On the Metric Completion of Immersed Open Curves
Abstract
The completeness properties of spaces of immersed curves endowed with Sobolev-type reparametrization-invariant Riemannian metrics have been studied extensively in recent years. Whereas constant-coefficient metrics of large enough order on the space of closed curves are known to be complete, this is not the case for open curves. In this work, we characterize the metric completion of the space of real-valued immersed open curves with respect to these metrics. We show that the set of additional limit points in the completion is homeomorphic to Hn-diffeomorphisms of a closed interval. This result answers a question of the structure of the completion, posed in (Bauer et al., Ann. Sc. Norm. Super. Pisa Cl. Sci. (2023)), in the negative, and suggests a more refined conjecture on the structure in the case of multi-dimensional ambient space.
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BibTeX
@misc{gelman2026metric,
title = {{On the Metric Completion of Immersed Open Curves}},
author = {Ronny Gelman and Cy Maor},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17719},
eprint = {2609.17719},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17719},
}
RIS
TY - UNPB
TI - On the Metric Completion of Immersed Open Curves
AU - Ronny Gelman
AU - Cy Maor
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17719
UR - https://arxiv.org/abs/2609.17719
AB - The completeness properties of spaces of immersed curves endowed with Sobolev-type reparametrization-invariant Riemannian metrics have been studied extensively in recent years. Whereas constant-coefficient metrics of large enough order on the space of closed curves are known to be complete, this is not the case for open curves. In this work, we characterize the metric completion of the space of real-valued immersed open curves with respect to these metrics. We show that the set of additional limit points in the completion is homeomorphic to H^n-diffeomorphisms of a closed interval. This result answers a question of the structure of the completion, posed in (Bauer et al., Ann. Sc. Norm. Super. Pisa Cl. Sci. (2023)), in the negative, and suggests a more refined conjecture on the structure in the case of multi-dimensional ambient space.
ER -