The Einstein Constraint Equations on Asymptotically Euclidean Manifolds

TL;DR

This dissertation investigates the Einstein constraint equations on asymptotically Euclidean manifolds, providing conditions for solutions, analyzing the conformal method, and characterizing scalar curvature, with implications for general relativity.

Contribution

It introduces new necessary and sufficient conditions for solutions, extends the conformal method to far-from-CMC data, and characterizes Yamabe classes on asymptotically Euclidean manifolds.

Findings

Conditions for Lichnerowicz equation solutions

Existence of global supersolutions for near- and far-CMC data

Characterization of Yamabe classes on AE manifolds

Abstract

In this dissertation, we prove a number of results regarding the conformal method of finding solutions to the Einstein constraint equations. These results include necessary and sufficient conditions for the Lichnerowicz equation to have solutions, global supersolutions which guarantee solutions to the conformal constraint equations for near-constant-mean-curvature (near-CMC) data as well as for far-from-CMC data, a proof of the limit equation criterion in the near-CMC case, as well as a model problem on the relationship between the asymptotic constants of solutions and the ADM mass. We also prove a characterization of the Yamabe classes on asymptotically Euclidean manifolds and resolve the (conformally) prescribed scalar curvature problem on asymptotically Euclidean manifolds for the case of nonpositive scalar curvatures. Many, though not all, of the results in this dissertation have been previously published in [Dilts13b], [DIMM14], [DL14], [DM15], and [DGI15]. This article is the author's Ph.D. dissertation, except for a few minor changes.