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| Format: | Preprint |
| Published: |
2023
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| Online Access: | https://arxiv.org/abs/2306.01838 |
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| _version_ | 1866916254851268608 |
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| author | Perry, Daniel |
| author_facet | Perry, Daniel |
| contents | We show that for any purely 2-unrectifiable metric space $M$, for example the Heisenberg group $\mathbb{H}^1$ equipped with the Carnot-Carathéodory metric, every homotopy class $[γ]$ of Lipschitz paths contains a length minimizing representative $γ_\infty$ that is unique up to reparametrization. The length minimizer $γ_\infty$ is the core of the homotopy class $[γ]$ in the sense that the image of $γ_\infty$ is a subset of the image of any path contained in $[γ]$. Furthermore, the existence of length minimizers guarantees that only the trivial class in the first Lipschitz homotopy group of $M$ with a base point can be represented by a loop within each neighborhood of the base point. The results detailed here are used in arXiv:2402.10420 to define and prove properties of a universal Lipschitz path space over $\mathbb{H}^1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_01838 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Existence of length minimizers in homotopy classes of Lipschitz paths in $\mathbb{H}^1$ Perry, Daniel Metric Geometry We show that for any purely 2-unrectifiable metric space $M$, for example the Heisenberg group $\mathbb{H}^1$ equipped with the Carnot-Carathéodory metric, every homotopy class $[γ]$ of Lipschitz paths contains a length minimizing representative $γ_\infty$ that is unique up to reparametrization. The length minimizer $γ_\infty$ is the core of the homotopy class $[γ]$ in the sense that the image of $γ_\infty$ is a subset of the image of any path contained in $[γ]$. Furthermore, the existence of length minimizers guarantees that only the trivial class in the first Lipschitz homotopy group of $M$ with a base point can be represented by a loop within each neighborhood of the base point. The results detailed here are used in arXiv:2402.10420 to define and prove properties of a universal Lipschitz path space over $\mathbb{H}^1$. |
| title | Existence of length minimizers in homotopy classes of Lipschitz paths in $\mathbb{H}^1$ |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2306.01838 |