  Motivated by the power of subregion/subregion duality for constraining the
bulk geometry in gauge/gravity duality, we pursue a comprehensive and
systematic approach to the behavior of extremal surfaces under perturbations.
Specifically, we consider modifications to their boundary conditions, to the
bulk metric, and to bulk quantum matter fields. We present a unified framework
for treating such perturbations for classical extremal surfaces, classify some
of their stability properties, and develop new technology to extend our
treatment to quantum extremal surfaces, culminating in an "equation of quantum
extremal deviation". The power of this formalism stems from its ability to map
geometric statements into the language of elliptic operators; to illustrate, we
show that various a priori disparate bulk constraints all follow from basic
consistency of subregion/subregion duality. These include familiar properties
such as (smeared) versions of the quantum focusing conjecture and the
generalized second law, as well as new constraints on (i) metric and matter
perturbations in spacetimes close to vacuum and (ii) the bulk stress tensor in
generic (not necessary close to vacuum) spacetimes. This latter constraint is
highly reminiscent of a quantum energy inequality.
