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

The conformal method has been effective for parametrizing solutions to the Einstein constraint equations on closed 3-manifolds. However, it is still not well-understood; for example, existence of solutions to the conformal equations for zero or negative Yamabe metrics is still unknown without the so-called ``CMC'' or ``near-CMC'' assumptions. The first existence results without such assumptions, termed the ``far-from-CMC'' case, were obtained by Holst, Nagy, and Tsogtgerel in 2008 for positive Yamabe metrics. However, their results are based on topological arguments, and as a result solution uniqueness is not known. Indeed, Maxwell gave evidence in 2011 that far-from-CMC solutions are not unique in certain cases. In this article, we provide further insight by establishing a type of alternative theorem for general far-from-CMC solutions. For a given manifold M that admits a metric of positive scalar curvature and scalar flat metric g(0) with no conformal Killing fields, we first prove existence 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-Schmidt reduction, we determine an analytic solution curve for F((phi,w),z) = 0 through a critical point where the linearization of F((phi,w),z) vanishes. The regularity of this curve, the definition of F((phi,w),z), and the earlier far-from-CMC results of Holst et al. allow us to then prove the following alternative theorem for far-from-CMC solutions: either (1) there exists a z_1 >0 such that (positive Yamabe) solutions to the z_1-parameterized conformal equations are non-unique; or (2) there exists z_2 < 0 such that (negative Yamabe) solutions to the z_2-parameterized conformal equations exist.