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

TL;DR

This paper investigates the existence and uniqueness of solutions to the Einstein constraint equations in the conformal formulation, especially for zero or negative Yamabe metrics, providing an alternative theorem for far-from-CMC cases.

Contribution

It establishes an alternative theorem for far-from-CMC solutions, showing either non-uniqueness for positive Yamabe solutions or existence of solutions for negative Yamabe metrics.

Findings

Proves existence of a family of metrics with scalar curvature z

Establishes an analytic solution curve for the conformal equations

Shows an alternative: either non-uniqueness or existence depending on Yamabe sign

Abstract

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.