  It was shown recently, building on work of Alexakis, Balehowksy, and Nachman
that the geometry of (some portion of) a manifold with boundary is uniquely
fixed by the areas of a foliation of two-dimensional disk-shaped surfaces
anchored to the boundary. In the context of AdS/CFT, this implies that (a
portion of) a four-dimensional bulk geometry can be fixed uniquely from the
entanglement entropies of disk-shaped boundary regions, subject to several
constraints. In this Note, we loosen some of these constraints, in particular
allowing for the bulk foliation of extremal surfaces to be local and removing
the constraint of disk topology; these generalizations ensure uniqueness of
more of the deep bulk geometry by allowing for e.g. surfaces anchored on
disconnected asymptotic boundaries, or HRT surfaces past a phase transition. We
also explore in more depth the generality of the local foliation requirement,
showing that even in a highly dynamical geometry like AdS-Vaidya it is
satisfied.
