Sharp geometric rigidity of isometries on Heisenberg group

TL;DR

This paper establishes that approximate isometries in the Heisenberg group are quantitatively close to true isometries, with explicit bounds, demonstrating the rigidity of geometric structures in sub-Riemannian geometry.

Contribution

It provides the first quantitative stability estimates for isometries on the Heisenberg group, showing how near-isometries are close to genuine isometries with explicit bounds.

Findings

Closeness bounds are of order () in uniform norm.

Closeness bounds are of order () in Sobolev norm.

Results are asymptotically sharp, confirmed by homogeneous dilations.

Abstract

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.