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arXiv · 2306.01838

Existence of length minimizers in homotopy classes of Lipschitz paths in $\mathbb{H}^1$

Daniel Perry
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Abstract

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 $\mathbb{H}^1$.

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Daniel Perry. 2024-05-21. Existence of length minimizers in homotopy classes of Lipschitz paths in $\mathbb{H}^1$. https://arxiv.org/abs/2306.01838

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