A Scale-Critical Trapped Surface Formation Criterion: A New Proof via Signature for Decay Rates

Abstract

We provide a simple, self-contained proof of a trapped surface formation theorem that sharpens the previous results both of Christodoulou and An-Luk. Our argument is based on a systematic extension of the scale-critical arguments in An-Luk, to connect Christodoulou’s short-pulse method and Klainerman-Rodnianski’s signature counting argument to the peeling properties previously used in small-data results such as Klainerman-Nicolo. This in particular allows us to avoid elliptic estimates and geometric renormalizations, and gives us our new technical simplifications.