Flames and other interfaces can be shown theoretically to have curvature-dependent propagation speeds that do not necessarily vary monotonically. Such behaviour has a number of consequences for the nature of any overall flame propagation, including a suggestion that a propagating interface might separate into different `phases' of curvature. In studying a blend of slowly varying and nearly equidiffusonal properties of a flame near stoichiometry, a high-order evolution equation is suggested. This is enormously complicated, at first sight, and some of the questions raised will be addressed. It shows, for example, that curvature-dependence may not only be non-monotonic but also multi-valued with bounded support.