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

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.