Measure contraction property, curvature exponent and geodesic dimension of sub-Finsler $\ell^p$-Heisenberg groups
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910952088141824 |
|---|---|
| author | Borza, Samuël Tashiro, Kenshiro |
| author_facet | Borza, Samuël Tashiro, Kenshiro |
| contents | We initiate the study of synthetic curvature-dimension bounds in sub-Finsler geometry. More specifically, we investigate the measure contraction property $\mathsf{MCP}(K, N)$, and the geodesic dimension on the Heisenberg group equipped with an $\ell^p$-sub-Finsler norm. We show that for $p\in(2,\infty]$, the $\ell^p$-Heisenberg group fails to satisfy any of the measure contraction properties. On the other hand, if $p\in(1,2)$, then it satisfies the measure contraction property $\mathsf{MCP}(K, N)$ if and only if $K \leq 0$ and $N \geq N_p$, where the curvature exponent $N_p$ is strictly greater than $2q+1$ ($q$ being the Hölder conjugate of $p$). We also prove that the geodesic dimension of the $\ell^p$-Heisenberg group is $\min(2q+2,5)$ for $p\in[1,\infty)$. As a consequence, we provide the first example of a metric measure space where there is a gap between the curvature exponent and the geodesic dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_16722 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Measure contraction property, curvature exponent and geodesic dimension of sub-Finsler $\ell^p$-Heisenberg groups Borza, Samuël Tashiro, Kenshiro Metric Geometry Differential Geometry 53C17, 26A33, 49N60, 49Q22 We initiate the study of synthetic curvature-dimension bounds in sub-Finsler geometry. More specifically, we investigate the measure contraction property $\mathsf{MCP}(K, N)$, and the geodesic dimension on the Heisenberg group equipped with an $\ell^p$-sub-Finsler norm. We show that for $p\in(2,\infty]$, the $\ell^p$-Heisenberg group fails to satisfy any of the measure contraction properties. On the other hand, if $p\in(1,2)$, then it satisfies the measure contraction property $\mathsf{MCP}(K, N)$ if and only if $K \leq 0$ and $N \geq N_p$, where the curvature exponent $N_p$ is strictly greater than $2q+1$ ($q$ being the Hölder conjugate of $p$). We also prove that the geodesic dimension of the $\ell^p$-Heisenberg group is $\min(2q+2,5)$ for $p\in[1,\infty)$. As a consequence, we provide the first example of a metric measure space where there is a gap between the curvature exponent and the geodesic dimension. |
| title | Measure contraction property, curvature exponent and geodesic dimension of sub-Finsler $\ell^p$-Heisenberg groups |
| topic | Metric Geometry Differential Geometry 53C17, 26A33, 49N60, 49Q22 |
| url | https://arxiv.org/abs/2305.16722 |