Sobolev metrics on spaces of manifold-valued curves

Martin Bauer, Cy Maor, Peter W. Michor

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


We study completeness properties of reparametrization-invariant Sobolev metrics of order n 2 on the space of open and closed immersed curves in a manifold. In particular, for several important classes 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).

Original languageAmerican English
Pages (from-to)1895-1948
Number of pages54
JournalAnnali della Scuola normale superiore di Pisa - Classe di scienze
Issue number4
StatePublished - 2023

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