Constant mean curvature solutions of the Einstein-scalar field constraint equations on asymptotically hyperbolic manifolds

TL;DR

This paper investigates the existence and non-existence of solutions to Einstein-scalar field constraint equations on asymptotically hyperbolic manifolds, extending previous work on closed manifolds.

Contribution

It generalizes the analysis of Einstein-scalar field constraints to asymptotically hyperbolic manifolds, providing new existence and non-existence results.

Findings

Established conditions for existence of solutions.

Identified scenarios where solutions do not exist.

Extended previous methods to new geometric settings.

Abstract

We follow the approach employed by Y. Choquet-Bruhat, J. Isenberg and D. Pollack in the case of closed manifolds and establish existence and non-existence results for the Einstein-scalar field constraint equations on asymptotically hyperbolic manifolds.