A limit equation associated to the solvability of the vacuum Einstein constraint equations by using the conformal method. (English) Zbl 1258.53037

Authors’ abstract: Let \((M,g)\) be a compact Riemannian manifold on which a trace-free and divergence-free \((0,2)\)-tensor \(\sigma\in W^{1,p}\) 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 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.


53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
35Q75 PDEs in connection with relativity and gravitational theory
53C80 Applications of global differential geometry to the sciences
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
Full Text: DOI arXiv Euclid


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