  It has been proposed that the areas of marginally trapped or anti-trapped
surfaces (also known as leaves of holographic screens) may encode some notion
of entropy. To connect this to AdS/CFT, we study the case of marginally trapped
surfaces anchored to an AdS boundary. We establish that such boundary-anchored
leaves lie between the causal and extremal surfaces defined by the anchor and
that they have area bounded below by that of the minimal extremal surface. This
suggests that the area of any leaf represents a coarse-grained von Neumann
entropy for the associated region of the dual CFT. We further demonstrate that
the leading area-divergence of a boundary-anchored marginally trapped surface
agrees with that for the associated extremal surface, though subleading
divergences generally differ. Finally, we generalize an argument of Bousso and
Engelhardt to show that holographic screens with all leaves anchored to the
same boundary set have leaf-areas that increase monotonically along the screen,
and we describe a construction through which this monotonicity can take the
more standard form of requiring entropy to increase with boundary time. This
construction is related to what one might call future causal holographic
information, which in such cases also provides an upper bound on the area of
the associated leaves.
