Inspired by proteins that generate membrane curvature, sense the underlying membrane geometry, and migrate driven by curvature gradients, we explore the question: Can colloids, adhered to lipid bilayers, also sense and respond to membrane geometry? We report the migration of Janus microparticles adhered to giant unilamellar vesicles elongated to present spatially varying curvatures. In our experiments, colloids migrate only when the membranes are tense, suggesting that they migrate to minimize membrane area. By determining the energy dissipated along a trajectory, the energy field is inferred to depend on the local deviatoric curvature, like curvature driven capillary migration on interfaces between immiscible fluids. In this latter system, energy gradients are larger, so colloids move deterministically, whereas the paths traced by colloids on vesicles have significant fluctuations. By addressing the role of Brownian motion, we show that the observed migration is analogous to curvature driven capillary migration, with membrane tension playing the role of interfacial tension. Since this motion is mediated by membrane shape, it can be turned on and off by dynamically deforming the vesicle. While particle-particle interactions on lipid membranes have been considered in many contributions, we report here an exciting and previously unexplored modality to actively direct the migration of colloids to desired locations on lipid bilayers.