  In this paper, we study the "minimal requirement" on the incoming radiation
that guarantees a trapped surface to form in vacuum. First, we extend the
region of existence in Christodoulou's theorem on the formation of trapped
surfaces and consequently show that the lower bound required to form a trapped
surface can be relaxed. Second, we demonstrate that trapped surfaces form
dynamically from a class of initial data which are large merely in a
scaling-critical norm. This result is motivated in part by the scaling in
Christodoulou's formation of trapped surfaces theorem for the Einstein-scalar
field system in spherical symmetry.
