Philipp Reiter (TU Chemnitz):
On a Complete Riemannian Metric on the Space of Closed Embedded Curves
Time and place
Tuesday, 29.10.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Abstract
On a Complete Riemannian Metric on the Space of Closed Embedded Curves\njoint work with Elias Döhrer and Henrik Schumacher (Chemnitz University\nof Technology / Univ. of Georgia)\n\nIn pursuit of choosing optimal paths in the manifold of closed embedded\nspace curves we introduce a Riemannian metric which is inspired by a\nself-contact avoiding functional, namely the tangent-point potential.\nThe latter blows up if an embedding degenerates which yields infinite\nbarriers between different isotopy classes.\n\nFor finite-dimensional Riemannian manifolds the Hopf—Rinow theorem\nstates that the Heine—Borel property (bounded sets are relatively\ncompact), geodesic completeness (long-time existence of geodesic\nshooting), and metric completeness of the geodesic distance are\nequivalent. Moreover, it states that existence of length-minimizing\ngeodesics follows from each of these statements. Albeit the Hopf—Rinow\ntheorem does not hold true in this generality for infinite-dimensional\nRiemannian manifolds, we can prove all its four assertions for a\nsuitably chosen Riemannian metric on the space of closed embedded\ncurves.\n