Absolute flux optimising curves of flows on a surface

Given a flow on a surface, we consider the problem of connecting two distinct trajectories by a curve of extremal (absolute) instantaneous flux. We develop a complete classification of flux optimal curves, accounting for the possibility of the flux having spatially and temporally varying weight. This weight enables modelling the flux of non-equilibrium distributions of tracer particles, pollution concentrations, or active scalar fields such as vorticity. Our results are applicable to all smooth autonomous flows, area preserving or not.