Marginally trapped surfaces in a perturbed Schwarzschild spacetime

The concept of a marginally trapped surface is important in the theory of general relativity. In the Schwarzschild black hole spacetime, its event horizon is foliated by marginally trapped surfaces. In a more general black hole spacetime, the concept of a marginally trapped surface is closely related to various sorts of horizon, for example, the apparent horizon, the trapping boundary, the isolated horizon and the dynamical horizon. In this paper, we study the set of marginally trapped surfaces in a perturbed Schwarzschild spacetime. We show that for every incoming null hypersurface which is nearly spherically symmetric, there exists a unique embedded marginally trapped surface. In order to prove this result, we develop a general method to study the geometry of spacelike surfaces in a double null coordinate system, which can be applied to study other problems for spacelike surfaces in a Lorentzian manifold.