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

Understand

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.