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

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 $[\gamma ]$ of Lipschitz paths contains a length minimizing representative ${\gamma }_{\infty }$ that is unique up to reparametrization. The length minimizer ${\gamma }_{\infty }$ is the core of the homotopy class $[\gamma ]$ in the sense that the image of ${\gamma }_{\infty }$ is a subset of the image of any path contained in $[\gamma ]$. 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 ${\mathbb{H}}^1$.