8 Asymptotic Expansions Using Fuchsian Methods, General T3-Case

The analysis in the polarized case, which we outlined above, illustrates the importance of being able to compute asymptotic expansions. Thus, in analyzing the general case it is natural to begin by trying to carry out a similar computation. As we observed in Section 5.2, there are two different ways to proceed. One is to derive expansions given the solution. The other is to prove the existence of solutions with specified asymptotics. In the present section, we are concerned with the latter perspective.

The natural starting point for the exposition is the formal expansions in the general T3-Gowdy case, which were derived in [40]. In [54Jump To The Next Citation Point], Fuchsian methods were then used to prove their existence. With respect to our parametrization of the metric, the expansions take the form

P(τ,𝜃) = va(𝜃)τ + ϕ(𝜃) + u(τ,𝜃) (37 ) −2va(𝜃)τ Q(τ,𝜃) = q(𝜃) + e [ψ (𝜃) + w(τ,𝜃)], (38 )
where w,u → 0 as τ → ∞ and 0 < va(𝜃) < 1. Let us comment on a few aspects of these expansions.

 8.1 Geometric interpretation of va
 8.2 Restriction on the velocity
 8.3 Geodesic loop
 8.4 Existence of expansions using Fuchsian methods, T3-case
 8.5 Existence of expansions using Fuchsian methods, S2 × S1 and S3 cases

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