Constrained metric variations and emergent equilibrium surfaces

• New variational framework for examining surface energies is presented.
• No explicit reference is made to the embedding of surface in the ambient space.
• Gauss–Codazzi–Mainardi equations are enforced using local Lagrange multipliers.
• Surfaces themselves are emergent entities.
• The divergence of the multipliers signals an instability of equilibrium surfaces.

Any surface is completely characterized by a metric and a symmetric tensor satisfying the Gauss–Codazzi–Mainardi equations (GCM), which identifies the latter as its curvature. We demonstrate that physical questions relating to a surface described by any Hamiltonian involving only surface degrees of freedom can be phrased completely in terms of these tensors without explicit reference to the ambient space: the surface is an emergent entity. Lagrange multipliers are introduced to impose GCM as constraints on these variables and equations describing stationary surface states derived. The behavior of these multipliers is explored for minimal surfaces, showing how their singularities correlate with surface instabilities.