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

TL;DR

This paper develops solutions to Einstein's vacuum constraint equations on compact 3-manifolds with a translational symmetry, especially for high-genus quotients, using the limit equation method.

Contribution

It extends the limit equation approach to vacuum Einstein constraints with translational Killing vectors on hyperbolic manifolds, providing new non-CMC solutions.

Findings

Solutions constructed for manifolds with genus > 2

Strong results obtained far from constant mean curvature

Application of limit equation criterion in hyperbolic setting

Abstract

We construct solutions to the constraint equations in general relativity using the limit equation criterion introduced by Dahl, Humbert and the first author. We focus on solutions over compact 3-manifolds admitting a $\bS^1$-symmetry group. When the quotient manifold has genus greater than 2, we obtain strong far from CMC results.