Spinor representation of surfaces and complex stresses on membranes and interfaces

TL;DR

This paper develops a spinor-based variational framework to analyze membrane and interface equilibrium, generalizing minimal surface representations to include non-zero mean curvature, with applications to fluid membrane models.

Contribution

It introduces a novel spinor representation for surface geometry that relaxes the minimal surface constraint, enabling new analysis of membrane stresses and energies.

Findings

Derived a spinor-based variational principle for membranes.

Constructed a conserved complex stress tensor for fluid membranes.

Applied the framework to the Canham-Helfrich energy model.

Abstract

Variational principles are developed within the framework of a spinor representation of the surface geometry to examine the equilibrium properties of a membrane or interface. This is a far-reaching generalization of the Weierstrass-Enneper representation for minimal surfaces, introduced by mathematicians in the nineties, permitting the relaxation of the vanishing mean curvature constraint. In this representation the surface geometry is described by a spinor field, satisfying a two-dimensional Dirac equation, coupled through a potential associated with the mean curvature. As an application, the mesoscopic model for a fluid membrane as a surface described by the Canham-Helfrich energy quadratic in the mean curvature is examined. An explicit construction is provided of the conserved complex-valued stress tensor characterizing this surface.