Non CMC Conformal Data Sets Which Do Not Produce Solutions of the Einstein Constraint Equations

TL;DR

This paper identifies specific non-CMC conformal data sets for Einstein constraint equations that provably do not admit solutions, advancing understanding of solution existence in general relativity.

Contribution

It introduces the first class of non-CMC data sets with proven non-existence of solutions to the Einstein constraint equations.

Findings

Identifies non-CMC data sets with no solutions

Advances understanding of Einstein constraint equations

Provides theoretical proof of non-existence

Abstract

The conformal formulation provides a method for constructing and parametrizing solutions of the Einstein constraint equations by mapping freely chosen sets of conformal data to solutions, provided a certain set of coupled, elliptic determined PDEs (whose expression depends on the chosen conformal data) admit a unique solution. For constant mean curvature (CMC) data, it is known in almost all cases which sets of conformal data allow these PDEs to have solutions, and which do not. For non CMC data, much less is known. Here we exhibit the first class of non CMC data for which we can prove that no solutions exist.