Title: Completeness properties of spaces of immersed curves

Abstract: When comparing 2D shapes, one way of doing so is to find an appropriate metric on the space of their boundaries, and to study the geodesic flow from one boundary to the other. We will describe a natural family of such Riemannian metrics of the (infinite-dimensional) space of immersed curves, and discuss their completeness properties: does the geodesic flow exist for all time? Can two shapes be connected by a minimal geodesic? We will see how the answer depends on the metric that we choose, and whether we consider closed or open curves.
Based on a joint work with Martin Bauer.