Non-uniqueness in conformal formulations of the Einstein constraints

Standard methods in nonlinear analysis are used to indicate that there exists a parabolic branching of solutions of the Lichnerowicz-York equation with an unscaled source. We also apply these methods to the extended conformal thin sandwich formulation and, by assuming that the linearized system develops a kernel solution for sufficiently large initial data, we reproduce the parabolic solution curves for the conformal factor, lapse and shift found numerically by Pfeiffer and York.