Inflection Points, Extatic Points and Curve Shortening

As the name suggests, Curve Shortening is a gradientflow for the length functional on the space of immersed curves in the surfaceM. One can therefore try to use Curve Shortening to prove existence of geodesics by variational methods. In my talk at S’Agarro I observed that geodesics always are curves without self-tangencies, and recalled that the space of such curves has many different connected components. I then discussed how one can try to exploit the nice behaviour of Curve Shortening with respect to self-intersections to prove existence of geodesics in each component.