Noncompact $n$-dimensional Einstein spaces as attractors for the Einstein flow

We prove that along with the Einstein flow, any small perturbations of an$n(n \geq 4)$-dimensional, non-compact negative Einstein space with some"non-positive Weyl tensor" lead to a unique and global solution, and thesolution will be attracted to a noncompact Einstein space that is close to thebackground one. The $n=3$ case has been addressed in [30], while in dimension$n \geq 4$, as we know, negative Einstein metrics in general have non-trivialmoduli spaces. This fact is reflected on the structure of Einstein equations,which further indicates no decay for the spatial Weyl tensor. Furthermore, itis suggested in the proof that the mechanic preventing the metric from flowingback to the original Einstein metric lies in the non-decaying character ofspatial Weyl tensor. In contrary to the compact case considered inAndersson-Moncrief [4], our proof is independent of the theory of infinitesimalEinstein deformations. Instead, we take advantage of the inherent geometricstructures of Einstein equations and develop an approach of energy estimatesfor a hyperbolic system of Maxwell type.