Asymptotically Kasner-like singularities

abstract: We prove existence, uniqueness and regularity of solutions to the Einstein vacuum equations taking the form ^ (4) g = -dt² + ₈, ₉=₁³ a₈₉t^2 p\{₈, ₉\}\, d xⁱ\, d xʲ on (0, T]ₜ³ₓ, where a₈₉ (t, x) and pᵢ (x) are regular functions without symmetry or analyticity assumptions. These metrics are singular and asymptotically Kasner-like as t 0^+. These solutions are expected to be highly non-generic, and our construction can be viewed as solving a singular initial value problem with Fuchsian-type analysis where the data are posed on the ``singular hypersurface'' \t=0\. This is the first such result without imposing symmetry or analyticity. To carry out the analysis, we study the problem in a synchronized coordinate system. In particular, we introduce a novel way to perform (weighted) energy estimates in such a coordinate system based on estimating the second fundamental forms of the constant-t hypersurfaces.