Constrained metric variations and emergent equilibrium surfaces

TL;DR

This paper reformulates surface physics problems using metric and curvature tensors, deriving equilibrium conditions and analyzing instabilities without referencing the surrounding space.

Contribution

It introduces a tensor-based framework for surface analysis, deriving stationary states and stability conditions purely from surface degrees of freedom.

Findings

Lagrange multipliers enforce GCM constraints on surface tensors.

Singularities in multipliers relate to surface instabilities.

Framework applies to minimal surfaces and their stability analysis.

Abstract

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.