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

TL;DR

This work establishes the global stability of a broad class of noncompact, $n\ge 4$ dimensional negative Einstein spaces under the Einstein flow, for backgrounds with non-positive Weyl tensor. By formulating a 1+ n decomposition in Gaussian normal coordinates, the authors derive a hyperbolic system of Maxwell type for the Weyl components $(\mathcal{E},\mathcal{H})$ and $(\mathcal{H},\mathcal{J})$, and couple this with elliptic equations for the spatial Weyl tensor $W$ to control high-derivative terms. The analysis yields decay estimates for the dynamical gravity quantities while the Weyl sector remains non-decaying, implying the flow converges to an Einstein metric $g_\infty$ nearby in moduli space, rather than to the background metric $\gamma$ itself in general. The approach is independent of infinitesimal Einstein deformation theory and provides a robust framework for noncompact stability, including an attractor that lies within the deformation space of the background metric. These results extend the stability picture from compact to certain noncompact settings and highlight the distinctive role of Weyl geometry in the late-time behavior of the Einstein flow.

Abstract

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 the solution will be attracted to a noncompact Einstein space that is close to the background 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-trivial moduli spaces. This fact is reflected on the structure of Einstein equations, which further indicates no decay for the spatial Weyl tensor. Furthermore, it is suggested in the proof that the mechanic preventing the metric from flowing back to the original Einstein metric lies in the non-decaying character of spatial Weyl tensor. In contrary to the compact case considered in Andersson-Moncrief [4], our proof is independent of the theory of infinitesimal Einstein deformations. Instead, we take advantage of the inherent geometric structures of Einstein equations and develop an approach of energy estimates for a hyperbolic system of Maxwell type.

Theorems & Definitions (50)

• Definition 1.1: Non-positive Weyl tensor
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Proposition 2.1
• proof
• Proposition 2.2
• Proposition 2.3