H\"{o}lder embeddings into the Heisenberg Group

The Heisenberg group, $\mathbb{H}^n$, is a sub-Riemannian manifold equipped with the Carnot-Caratheodory metric $d_c$. It is homeomorphic to $\mathbb{R}^{2n+1}$ but it has Hausdorff dimension of $2n+2$.  In this talk, we will discuss some recent progress about H\"{o}lder embeddings into $\mathbb{H}^n$. In particular, Wenger and Young developed a variant of Dehn function, then proved that for any $\alpha \in (0,2/3)$, an $\alpha-$H\"{o}lder map $f: S^1 \to \mathbb{H}$  admits an $\alpha-$H\"{o}lder extension to $D^2$. Moreover, we will introduce a  theorem about non-existence of H\"{o}lder embeddings by Haj\l{}asz and Schikorra, which generalizes a result of Gromov.