Stability of the Einstein–Lichnerowicz constraint system

Abstract

We study the Einstein-Lichnerowicz constraint system, obtained through the conformal method when addressing the initial data problem for the Einstein equations in a scalar field theory. We prove that this system is stable with respect to the physics data when posed on the standard 3-sphere.

Schrödinger-Poisson systems in $4$-dimensional closed manifolds

“…Now we use a variant of the bounded stability argument (see Li and Zhu [25] and Hebey [20]) which is inspired by Druet and Premoselli [17]. For any α, we let r α be defined by…”

Non-compactness and infinite number of conformal initial data sets in high dimensions

“…Stability issues for equation (1.1) in the focusing case have been investigated in Druet-Hebey [10], Hebey-Veronelli [15] and Premoselli [25]. The stability of the general conformal constraint system in all dimensions has been investigated by Druet-Premoselli [11] and Premoselli [23]. In the specific case of equation (1.1), the results of Druet-Hebey [10] and Premoselli [23] yield in particular that in dimensions n ≥ 6, equation (1.1) is stable with respect to perturbations of its coefficients as soon as there holds in M :…”

Stability and Instability of the Einstein–Lichnerowicz Constraint System

“…Only recently the more involved focusing case, which is the case of interest in Theorem 1.1, has been addressed: we refer to Hebey-Pacard-Pollack [25], Premoselli [35,34] and HolstMeier [27]. Stability results for system (C D ) had been obtained in specific cases, see Druet-Hebey [15], Premoselli [34] and Druet-Premoselli [19]. Note also that, unlike all the known existence results, Theorem 1.1 requires no smallness assumptions on the physics data and allows, in full generality, the manifold M to possess non-trivial conformal Killing 1-forms in M .…”