Interaction between the bilayer shape and surface flow is important for capturing the flow of lipids in many biological membranes. Recent microscopy evidence has shown that minimal surfaces (planes, catenoids, and helicoids) occur often in cellular membranes. In this study, we explore lipid flow in these geometries using a ‘stream function’ formulation for viscoelastic lipid bilayers. Using this formulation, we derive two-dimensional lipid flow equations for the commonly occurring minimal surfaces in lipid bilayers. We show that for three minimal surfaces (planes, catenoids, and helicoids), the surface flow equations satisfy Stokes flow equations. In helicoids and catenoids, we show that the tangential velocity field is a Killing vector field. Thus, our analysis provides fundamental insight into the flow patterns of lipids on intracellular organelle membranes that are characterized by fixed shapes reminiscent of minimal surfaces.