The time evolution of marginally trapped surfaces

In previous work, we have shown the existence of a dynamical horizon or marginally outer trapped tube (MOTT) containing a given strictly stable marginally outer trapped surface (MOTS). In this paper, we show some results on the global behavior of MOTTs assuming the null energy condition. In particular, we show that MOTSs persist in the sense that every Cauchy surface in the future of a given Cauchy surface containing a MOTS also must contain a MOTS. We describe a situation where the evolving outermost MOTS must jump during the coalescence of two separate MOTSs. We furthermore characterize the behavior of MOTSs in the case that the principal eigenvalue vanishes under a genericity assumption. This leads to a regularity result for the tube of outermost MOTSs under the genericity assumption. This tube is then smooth up to finitely many jump times. Finally we discuss the relation of MOTSs to singularities of a spacetime.