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

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-00783632
Contributor : Romain Gicquaud <>
Submitted on : Friday, February 1, 2013 - 1:38:04 PM
Last modification on : Thursday, March 5, 2020 - 5:33:28 PM

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  • HAL Id : hal-00783632, version 1
  • ARXIV : 1012.2188

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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, Duke University Press, 2012, 161 (14), pp.2669-2698. ⟨hal-00783632⟩

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