  We present a generic condition for Lorentzian manifolds to have a barrier
that limits the reach of boundary-anchored extremal surfaces of arbitrary
dimension. We show that any surface with nonpositive extrinsic curvature is a
barrier, in the sense that extremal surfaces cannot be continuously deformed
past it. Furthermore, the outermost barrier surface has nonnegative extrinsic
curvature. Under certain conditions, we show that the existence of trapped
surfaces implies a barrier, and conversely. In the context of AdS/CFT, these
barriers imply that it is impossible to reconstruct the entire bulk using
extremal surfaces. We comment on the implications for the firewall controversy.
