A class of solutions of the vacuum Einstein constraint equations with freely specified mean curvature

Abstract

We give a sufficient condition, with no restrictions on the mean curvature, under which the conformal method can be used to generate solutions of the vacuum Einstein constraint equations on compact manifolds. The condition requires a so-called global supersolution but does not require a global subsolution. As a consequence, we construct a class of solutions of the vacuum Einstein constraint equations with freely specified mean curvature, extending a recent result [HNT07] which constructed similar solutions in the presence of matter. We give a second proof of this result showing that vacuum solutions can be obtained as a limit of [HNT07] non-vacuum solutions. Our principal existence theorem is of independent interest in the near-CMC case, where it simplifies previously known hypotheses required for existence.

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

“…This method has several applications. In particular, it greatly simplifies the proof of the main theorem in [8] (see Theorem 3.3) and allows to recover an existence result provided σ is small enough in L ∞ (depending only on g and τ) as noticed in [11] and [15] (see Proposition 3.9). Furthermore, it gives an unifying point of view of these results.…”

“…It allows to show that the assumption of the existence of global supersolutions used in [8], [11] and [15] to solve (4) can be weakened: the existence of local supersolutions, whose definition is given in Section 4, is sufficient here. As applications of this method, we prove the solvability of a modification of the system (4) when τ has some zeros and we show that the smallness of σ in L 2 leads to the solvability of (4).…”

“…The first one is based on Schaefer's fixed points which turns out to be more efficient in this situation than an application of Schauder's fixed point theorem as used in [8], [11] and [15]. This method has several applications.…”

“…• Holst-Nagy-Tsogtgerel [11] and Maxwell [15] for far from CMC-results with a smallness assumption on σ, depending only on g and τ.…”

On the stability of solutions of the Lichnerowicz–York equation

“…One motivation for that study was to examine the uniqueness of solutions in the far-fromconstant mean curvature regime now that existence results for this case exist (see [8], [14]). He found interesting non-existence and non-uniqueness results showing that the constraints are ill-posed beyond the non-uniqueness that is introduced when one couples the lapse fixing equation to the four constraint equations, as in the extended conformal thin sandwich (XCTS) formulation (see [15], [2], [19]) and some constrained evolution schemes (see [16], [6] for a resolution of this scaling problem).…”