Hausdorff dimension and failure of synthetic curvature bounds in the sub-Lorentzian Heisenberg group
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908524654624768 |
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| author | Borza, Samuël Rigoni, Chiara Zoghlami, Omar |
| author_facet | Borza, Samuël Rigoni, Chiara Zoghlami, Omar |
| contents | We study the geodesics, Hausdorff dimension, and curvature bounds of the sub-Lorentzian Heisenberg group. Through an elementary variational approach, we provide a new proof of the structure of its maximizing geodesics, showing that they are lifts of hyperbolae coming from a Lorentzian isoperimetric problem in the Minkowski plane. We prove that the Lorentzian Hausdorff dimension of the space is $4$ and that the corresponding measure coincides with the Haar measure. We further establish a novel result in the spirit of the Ball-Box theorem, giving a uniform estimate of causal diamonds by anisotropic boxes. Finally, we show that the Heisenberg group satisfies neither the timelike curvature-dimension condition $\mathsf{TCD}(K,N)$ nor the timelike measure contraction property $\mathsf{TMCP}(K,N)$ for any values of the parameters $K$ and $N$, in sharp contrast with its sub-Riemannian counterpart. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06563 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hausdorff dimension and failure of synthetic curvature bounds in the sub-Lorentzian Heisenberg group Borza, Samuël Rigoni, Chiara Zoghlami, Omar Differential Geometry Metric Geometry 53C50, 53C17, 49Q22, 28A75, 51K10 We study the geodesics, Hausdorff dimension, and curvature bounds of the sub-Lorentzian Heisenberg group. Through an elementary variational approach, we provide a new proof of the structure of its maximizing geodesics, showing that they are lifts of hyperbolae coming from a Lorentzian isoperimetric problem in the Minkowski plane. We prove that the Lorentzian Hausdorff dimension of the space is $4$ and that the corresponding measure coincides with the Haar measure. We further establish a novel result in the spirit of the Ball-Box theorem, giving a uniform estimate of causal diamonds by anisotropic boxes. Finally, we show that the Heisenberg group satisfies neither the timelike curvature-dimension condition $\mathsf{TCD}(K,N)$ nor the timelike measure contraction property $\mathsf{TMCP}(K,N)$ for any values of the parameters $K$ and $N$, in sharp contrast with its sub-Riemannian counterpart. |
| title | Hausdorff dimension and failure of synthetic curvature bounds in the sub-Lorentzian Heisenberg group |
| topic | Differential Geometry Metric Geometry 53C50, 53C17, 49Q22, 28A75, 51K10 |
| url | https://arxiv.org/abs/2509.06563 |