Annales Henri Poincaré

, Volume 16, Issue 12, pp 2919–2954 | Cite as

Conformal Parameterizations of Slices of Flat Kasner Spacetimes

  • David MaxwellEmail author


The Kasner metrics are among the simplest solutions of the vacuum Einstein equations, and we use them here to examine the conformal method of finding solutions of the Einstein constraint equations. After describing the conformal method’s construction of constant mean curvature (CMC) slices of Kasner spacetimes, we turn our attention to non-CMC slices of the smaller family of flat Kasner spacetimes. In this restricted setting we obtain a full description of the construction of certain U n-1 symmetric slices, even in the far-from-CMC regime. Among the conformal data sets generating these slices we find that most data sets construct a single flat Kasner spacetime, but that there are also far-from-CMC data sets that construct one-parameter families of slices. Although these non-CMC families are analogues of well-known CMC one-parameter families, they differ in important ways. Most significantly, unlike the CMC case, the condition signaling the appearance of these non-CMC families is not naturally detected from the conformal data set itself. In light of this difficulty, we propose modifications of the conformal method that involve a conformally transforming mean curvature.


Constraint Equation Fundamental Form Conformal Factor Conformal Killing Constant Mean Curvature 
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© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of Alaska FairbanksFairbanksUSA

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