Shape of a Membrane on a Liquid Interface with Arbitrary Curvatures

We study the deformation of a liquid interface with arbitrary principal curvatures by a flat circular sheet. Working first at small slopes, we determine the shape of the sheet analytically in the membrane limit, where the sheet is inextensible yet free to bend and compress. We find that the sheet takes a cylindrical shape on interfaces with negative Gaussian curvature. On interfaces with positive Gaussian curvature, an inner region still adopts a cylindrical shape while the outer region is under azimuthal compression. Numerical energy minimization confirms our predictions and shows that this behavior holds for finite slopes. Experiments on a thin polystyrene film at an anisotropic air-water interface show consistent behaviors.