Statistical Mechanics of Doubly Curved Membranes

Doubly curved surfaces, characterized by two principal curvatures, are widely present in biophysics and materials science, such as erythrocyte membranes, endoplasmic reticula, and bicontinuous nanoporous graphene. The effective mechanical properties of these intrinsically curved surfaces can be renormalized by thermal fluctuations, akin to how flat solid membranes exhibit scale-dependent increased bending rigidity and reduced in-plane elastic moduli. The intrinsic curvature of curved surfaces introduces additional interactions in the free energy, potentially leading to different renormalized behavior compared to flat membranes. For example, thermal fluctuations induce an effective compressive stress in spherical shells, which can crush the shell if the diameter is sufficiently large.



This work focuses on a more general case of doubly curved shells with arbitrary curvatures. We investigate how thermal fluctuations renormalize the mechanical properties of the curved membrane and how distinct signs of the Gaussian curvature give rise to different physical phenomena. These findings are expected to provide theoretical guidance for understanding the statistical mechanics of a wide variety of curved biological membranes, as well as for the design of novel two-dimensional materials.