A limit equation associated to the solvability of the vacuum Einstein constraint equations using the conformal method - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Duke Mathematical Journal Année : 2012

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

Résumé

Let $(M,g)$ be a compact Riemannian manifold on which a trace-free and divergence-free $\sigma \in W^{1,p}$ and a positive function $\tau \in W^{1,p}$, $p > n$, are fixed. In this paper, we study the vacuum Einstein constraint equations using the well known conformal method with data $\sigma$ and $\tau$. 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.

Dates et versions

hal-00783632 , version 1 (01-02-2013)

Identifiants

Citer

Mattias Dahl, Romain Gicquaud, Emmanuel Humbert. A limit equation associated to the solvability of the vacuum Einstein constraint equations using the conformal method. Duke Mathematical Journal, 2012, 161 (14), pp.2669-2698. ⟨hal-00783632⟩
128 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More