Sharp geometric rigidity of isometries on Heisenberg groups

TL;DR

This paper establishes precise geometric rigidity estimates for isometries in Heisenberg groups, demonstrating that near-isometries are close to true isometries with quantifiable bounds.

Contribution

It provides the first sharp quantitative bounds on how $(1+ ext{small})$-quasi-isometries approximate true isometries in Heisenberg groups.

Findings

Near-isometries are close to true isometries within specific bounds.

The bounds are proven to be asymptotically sharp.

Results apply to John domains in Heisenberg groups.

Abstract

We prove sharp geometric rigidity estimates for isometries on Heisenberg groups. Our main result asserts that every $(1+\varepsilon)$-quasi-isometry on a John domain of the Heisenberg group $\mathbb{H}^n$, $n>1$, is close to some isometry up to proximity order $\sqrt{\varepsilon}+\varepsilon$ in the uniform norm, and up to proximity order $\varepsilon$ in the $L_p^1$-norm. We give examples showing the asymptotic sharpness of our results.