Sharp Estimates for Geometric Rigidity of Isometries on the First Heisenberg Group

Abstract

We prove the 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 the order of closeness \(\sqrt \varepsilon + \varepsilon \) in the uniform norm and with the order of closeness \(\varepsilon \) in the Sobolev norm. An example demonstrating the asymptotic sharpness of the results is given.