Sobolev metrics on spaces of manifold valued curves

TL;DR

This paper investigates the mathematical properties of Sobolev metrics on manifold-valued curves, establishing conditions for completeness and well-posedness of geodesic equations, extending previous Euclidean space results.

Contribution

It proves metric and geodesic completeness for certain Sobolev metrics on manifold-valued curves, and characterizes incompleteness in constant coefficient cases.

Findings

Sobolev immersions are metrically and geodesically complete for several metrics

Constant coefficient Sobolev metrics on open curves are metrically incomplete due to curves vanishing

Previous results were limited to Euclidean space, now extended to manifold values

Abstract

We study completeness properties of reparametrization invariant Sobolev metrics of order $n\ge 2$ on the space of manifold valued open and closed immersed curves. In particular, for several important cases of metrics, we show that Sobolev immersions are metrically and geodesically complete (thus the geodesic equation is globally well-posed). These results were previously known only for closed curves with values in Euclidean space. For the class of constant coefficient Sobolev metrics on open curves, we show that they are metrically incomplete, and that this incompleteness only arises from curves that vanish completely (unlike "local" failures that occur in lower order metrics).