are violated.

Findings

Proves stability of the system under generic conditions.

Constructs explicit examples of instability.

Results apply to a broader class of constraint systems.

Abstract

We investigate the relevance of the conformal method by investigating stability issues for the Einstein-Lichnerowicz conformal constraint system in a nonlinear scalar-field setting. We prove the stability of the system with respect to arbitrary perturbations of generic focusing physics data on closed locally conformally flat manifolds, in any dimension. We also show that our stability result is sharp by constructing explicit instability examples when its assumptions are not satisfied. Our results apply to a more general class of constraint-like systems.