## Journal of Differential Geometry

- J. Differential Geom.
- Volume 89, Number 1 (2011), 1-47.

### Einstein spaces as attractors for the Einstein flow

Lars Andersson and Vincent Moncrief

#### Abstract

In this paper we prove a global existence theorem, in the direction
of cosmological expansion, for sufficiently small perturbations
of a family of $n + 1$-dimensional, spatially compact spacetimes,
which generalizes the $k = -1$ Friedmann-Lemaître-Robertson-Walker vacuum spacetime. This work extends the result from *Future complete vacuum spacetimes*.
The background spacetimes we consider are Lorentz cones over
negative Einstein spaces of dimension $n \ge 3$.

We use a variant of the constant mean curvature, spatially harmonic
(CMCSH) gauge introduced in *Elliptic-hyperbolic systems and the Einstein
equations*. An important difference
from the $3+1$ dimensional case is that one may have a nontrivial
moduli space of negative Einstein geometries. This makes it necessary
to introduce a time-dependent background metric, which is
used to define the spatially harmonic coordinate system that goes
into the gauge.

Instead of the Bel-Robinson energy used in *Future complete vacuum spacetimes*, we here use an
expression analogous to the wave equation type of energy introduced
in *Elliptic-hyperbolic systems and the Einstein
equations* for the Einstein equations in CMCSH gauge. In order
to prove energy estimates, it turns out to be necessary to assume
stability of the Einstein geometry. Further, for our analysis it is
necessary to have a smooth moduli space. Fortunately, all known
examples of negative Einstein geometries satisfy these conditions.
We give examples of families of Einstein geometries which have
nontrivial moduli spaces. A product construction allows one to
generate new families of examples.

Our results demonstrate causal geodesic completeness of the perturbed spacetimes, in the expanding direction, and show that the scale-free geometry converges toward an element in the moduli space of Einstein geometries, with a rate of decay depending on the stability properties of the Einstein geometry.

#### Article information

**Source**

J. Differential Geom., Volume 89, Number 1 (2011), 1-47.

**Dates**

First available in Project Euclid: 21 December 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.jdg/1324476750

**Digital Object Identifier**

doi:10.4310/jdg/1324476750

**Mathematical Reviews number (MathSciNet)**

MR2863911

**Zentralblatt MATH identifier**

1256.53035

#### Citation

Andersson, Lars; Moncrief, Vincent. Einstein spaces as attractors for the Einstein flow. J. Differential Geom. 89 (2011), no. 1, 1--47. doi:10.4310/jdg/1324476750. https://projecteuclid.org/euclid.jdg/1324476750