  It is well-known that solutions to the conformal formulation of the Einstein
constraint equations are unique in the cases of constant mean curvature (CMC)
and near constant mean curvature (near-CMC). However, the new far-from-constant
mean curvature (far-from-CMC) existence results due to Holst, Nagy, and
Tsogtgerel in 2008, to Maxwell in 2009, and to Dahl, Gicquaud and Humbert in
2010, are based on degree theory rather than on the (uniqueness-providing)
contraction arguments that had been used for all non-CMC existence results
prior to 2008. In fact, Maxwell demonstrated in 2011 that solutions are
non-unique in the far-from-CMC case for certain types of low-regularity mean
curvature. In this article, we investigate uniqueness properties of solutions
to the Einstein constraint equations on closed manifolds using tools from
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
Liapunov-Schmidt reduction and standard tools from nonlinear analysis, we
demonstrate that solutions to the conformal formulation with unscaled data are
non-unique by determining an explicit solution curve, and by analyzing its
behavior in the neighborhood of a particular solution.
