  The dynamics of surfaces and interfaces describe many physical systems,
including fluid membranes, entanglement entropy and the coupling of defects to
quantum field theories. Based on the formulation of submanifold calculus
developed by Carter, we introduce a new variational principle for (entangling)
surfaces. This principle captures all diffeomorphism constraints on
surface/interface actions and their associated spacetime stress tensor. The
different couplings to the geometric tensors appearing in the surface action
are interpreted in terms of response coefficients within elasticity theory. An
example of a surface action with edges at the two-derivative level is studied,
including both the parity-even and parity-odd sectors. Its conformally
invariant counterpart restricts the type of conformal anomalies that can appear
in two-dimensional submanifolds with boundaries. Analogously to hydrodynamics,
it is shown that classification methods can be used to constrain the stress
tensor of (entangling) surfaces at a given order in derivatives. This analysis
reveals a purely geometric parity-odd contribution to the Young modulus of a
thin elastic membrane. Extending this novel variational principle to BCFTs and
DCFTs in curved spacetimes allows to obtain the Ward identities for
diffeomorphism and Weyl transformations. In this context, we provide a formal
derivation of the contact terms in the stress tensor and of the displacement
operator for a broad class of actions.
