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The conformal method has been effective for parametrizing solutions to theEinstein constraint equations on closed 3-manifolds. However, it is still notwell-understood; for example, existence of solutions to the conformal equationsfor zero or negative Yamabe metrics is still unknown without the so-called``CMC'' or ``near-CMC'' assumptions. The first existence results without suchassumptions, termed the ``far-from-CMC'' case, were obtained by Holst, Nagy,and Tsogtgerel in 2008 for positive Yamabe metrics. However, their results arebased on topological arguments, and as a result solution uniqueness is notknown. Indeed, Maxwell gave evidence in 2011 that far-from-CMC solutions arenot unique in certain cases. In this article, we provide further insight byestablishing a type of alternative theorem for general far-from-CMC solutions.For a given manifold M that admits a metric of positive scalar curvature andscalar flat metric g(0) with no conformal Killing fields, we first proveexistence of an analytic, one-parameter family of metrics g(z) through g(0)such that R(g(z)) = z. Using this family of metrics and given data(tau,sigma,rho,j), we form a one-parameter family of operators F((phi,w),z)whose zeros satisfy the conformal equations. Applying Liapnuov-Schmidtreduction, we determine an analytic solution curve for F((phi,w),z) = 0 througha critical point where the linearization of F((phi,w),z) vanishes. Theregularity of this curve, the definition of F((phi,w),z), and the earlierfar-from-CMC results of Holst et al. allow us to then prove the followingalternative theorem for far-from-CMC solutions: either (1) there exists a z_1>0 such that (positive Yamabe) solutions to the z_1-parameterized conformalequations are non-unique; or (2) there exists z_2 < 0 such that (negativeYamabe) solutions to the z_2-parameterized conformal equations exist.

An Alternative Between Non-unique and Negative Yamabe Solutions to the Conformal Formulation of the Einstein Constraint Equations