Linearization stability of the Einstein constraint equations on an asymptotically hyperbolic manifold

TL;DR

This paper investigates the stability of Einstein constraint equations on asymptotically hyperbolic manifolds, showing stability near certain vacuum solutions and identifying decay conditions where stability fails.

Contribution

It proves linearization stability for Einstein constraints near vacuum solutions with specific decay rates and constructs counterexamples for faster decays using new TT-tensor constructions.

Findings

Stability holds near vacuum solutions with certain decay rates.

Stability fails for faster decay rates, with counterexamples constructed.

New TT-tensor constructions on Euclidean space underpin the results.

Abstract

We study the linearization stability of the Einstein constraint equations on an asymptotically hyperbolic manifold. In particular we prove that these equations are linearization stable in the neighborhood of vacuum solutions for a non-positive cosmological constant and of Friedman--Lema\^itre--Robertson--Walker spaces in a certain range of decays. We also prove that this result is no longer true for faster decays. The construction of the counterexamples is based on a new construction of TT-tensors on the Euclidean space and on positive energy theorems.