The one-loop elastic coefficients for the Helfrich membrane in higher dimensions

TL;DR

This paper derives the one-loop elastic coefficients for a Helfrich membrane in higher dimensions using a covariant geometric approach, extending understanding of membrane mechanics beyond three dimensions.

Contribution

It introduces a covariant method to compute effective bending couplings for membranes embedded in higher-dimensional spaces, generalizing previous 3D results.

Findings

Thermal fluctuations soften the membrane in higher dimensions.

Derived explicit formulas for elastic coefficients in higher-dimensional embeddings.

Confirmed consistency with known 3D membrane behavior.

Abstract

Using a covariant geometric approach we obtain the effective bending couplings for a 2-dimensional rigid membrane embedded into a $(2+D)$-dimensional Euclidean space. The Hamiltonian for the membrane has three terms: The first one is quadratic in its mean extrinsic curvature. The second one is proportional to its Gaussian curvature, and the last one is proportional to its area. The results we obtain are in agreement with those finding that thermal fluctuations soften the 2-dimensional membrane embedded into a 3-dimensional Euclidean space.