A LIMIT EQUATION ASSOCIATED TO THE SOLVABILITY OF THE VACUUM EINSTEIN CONSTRAINT EQUATIONS BY USING THE CONFORMAL METHOD

作者：Dahl Mattias^*; Gicquaud Romain; Humbert Emmanuel

来源：Duke Mathematical Journal, 2012, 161(14): 2669-2697.

DOI：10.1215/00127094-1813182

Let (M, g) be a compact Riemannian manifold on which a trace-free and divergence-free sigma is an element of W-1,W-p and a positive function tau is an element of W-1,W-p, p %26gt; n are fixed. In this paper, we study the vacuum Einstein constraint equations by using the well-known conformal method with data sigma and tau. We show that if no solution exists, then there is a nontrivial solution of another nonlinear limit equation on 1-forms. This last equation can be shown to be without solutions in many situations. As a corollary, we get the 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.

• 出版日期2012-11-1