A model problem for conformal parameterizations of the Einstein constraint equations

TL;DR

This paper explores the effectiveness of conformal and CTS methods in solving Einstein constraint equations using a model problem on conformally flat tori, revealing new phenomena and conditions for solution existence.

Contribution

It introduces a model problem for conformal parameterizations, demonstrating new phenomena and establishing existence criteria, especially when mean curvature changes sign.

Findings

Solutions exist if mean curvature does not change sign.

Solutions exist with small transverse-traceless tensor when mean curvature changes sign.

Multiple solutions can occur under certain conditions.

Abstract

We investigate the possibility that the conformal and conformal thin sandwich (CTS) methods can be used to parameterize the set of solutions of the vacuum Einstein constraint equations. To this end we develop a model problem obtained by taking the quotient of certain symmetric data on conformally flat tori. Specializing the model problem to a three-parameter family of conformal data we observe a number of new phenomena for the conformal and CTS methods. Within this family, we obtain a general existence theorem so long as the mean curvature does not change sign. When the mean curvature changes sign, we find that for certain data solutions exist if and only if the transverse-traceless tensor is sufficiently small. When such solutions exist, there are generically more than one. Moreover, the theory for mean curvatures changing sign is shown to be extremely sensitive with respect to the value of a coupling constant in the Einstein constraint equations.