Stability and instability for Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds (Q834784)

Using the conformal method, \textit{Y. Choquet-Bruhat, J. Isenberg} and \textit{D. Pollack} [Chin. Ann. Math., Ser. B 27, No. 1, 31--52 (2006; Zbl 1112.83008)] reformulated the constraint equations for the Einstein-scalar field system as a determined system of nonlinear partial differential equations, consisting of the (linear) conformal momentum constraint and the (nonlinear) conformal Hamiltonian constraint, called the Einstein-scalar field Lichnerowicz equations. The authors investigate the stability of the latter equations for an inhomogeneous compact \(n\)-dimensional Riemannian manifold on which the conformal initial data are defined, building on the variational analysis of \textit{E. Hebey, F. Pacard} and \textit{D. Pollack} [Commun. Math. Phys. 278, No. 1, 117--132 (2008; Zbl 1218.53042)]. They prove that stability holds for \(n\leq5\) but fails to hold in general when \(n\geq6\).