A Model Problem for Conformal Parameterizations of the Einstein Constraint Equations

We study the conformal and conformal thin sandwich (CTS) methods as candidates for parameterizing the set vacuum initial data for the Cauchy problem of general relativity. To this end we consider a small family of symmetric conformal data. Within this family we obtain an existence result so long as the mean curvature has constant sign. When the mean curvature changes sign we find that solutions either do not exist, or they are not unique. In some cases solutions are shown to be non-unique. Moreover, the theory for mean curvatures with changing sign is shown to be extremely sensitive with respect to the value of a coupling constant in the Einstein constraint equations.