Second variation of the Helfrich-Canham Hamiltonian and reparametrization invariance

TL;DR

This paper develops a covariant framework to analyze first and second variations of the Helfrich-Canham Hamiltonian, clarifying the roles of tangential and normal deformations and their relation to reparametrization invariance in membrane models.

Contribution

It introduces a covariant method to compute second variations of membrane energy, explicitly distinguishing tangential deformations from reparametrizations and analyzing their coupling.

Findings

Derived second-order expansion of geometrical invariants.

Clarified the distinction between tangential deformations and reparametrizations.

Provided formulas applicable to general membrane models.

Abstract

A covariant approach towards a theory of deformations is developed to examine both the first and second variation of the Helfrich-Canham Hamiltonian -- quadratic in the extrinsic curvature -- which describes fluid vesicles at mesoscopic scales. Deformations are decomposed into tangential and normal components; At first order, tangential deformations may always be identified with a reparametrization; at second order, they differ. The relationship between tangential deformations and reparametrizations, as well as the coupling between tangential and normal deformations, is examined at this order for both the metric and the extrinsic curvature tensors. Expressions for the expansion to second order in deformations of geometrical invariants constructed with these tensors are obtained; in particular, the expansion of the Hamiltonian to this order about an equilibrium is considered. Our approach applies as well to any geometrical model for membranes.