On a Complete Riemannian Metric on the Space of Embedded Curves

TL;DR

This paper introduces a new complete Riemannian metric on the space of embedded curves, ensuring well-posedness of geodesics and compactness properties, with applications to tangent-point energies.

Contribution

It defines a novel strong Riemannian metric on embedded curves that guarantees completeness and smoothness properties, advancing geometric analysis of curve spaces.

Findings

Bounded sets are relatively compact in the weak $H^s$ topology.

Every Cauchy sequence with respect to the geodesic distance converges.

Geodesic initial-value problems have solutions for all times.

Abstract

We propose a new strong Riemannian metric on the manifold of (parametrized) embedded curves of regularity $H^s$, $s\in(3/2,2)$. We highlight its close relationship to the (generalized) tangent-point energies and employ it to show that this metric is complete in the following senses: (i) bounded sets are relatively compact with respect to the weak $H^s$ topology; (ii) every Cauchy sequence with respect to the induced geodesic distance converges; (iii) solutions of the geodesic initial-value problem exist for all times; and (iv) there are length-minimizing geodesics between every pair of curves in the same path component (i.e., in the same knot class). As a by-product, we show $C^\infty$-smoothness of the tangent-point energies in the Hilbert case.