Completeness Properties of Sobolev Metrics on the Space of Curves

TL;DR

This paper investigates the completeness of Sobolev metrics on the space of immersed and unparametrized curves, establishing conditions under which these metrics are complete and geodesics exist.

Contribution

It proves that Sobolev metrics of order n≥2 are metrically complete on Sobolev immersion spaces and that the shape space of unparametrized curves forms a complete length space.

Findings

Sobolev metrics of order n≥2 are metrically complete.

Any two curves in the same connected component can be joined by a minimizing geodesic.

Shape space of unparametrized curves is a complete length space.

Abstract

We study completeness properties of Sobolev metrics on the space of immersed curves and on the shape space of unparametrized curves. We show that Sobolev metrics of order $n\geq 2$ are metrically complete on the space $\mathcal I^n(S^1,\mathbb R^d)$ of Sobolev immersions of the same regularity and that any two curves in the same connected component can be joined by a minimizing geodesic. These results then imply that the shape space of unparametrized curves has the structure of a complete length space.