What Uniqueness for the Holst-Nagy-Tsogtgerel--Maxwell Solutions to the Einstein Conformal Constraint Equations?

This paper addresses the issue of uniqueness of solutions in the conformal method for solving the constraint equations in general relativity with arbitrary mean curvature as developed initially by Holst, Nagy, Tsogtegerel and Maxwell. We show that the solution they construct is unique amongst those having volume below a certain threshold.