A large class of non-constant mean curvature solutions of the Einstein constraint equations on an asymptotically hyperbolic manifold. (English) Zbl 1247.83010
Summary: We construct solutions of a constraint equation with non constant mean curvature on an asymptotically hyperbolic manifold by the conformal method. Our approach consists in decreasing a certain exponent appearing in the equations, constructing solutions of these sub-critical equations and then in letting the exponent tend to its true value. We prove that the solutions of the sub-critical equations remain bounded which yields solutions of the constraint equation unless a certain limit equation admits a non-trivial solution. Finally, we give conditions which ensure that the limit equation admits no non-trivial solution.
MSC:
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
| 35L15 | Initial value problems for second-order hyperbolic equations |
| 83C15 | Exact solutions to problems in general relativity and gravitational theory |
| 53Z05 | Applications of differential geometry to physics |
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