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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.10954 |
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| _version_ | 1866916948908965888 |
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| author | Gelman, Ronny |
| author_facet | Gelman, Ronny |
| contents | The completeness properties of spaces of immersed curves equipped with reparametrization-invariant Riemannian metrics have recently been the subject of active research. This thesis studies the metric completion of spaces of immersed open curves endowed with Sobolev-type metrics and examines a previously proposed conjecture that suggests the metric completion consists of a single additional point, representing all vanishing length Cauchy sequences. We disprove the conjecture in the setting of real-valued immersed curves by demonstrating the existence of multiple distinct limit points. Furthermore, we provide a nearly complete characterization of the metric structure of the metric completion in this case. These results lead to a revised conjecture regarding the structure of the metric completion in more general settings |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_10954 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterization of the Metric Completion of Immersed Open-Curve Spaces Gelman, Ronny Differential Geometry The completeness properties of spaces of immersed curves equipped with reparametrization-invariant Riemannian metrics have recently been the subject of active research. This thesis studies the metric completion of spaces of immersed open curves endowed with Sobolev-type metrics and examines a previously proposed conjecture that suggests the metric completion consists of a single additional point, representing all vanishing length Cauchy sequences. We disprove the conjecture in the setting of real-valued immersed curves by demonstrating the existence of multiple distinct limit points. Furthermore, we provide a nearly complete characterization of the metric structure of the metric completion in this case. These results lead to a revised conjecture regarding the structure of the metric completion in more general settings |
| title | Characterization of the Metric Completion of Immersed Open-Curve Spaces |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2509.10954 |