Completeness of Length-Weighted Sobolev Metrics on the Space of Curves

TL;DR

This paper establishes conditions under which Sobolev metrics with variable coefficients on the space of curves are complete, extending previous results from constant coefficient cases and providing examples of incomplete metrics.

Contribution

It provides necessary and sufficient conditions for the completeness of length-weighted Sobolev metrics on curve spaces, advancing the understanding of their geometric properties.

Findings

Identified conditions for metric completeness

Constructed examples of incomplete metrics

Extended previous work on constant coefficient Sobolev metrics

Abstract

In this article we prove completeness results for Sobolev metrics with nonconstant coefficients on the space of immersed curves and on the space of unparametrized curves. We provide necessary as well as sufficient conditions for the coefficients of the Riemannian metric for the metric to be metrically complete and we construct examples of incomplete metrics. This work is an extension of previous work on completeness of Sobolev metrics with constant coefficients.