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 T^{3}-Gowdy case,
which were derived in [40]. In [54], Fuchsian methods were then used to prove their existence. With respect
to our parametrization of the metric, the expansions take the form

8.1 Geometric interpretation of v_{a}

8.2 Restriction on the velocity

8.3 Geodesic loop

8.4 Existence of expansions using Fuchsian methods, T^{3}-case

8.5 Existence of expansions using Fuchsian methods, S^{2} × S^{1}
and S^{3} cases

8.2 Restriction on the velocity

8.3 Geodesic loop

8.4 Existence of expansions using Fuchsian methods, T

8.5 Existence of expansions using Fuchsian methods, S

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