Polymers in curved boxes

We apply results derived in other contexts for the spectrum of the Laplace operator in curved geometries to the study of an ideal polymer chain confined to a spherical annulus in arbitrary space dimension D and conclude that the free energy compared to its value for an uncurved box of the same thickness and volume is lower when D < 3, stays the same when D = 3, and is higher when D > 3. Thus, confining an ideal polymer chain to a cylindrical shell lowers the effective bending elasticity of the walls and might induce spontaneous symmetry breaking, i.e. bending. (Actually, the above mentioned results show that any shell in D = 3 induces this effect, except for a spherical shell.) We compute the contribution of this effect to the bending rigidities in the Helfrich free energy expression.