Rough Solutions of the Einstein Constraint Equations on Compact Manifolds

TL;DR

This paper develops low regularity solutions for the vacuum Einstein constraint equations on compact 3-manifolds, extending previous work to include metrics with Sobolev regularity greater than 3/2, using the CMC conformal method.

Contribution

It introduces a method to construct all CMC initial data with low regularity on compact manifolds, expanding prior results from asymptotically Euclidean cases.

Findings

Solutions with metrics in H^s for s>3/2 on 3-manifolds

Extension of previous constructions to compact manifolds

All CMC initial data with this regularity level can be constructed

Abstract

We construct low regularity solutions of the vacuum Einstein constraint equations on compact manifolds. On 3-manifolds we obtain solutions with metrics in $H^s$ where $s>3/2$. The constant mean curvature (CMC) conformal method leads to a construction of all CMC initial data with this level of regularity. These results extend a construction from Ma04 that treated the asymptotically Euclidean case.