On r-trapped immersions in Lorentzian spacetimes and a weighted inequality for tensors

Abstract

This work was divided into two moments: at first, we set out to study spacelike sub manifolds Σn immersed in Lorentz spacetimes Mn+p+1. So, we introduce the notion of r-trapped submanifolds as a generalization of the trapped submanifolds introduced by Penrose. In the case where the ambient space is a GRW −I ×ρ Mn+p, considering some properties such as parabolicity and stochastic completeness we prove rigidity and nonexistence results for r-trapped in some configurations of GRW spacetimes and, lastly, we provide examples of r-trapped submanifolds, some of them are also simultaneously trapped, but we provided examples proving that the notion of r-trapped submanifolds are different accordingly to the number r. On the other hand, in the case where the ambient space is an standard static spacetime (SSST) Mn+p ×ρ R1, we calculate the differential operators Lr and Lr,φ applied to the height function h = πR ◦ψ of the immersion ψ : Σn → Mn+p ×ρ R1 and we consider some properties on Σn such as parabolicity and maximum principles. In this setting, we prove rigidity and nonexistence results for r-trapped spacelike submanifolds. After, we obtain some De Lellis-Topping type inequalities for general tensors under constraints in the Bakry-Émery Ricci tensor. In particular, we provide new results on manifolds with convex boundary, improving some known results given on manifolds with totally geodesic boundary. Furthemore, we apply our results in a class of locally conserved tensors.