Nonexistence and Nonuniqueness Results for Solutions to the Vacuum Einstein Conformal Constraint Equations

TL;DR

This paper investigates the vacuum Einstein conformal constraint equations, establishing conditions for nonexistence and nonuniqueness of solutions, and providing counterexamples to previous assumptions in the far-from-CMC case.

Contribution

It presents new nonexistence and nonuniqueness results for the equations, and shows that certain equations with positive Yamabe metrics and zero TT-tensor have solutions, answering a question by D. Maxwell.

Findings

Nonexistence results in the far-from-CMC case.

Nonuniqueness results for certain conformal equations.

Existence of solutions for specific Yamabe metrics with TT-tensor zero.

Abstract

In this article, we give nonexistence and nonuniqueness results for the vacuum Einstein conformal constraint equations in the far-from-CMC case and also show that in some cases the equations of the conformal method for positive Yamabe metrics and with TT-tensor $\sigma$ = 0 have a non-trivial solution, and thus answer a question by D. Maxwell.