## 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 H^{q}-immersed curves with the distance induced by the Riemannian metric.

Original language | American English |
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Article number | 214 |

Journal | Journal of Geometric Analysis |

Volume | 34 |

Issue number | 7 |

DOIs | |

State | Published - 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