Asymptotically Kasner-like singularities

Abstract

We prove existence, uniqueness and regularity of solutions to the Einstein vacuum equations taking the form $${}^{(4)}g=-d{t}^2+\sum _{i,j=1}^3{a}_{ij}{t}^{2p_{\max \{i,j\}}}\mathrm{d}{x}^i\mathrm{d}{x}^j$$ on $(0,T{]}_t\times {\mathbb{T}}_x^3$, where $a_{ij}(t,x)$ and $p_i(x)$ are regular functions without symmetry or analyticity assumptions. These metrics are singular and asymptotically Kasner-like as $t\to 0^{+}$. 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.

57 pages; minor corrections

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