A large class of non constant mean curvature solutions of the Einstein constraint equations on an asymptotically hyperbolic manifold

TL;DR

This paper develops a method to construct non-constant mean curvature solutions to Einstein's constraint equations on asymptotically hyperbolic manifolds by using a sub-critical approximation approach and analyzing the limit behavior.

Contribution

It introduces a novel approach of decreasing an exponent in the equations, solving sub-critical equations, and then passing to the limit to obtain solutions for the original problem.

Findings

Solutions of sub-critical equations remain bounded

Conditions are provided to ensure the limit equation has no non-trivial solutions

The method applies to a broad class of non-constant mean curvature solutions

Abstract

We construct solutions of the 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.