We set up the singular initial value problem for quasilinear hyperbolic Fuchsian systems of first order and establish an existence and uniqueness theory for this problem with smooth data and smooth coefficients (and with even lower regularity). We apply this theory in order to show the existence of smooth (generally not analytic) T 2 -symmetric solutions to the vacuum Einstein equations, which exhibit AVTD (asymptotically velocity term dominated) behavior in the neighborhood of their singularities and are polarized or half-polarized.
We prove in detail a theorem which completes the evaluation and parametrization of the space of constant mean curvature (CMC) solutions of the Einstein constraint equations on a closed manifold. This theorem determines which sets of CMC conformal data allow the constraint equations to be solved, and which sets of such data do not. The tools we describe and use here to prove these results have also been found to be useful for the study of non-constant mean curvature solutions of the Einstein constraints.
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