English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We show that for any purely 2-unrectifiable metric space $M$, for example theHeisenberg group $\mathbb{H}^1$ equipped with the Carnot-Carath\'{e}odorymetric, every homotopy class $[\alpha]$ of Lipschitz paths contains a lengthminimizing representative $\alpha_\infty$ that is unique up toreparametrization. The length minimizer $\alpha_\infty$ is the core of thehomotopy class $[\alpha]$ in the sense that the image of $\alpha_\infty$ is asubset of the image of any path contained in $[\alpha]$. Furthermore, theexistence of length minimizers guarantees that only the trivial class in thefirst Lipschitz homotopy group of $M$ with a base point can be represented by aloop within each neighborhood of the base point. The results detailed here willbe used to define and prove properties of a universal Lipschitz path space over$\mathbb{H}^1$ in a subsequent paper.

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