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The Einstein constraint equations have been the subject of study for morethan fifty years. The introduction of the conformal method in the 1970's as aparameterization of initial data for the Einstein equations led to increasedinterest in the development of a complete solution theory for the constraints,with the theory for constant mean curvature (CMC) spatial slices and closedmanifolds completely developed by 1995. The first general non-CMC existenceresult was establish by Holst et al. in 2008, with extensions to rough data byHolst et al. in 2009, and to vacuum spacetimes by Maxwell in 2009. The non-CMCtheory remains mostly open; moreover, recent work of Maxwell on specificsymmetry models sheds light on fundamental non-uniqueness problems with theconformal method as a parameterization in non-CMC settings. In parallel withthese mathematical developments, computational physicists have uncoveredsurprising behavior in numerical solutions to the extended conformal thinsandwich formulation of the Einstein constraints. In particular, numericalevidence suggests the existence of multiple solutions with a quadratic fold,and a recent analysis of a simplified model supports this conclusion. In thisarticle, we examine this apparent bifurcation phenomena in a methodical way,using modern techniques in bifurcation theory and in numerical homotopymethods. We first review the evidence for the presence of bifurcation in theHamiltonian constraint in the time-symmetric case. We give a brief introductionto the mathematical framework for analyzing bifurcation phenomena, and thendevelop the main ideas behind the construction of numerical homotopy, orpath-following, methods in the analysis of bifurcation phenomena. We then applythe continuation software package AUTO to this problem, and verify the presenceof the fold with homotopy-based numerical methods.

Numerical Bifurcation Analysis of Conformal Formulations of the Einstein Constraints