A limit equation associated to the solvability of the vacuum Einstein constraint equations by using the conformal method

Abstract

Let (M, g) be a compact Riemannian manifold on which a tracefree and divergence-free σ ∈ W 1,p and a positive function τ ∈ W 1,p , p > n, are fixed. In this paper, we study the vacuum Einstein constraint equations using the well known conformal method with data σ and τ . We show that if no solution exists then there is a non-trivial solution of another non-linear limit equation on 1-forms. This last equation can be shown to be without solutions no solution in many situations. As a corollary, we get existence of solutions of the vacuum Einstein constraint equation under explicit assumptions which in particular hold on a dense set of metrics g for the C 0 -topology.

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.…”

“…As proven in [8], G is continuous compact, so the continuity and compactness of T ′ and hence that of T , will follow from the continuity of T 1 . Actually, we prove more:…”

“…In this section we show how Schaefer's fixed point theorem can be applied to give a simpler proof of the main result in [8]. We first recall its statement (see [4,Theorem 3.4.8] or [9,Theorem 11.6]).…”

“…We first recall its statement (see [4,Theorem 3.4.8] or [9,Theorem 11.6]). We now state the main theorem in [8] and give an alternative proof. (5) and assume that (6) holds.…”

“…These methods allow to: unify some recent existence results, simplify many proofs (for instance, the one of the main theorem in [8]) and weaken the assumptions of many recent results. …”