Helfrich-Canham bending energy as a constrained non-linear sigma model

TL;DR

This paper links the Helfrich-Canham bending energy to a constrained non-linear sigma model, revealing how an orthogonality constraint affects stress tensor conservation and shape equations in surface physics.

Contribution

It introduces a novel interpretation of bending energy as a constrained sigma model, clarifying the role of orthogonality constraints in stress tensor conservation.

Findings

Conserved stress tensor reproduces the shape equation.

Orthogonality constraint introduces a source in stress divergence.

Identification of the bending energy with a non-linear sigma model.

Abstract

The Helfrich-Canham bending energy is identified with a non-linear sigma model for a unit vector. The identification, however, is dependent on one additional constraint: that the unit vector be constrained to lie orthogonal to the surface. The presence of this constraint adds a source to the divergence of the stress tensor for this vector so that it is not conserved. The stress tensor which is conserved is identified and its conservation shown to reproduce the correct shape equation.