Applied Analysis Seminar The Geodesic Curvature Flow with Forcing on 𝕊²

In this talk, we introduce the geodesic curvature flow with forcing as a geometric flow on the space of smooth immersed loops from the circle S¹ to a Riemannian sphere (S²,g). We begin with exploring the variational structure of closed but non-exact one-forms on manifolds, which lead to the study of gradient flows on covering spaces. Projections of the corresponding flow lines may exhibit periodic behavior, a trait which seems to contradict the nature of gradient flows. We then provide a brief physical motivation for the introduction of the flow, which is the question of existence of so-called magnetic geodesics. After mentioning a few properties of the flow such as short-time existence and short-time smoothing, which are inherited from its cousin, the mean curvature flow, we proceed to show that the flow is indeed the gradient flow of a branched functional defined on the space of smooth free immersions equipped with the strong C¹-Whitney-topology. If time permits, we take a glimpse at some original conjectures concerning the long-time behavior of the flow, based on simple numerical experiments.