Completeness and Geodesic Distance Properties for Fractional Sobolev Metrics on Spaces of Immersed Curves

Martin Bauer, Patrick Heslin, Cy Maor*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We investigate the geometry of the space of immersed closed curves equipped with reparametrization-invariant Riemannian metrics; the metrics we consider are Sobolev metrics of possible fractional-order q∈[0,∞). We establish the critical Sobolev index on the metric for several key geometric properties. Our first main result shows that the Riemannian metric induces a metric space structure if and only if q>1/2. Our second main result shows that the metric is geodesically complete (i.e., the geodesic equation is globally well posed) if q>3/2, whereas if q<3/2 then finite-time blowup may occur. The geodesic completeness for q>3/2 is obtained by proving metric completeness of the space of Hq-immersed curves with the distance induced by the Riemannian metric.

Original languageAmerican English
Article number214
JournalJournal of Geometric Analysis
Volume34
Issue number7
DOIs
StatePublished - Jul 2024

Bibliographical note

Publisher Copyright:
© The Author(s) 2024.

Keywords

  • 35A01
  • 35G55
  • 58B20
  • 58D10
  • Completeness
  • Fractional Sobolev space
  • Geodesic distance
  • Global well-posedness
  • Immersions
  • Infinite-dimensional Riemannian geometry

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