### 5.4 Stability of the Bianchi type III form of flat spacetime

Another vacuum cosmological model whose nonlinear stability has been investigated is the Bianchi III
form of flat spacetime. To obtain this model, first do the construction described in Section 5.3 with the
difference that the starting solution is three-dimensional Minkowski space. Then, take the metric product of
the resulting three-dimensional Lorentz manifold with a circle. This defines a flat spacetime that has one
Killing vector, which is the generator of rotations of the circle. It has been shown by Choquet-Bruhat
and Moncrief [94] that this solution is stable under small vacuum perturbations preserving
the one-dimensional symmetry. More precisely, they proved the result only for the polarized
case. This restriction was lifted in [88]. As in the case of the Milne model, a natural task is to
generalize this result to spacetimes with suitable matter content. It has been generalized to the
Einstein-Maxwell-Higgs system in [89]. The reasons it is necessary to restrict to symmetric perturbations
in this analysis, in contrast to what happens with the Milne model, are discussed in detail
in [94].
One of the main techniques used is a method of modified energy estimates that is likely to be of more
general applicability. The Bel-Robinson tensor plays no role. The other main technique is based on the fact
that the problem under study is equivalent to the study of the 2 + 1-dimensional Einstein equations coupled
to a wave map (a scalar field in the polarized case). This helps to explain why the use of the
Dirichlet energy could be imported into this problem from the work of [12] on 2 + 1 vacuum
gravity.