Formation of Trapped Surfaces in Geodesic Foliation

We revisit the classical results of the formation of trapped surfaces for the Einstein vacuum equation relying on the geodesic foliation, rather than the double null foliation used in all previous results, starting with the seminal work of Christodoulou (Christodoulou, D.: European Mathematical Society, 2009) and continued in (An, X.: Ann. of Math. 201(3), 775–908 2025; An, X.: Ann. PDE 8, 3 2022; An, X. and Luk, J.: Adv. Theor. Math. Phys. 21(1), 1–120 2017; Klainerman, S.,et al.: Invent. Math. 198(1), 1–26 2014; Klainerman, S. and Rodnianski, I.: Acta Math. 208, 211–333 2012). The main advantage of the method is that it only requires information on the incoming curvature along the incoming initial null hypersurface, which is more along the lines of (Christodoulou, D.: Commun. Pure Appl. Math. 44(3), 339–373 1991) on the formation of trapped surfaces in spherical symmetry. Therefore, the methods used here may be better suited for studying the Weak Cosmic Censorship conjecture in the spirit of (Christodoulou, D.: Ann. Math. 183–217, 149 1999). Another important advantage, which we plan to bring to fruition in a forthcoming paper, is that it is appropriate to the study of the formation of trapped surfaces from more general Cauchy data than treated in (Li, J. and Yu, P.: Ann. of Math. 181, 699–768 2015). Our paper is based on a version of the non-integrable PT frame introduced in (Giorgi, E.et al.: Pure Appl. Math. Q. 20(7), 2865–3849 2024) and (KS: Kerr Klainerman, S. and Szeftel, J.: Pure Appl. Math. Q. 19(3), 791–1678 2023), associated to the geodesic foliation.