We discuss a construction of polarized $U(1)$ space-times of a “fixed size”, where the initial data contains a given amount $\eta\gt 0$ of incoming gravitational energy which is allowed to be supported at arbitrarily small scales, without affecting the time of existence. More specialized examples can be constructed where any chosen fraction of this initial incoming energy can stay focused along a null geodesic, in the $U(1)$ picture. We discuss the relationship of this result with one direction of Thorne’s hoop conjecture. The work relies on a special structure of the Einstein equations in this symmetry class, a new re-scaling of the equations and large-time control, along with a new local Klainerman-Sobolev estimate. Time permitting, some comparisons of the result and the methods with large-data constructions for the Einstein equations over the last decade will also be presented.