Einstein spaces as attractors for small perturbations of noncompact n(n≥4)-dim Einstein manifold under the Einstein flow
摘要：We prove that along with the Einstein flow, any small perturbations of an n-dimensional, non-compact negative Einstein space with certain non-positive Weyl tensor will be attracted to an Einstein space that is close to the original one. As we know, in dimension n≥4, a negative Einstein metric has non-trivial moduli space. This fact is reflected in the structure of Einstein equations, which further implies no decay for the spatial Weyl tensor. Our proof suggests the mechanics preventing the metric flowing back to the original Einstein metric lie in the non-decaying character of the spatial Weyl tensors. In addition, we take advantage of the inherent geometric structures of Einstein equations and develop an approach of energy estimates for hyperbolic systems of Maxwell type coupled with elliptic equations.