Topological dynamics and morphogenesis

The dynamic morphogenesis of surfaces in biology is a common theme with variations, and persists across multiple scales, and includes the mechanics of sub-cellular vesicles, the dynamics of visceral organs, tissue organoids and whole organisms such as Hydra. Since morphogenesis involves changes in geometry and topology, a natural question is that of an appropriate mathematical description. Here, we suggest a minimal generalization of Ricci flow by coupling it to a scalar field, e.g. a morphogen, that allows us to consider the appearance and disappearance of holes. We prove the conditions under which topological changes can occur, and provide explicit solutions in simple axisymmetric geometries associated with topological transitions realized as finite time singularities.