Applied Analysis Seminar On a complete Riemannian metric on the space of closed embedded curves

In order to find optimal paths in the manifold of closed embedded space curves we define a Riemannian metric which is inspired by the tangent-point potential. This functional prevents self-contact by creating infinite barriers between different isotopy classes. More precisely, it blows up on sequences of embedded curves converging to a non-embedded limit curve. 

The Hopf—Rinow theorem states that, for finite-dimensional Riemannian manifolds, 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. As the manifold of closed embedded curves is infinite dimensional, we will separately prove that our metric satisfies all those four assertions. 

This is joint work with Elias Döhrer and Henrik Schumacher (Chemnitz University of Technology / RWTH Aachen University).