Trapped submanifolds into the light cone

Abstract: The concept of trapped surfaces was originally formulated by Penrose in his seminal paper published in 1965, very recently laureated with the Nobel Prize in Physics 2020. In that paper Penrose introduced 2-dimensional trapped surfaces in 4-dimensional spacetimes in terms of the signs or the vanishing of the so-called null expansions. This is obviously related to the causal orientation of the mean curvature vector of the surface, which provides a better and powerful characterization of the trapped surfaces and allows the generalization of this concept to codimension two spacelike submanifolds of arbitrary dimension.

In this lecture we consider codimension two trapped submanifolds contained into the light cone of de Sitter spacetime and into the light cone of the Lorentz-Minkowski spacetime. In particular, for the case of compact submanifolds into the light cone of de Sitter space, we show that they are conformally diffeomorphic to the round sphere. This fact enables us to deduce that the problem of characterizing compact marginally trapped submanifolds into the light cone is equivalent to solving the Yamabe problem on the round sphere, allowing us to obtain our main classification result for such submanifolds. As for the case of submanifolds into the light cone of the Lorentz-Minkowski space, we obtain a non-existence result for complete, non-compact, weakly trapped submanifolds.

These results have been obtained in collaboration with Verónica L. Cánovas (from Universidad de Murcia) and Marco Rigoli (from Università degli Studi di Milano), and they can be found in the following papers: Trapped submanifolds contained into a null hypersurface of de Sitter spacetime, Commun. Contemp. Math. 20 (2018), no. 8, 1750059, 30 pp.; and Codimension two spacelike submanifolds of the Lorentz-Minkowski spacetime into the light cone, Proc. Roy. Soc. Edinburgh Sect. A 149 (2019), no. 6, 1523-1553.

PortugueseMathematics

Audience: general audience

Seminários de Matemática da UFPB