Sharp regularity of sub-Riemannian length-minimizing curves
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917224123465728 |
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| author | Socionovo, Alessandro |
| author_facet | Socionovo, Alessandro |
| contents | A longstanding open question in sub-Riemannian geometry is the smoothness of (the arc-length parameterization of) length-minimizing curves. In [6], this question is negative answered, with an example of a $C^2$ but not $C^3$ length-minimizer of a real-analytic (even polynomial) sub-Riemannian structure. In this paper, we study a class of examples of sub-Riemannian structures that generalizes that presented in [6], and we prove that length-minimizing curves must be at least of class $C^2$ within these examples. In particular, we prove that Theorem 1.1 in [6] is sharp. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_00403 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp regularity of sub-Riemannian length-minimizing curves Socionovo, Alessandro Differential Geometry Metric Geometry Optimization and Control 53C17, 49K21, 28A75 A longstanding open question in sub-Riemannian geometry is the smoothness of (the arc-length parameterization of) length-minimizing curves. In [6], this question is negative answered, with an example of a $C^2$ but not $C^3$ length-minimizer of a real-analytic (even polynomial) sub-Riemannian structure. In this paper, we study a class of examples of sub-Riemannian structures that generalizes that presented in [6], and we prove that length-minimizing curves must be at least of class $C^2$ within these examples. In particular, we prove that Theorem 1.1 in [6] is sharp. |
| title | Sharp regularity of sub-Riemannian length-minimizing curves |
| topic | Differential Geometry Metric Geometry Optimization and Control 53C17, 49K21, 28A75 |
| url | https://arxiv.org/abs/2502.00403 |