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

We show that for any purely 2-unrectifiable metric space $M$, for example the Heisenberg group $\mathbb{H}^1$ equipped with the Carnot-Carath\'{e}odory 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 $\mathbb{H}^1$.