Vortices on curved surfaces

Topological defects in thin films coating a deformed substrate interact with the underlying curvature. This coupling mechanism influences the shape of biological structures and provides a new strategy for the design of interfaces with prescribed functionality. In this article, a mathematical formalism based on the method of conformal mapping that is presented permits the calculation of the energetics of disclinations, dislocations, and vortices on rigid substrates of spatially varying Gaussian curvature. Special emphasis is placed on determining the geometric force exerted on vortices in curved superfluid films. This force, which attracts (repels) vortices towards regions of negative (positive) Gaussian curvature, is an illustration of how material shape can influence quantum mechanical degrees of freedom.