Localization of thermal fluctuations on membranes with complex geometry*

The geometry of elastic membranes in their stress-free state affects both their mechanics and the spatial distribution of their thermal fluctuations. This effect results from the fact that membrane curvature introduces a geometric coupling between the soft out-of-plane bending modes and stiffer, in-plane elastic deformation. As a result, curved membranes have complex distributions of thermal fluctuations in equilibrium. Using shallow shell theory combined with equilibrium statistical physics, we demonstrate that thermalized membranes containing regions of negative Gaussian curvature develop anomalously large fluctuations. Moreover, the existence of special curves, “singular lines,” leads to a breakdown of linear membrane mechanics. These singular lines effectively partition the membrane into regions whose fluctuations are only weakly coupled. We test these predictions using high-resolution microscopy of human red blood cells as a case study. Our observations show geometry-dependent localization of thermal fluctuations demonstrating the important role of geometry in membrane mechanics and fluctuations.