Asymptotically Kasner-like singularities

abstract: We prove existence, uniqueness and regularity of solutions to the Einstein vacuum equations taking form $$ {}^{(4)}g = -dt^2 + \sum_{i,j=1}^3 a_{ij}t^{2 p_{\max\{i,j\}}}\,{\rm d} x^i\,{\rm x^j on $(0,T]_t\times\Bbb{T}^3_x$, where $a_{ij}(t,x)$ $p_i(x)$ are regular functions without symmetry or analyticity assumptions. These metrics singular asymptotically Kasner-like as $t\to 0^+$. expected be highly non-generic, our construction can viewed solving a initial value problem with Fuchsian-type analysis data posed ``singular hypersurface'' $\{t=0\}$. This is first such result imposing analyticity. To carry out analysis, we study in synchronized coordinate system. In particular, introduce novel way perform (weighted) energy estimates system based estimating second fundamental forms constant-$t$ hypersurfaces.