Focussed Solutions to the Einstein Vacuum Equations

We construct solutions to the Einstein vacuum equations in polarised translational symmetry in $3 + 1$ dimensions which have $H^1$ energy concentrated in an arbitrarily small region around a two-dimensional null plane and large $H^2$ initial data. Specifically, there is a parameter $k$ and coordinates $s$, $x$, $v$, $y$ such that the null plane is given by $x = k^{-1/2}/2$, $v = T\sqrt{2} - k^{-1}/2$ for some $T$ independent of $k$, the $H^1$ energy of the solution is concentrated on the region $[0, T'] \times [0, k^{-1/2}] \times [T\sqrt{2} - k^{-1}, T\sqrt{2}] \times \R^1$, and the $H^2$ norm of the initial data is bounded below by a multiple of $k^{3/4}$. The time $T'$ has a lower bound independent of $k$. This result relies heavily on a new existence theorem for the Einstein vacuum equations with characteristic initial data which is large in $H^2$. This result is proved using parabolically scaled coordinates in a null geodesic gauge.Ph.D