Prof. Dr. Julian Scheuer

Title: The gradient flow of the elastic and p-elastic energy for curves and fractional versions thereof
Abstract: During the last couple of years a surprising large number of new research papers appeared dealing with the elastic energy of curves and similar geometric equations.
In this survey talk we give an overview over new and old results in the context of gradient flows of the p-elastic energy for curves and fractional versions thereof. We discuss both, $L^2$ gradient flows and the more recently studied $H^2$-gradient flows for these elastic energies and try to highlight the main differences between the resulting evolution equations.

Title: Towards geodesic completeness of spaces of embedded curves joint work with Elias Döhrer and Henrik Schumacher (Chemnitz University of Technology)
Abstract: In pursuit of finding optimal paths in the manifold of closed embedded space curves we study a Riemannian metric which is inspired by the tangent-point energy. The latter is a self-repulsive functional defined on the space of curves which blows up if an embedding degenerates. Thereby it erects infinite barriers between distinct isotopy classes. For finite-dimensional Riemannian manifolds the Hopf—Rinow theorem states that the Heine—Borel property (bounded sets are relatively compact), geodesic completeness (long-time existence of geodesic shooting), and metric completeness of the geodesic distance are equivalent. Moreover, it states that existence of length-minimizing geodesics follows from each of these statements. Albeit the Hopf—Rinow theorem does not hold true in this generality for infinite-dimensional Riemannian manifolds, we can prove all its four assertions for a suitably chosen Riemannian metric on the space of closed embedded curves.