Energy estimates and gravitational collapse

Abstract. In this note, we study energy estimates for Einstein vacuum equations in order to prove the formation of black holes along evolutions. The novelty of the paper is that, we completely avoid using rotation vector fields to establish the global exis-tence theorem of the solution. More precisely, we use only canonical null directions as commutators to derive energy estimates at the level of one derivatives of null curvature components. We show that, thanks to the special cancelations coming from the null structure of non-linear interactions, desirable estimates on curvatures can be derived under the short pulse ansatz due to Klainerman and Rodnianski [4] (which is originall