Sharp geometric rigidity of isometries on Heisenberg group

We prove quantitative stability of isometries on the first Heisenberg group with sub-Riemannian geometry: every $(1+ \varepsilon)$-quasi-isometry of the John domain of the Heisenberg group $\mathbb {H}$ is close to some isometry with order of closeness $\sqrt{\varepsilon} + \varepsilon$ in the uniform norm and with order of closeness $\varepsilon$ in the Sobolev norm $L_2^1$. Homogeneous dilations show the asymptotic sharpness of the results.