Book Description

"This thesis constitutes an exposition of much of what is currently known about the conformal method of parameterizing solutions to the Einstein constraint equations. First, the relevant background information is presented, including the Einstein field equations. Then the problem of coming up with a well-posed initial value problem is discussed, including the relevant theorems of Y. Choquet-Bruhat, leading into a discussion and derivation of the Einstein constraint equations.Next, the conformal method for parameterizing solutions to the constraint equations is motivated and discussed, concluding the first part of the thesis. After this the Yamabe problem is discussed and the relevant results about the Yamabe invariant proven, which plays a very important role in the solving of the conformal constraint equations.The next part of this thesis will focus on the solving of the conformal constraint equations, and the main influence is Maxwell's work ([9]). We begin with the Lichnerowicz equation and move onto solving the coupled system by using techniquesincluding the method of global supersolutions ([9]) and the limit equation ([2], [10]). Nonexistence results are also discussed, mainly towards the end, as well as some minor new results." --