Solutions to the Einstein constraint equations with a small TT-tensor and vanishing Yamabe invariant

TL;DR

This paper proves an existence result for Einstein constraint equations with metrics having zero Yamabe invariant, assuming the transverse-traceless tensor is sufficiently small in L^2 norm.

Contribution

It establishes the existence of solutions under the condition of a small TT-tensor for metrics with vanishing Yamabe invariant, extending previous results.

Findings

Existence of solutions proven for small TT-tensors

Results apply to metrics with zero Yamabe invariant

Provides conditions under which Einstein constraints are solvable

Abstract

In this note we prove an existence result for the Einstein conformal constraint equations for metrics with vanishing Yamabe invariant assuming that the TT-tensor is small in $L^2$.