Metric variations become a surface

Any surface is completely characterized by a metric and a symmetric tensor field satisfying the Gauss-Codazzi-Mainardi equations (GCM). These equations identify the latter tensor as the extrinsic curvature. It should thus be possible, in principle, to phrase physical questions relating to a surface described by an action or Hamiltonian involving only surface degrees of freedom--be it a soap film or a relativistic string--in terms of a metric on a Riemannian manifold, coupling to a second independent symmetric tensor, without any explicit reference to the environment into which it is embedded: the surface itself arises as an emergent entity. Introducing Lagrange multipliers to impose the GCM equations as constraints on these variables, we demonstrate how the Euler-Lagrange equation describing the stationary states of any two-dimensional surface Hamiltonian is derived. The behavior of the multipliers is explored in detail for area minimizing surfaces and their connection to the instabilities of such surfaces pointed out.