A limit equation criterion for applying the conformal method to asymptotically cylindrical initial data sets

Abstract

We prove that in a certain class of conformal data on an asymptotically cylindrical manifold, if the conformally decomposed Einstein constraint equations do not admit a solution, then one can always find a nontrivial solution to the limit equation first explored by Dahl, Gicquaud, and Humbert in [DGH11]. We also give an example of a Ricci curvature condition on the manifold which precludes the existence of a solution to this limit equation, showing that such a limit criterion can be a useful tool for studying the Einstein constraint equations on manifolds with asymptotically cylindrical ends.

A new point of view on the solutions to the Einstein constraint equations with arbitrary mean curvature and small TT-tensor

“…This method was adapted to several other contexts such as asymptotically hyperbolic manifolds in [12] and asymptotically cylindrical manifolds in [8]. In particular, strong results are obtained for negatively curved manifolds; see [12, proposition 6.2 and remark 6.3].…”

Solutions to the Einstein-scalar field constraint equations with a small TT-tensor

“…This method turns out to be particularly efficient with negatively (Ricci) curved metrics, see [16]. See also [12] for the asymptotically cylindrical case.…”

Limit equation for vacuum Einstein constraints with a translational Killing vector field in the compact hyperbolic case

“…This approach has been generalized to the asymptotically hyperbolic case in [9] and to the asymptotically cylindrical case in [6]. The asymptotically Euclidean case [5] and the case of compact manifolds with boundary [7] are currently work in progress since new ideas have to be found to get the criterion.…”