Philipp Reiter - On a complete Riemannian metric on the space of closed embedded curves

joint work with Elias Döhrer and Henrik Schumacher (Chemnitz University of
Technology / Univ. of Georgia)

In pursuit of choosing optimal paths in the manifold of closed embedded
space curves we introduce a Riemannian metric which is inspired by a
self-contact avoiding functional, namely the tangent-point potential. The
latter blows up if an embedding degenerates which yields infinite barriers
between different 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.