Non-uniqueness of Solutions to the Conformal Formulation

It is well known that solutions to the conformal formulation of the Einstein constraint equations are unique in the event that the mean curvature is constant (CMC) or near constant (near-CMC). However, the new far-from-constant mean curvature (farfrom-CMC) solution constructions due to Holst, Nagy, and Tsogtgerel and to Maxwell in 2009, and to Gicquad and collaborators in 2010, are based on degree theory rather than the contraction arguments used originally by Isenberg and Moncrief in 1996 for the near-CMC case, and hence little is known about the uniqueness of far-from-CMC constructions. In fact, Maxwell recently demonstrated that solutions are non-unique in the far-from-CMC case for certain families of low regularity mean curvatures. In this article, we investigate uniqueness properties of solutions to the Einstein constraint equations on a closed manifold using standard tools in bifurcation theory. For positive, constant scalar curvature and constant mean curvature, we first demonstrate existence of a critical energy density for the Hamiltonian constraint with unscaled matter sources. We then show that for this choice of energy density, the linearization of the elliptic system develops a one-dimensional kernel in both the CMC and non-CMC (near and far) cases. Using a Liapunov-Schmidt reduction and some standard techniques from nonlinear analysis, we demonstrate that solutions to the conformal formulation with unscaled data are non-unique by determining an explicit solution curve and analyzing its behavior in the neighborhood of a particular solution.