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

We prove the quantitative stability of isometries on the first Heisenberg group with sub-Riemannian geometry: every (1 + ε)-quasi-isometry of the John domain of the Heisenberg group H is close to some isometry with the order of closeness (Formula presented.)ε + ε in the uniform norm and with the order of closeness ε in the Sobolev norm. An example demonstrating the asymptotic sharpness of the results is given.