Sharp geometric rigidity of isometries on Heisenberg groups

2012/04/16 by Д. В. Исангулова, Isangulova, D. V., S. K. Vodopyanov +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.1204.3441

openalex publication_date 2012/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove sharp geometric rigidity estimates for isometries on Heisenberg groups. Our main result asserts that every (1+ε)-quasi-isometry on a John domain of the Heisenberg group ℍⁿ, n>1, is close to some isometry up to proximity order √(ε)+ε in the uniform norm, and up to proximity order ε in the L_p¹-norm. We give examples showing the asymptotic sharpness of our results.

Citations

• Balls and metrics defined by vector fields I: Basic properties
• Minimal Surfaces in the Roto-Translation Group with Applications to a Neuro-Biological Image Completion Model
• A theorem on geometric rigidity and the derivation of nonlinear plate theory from three‐dimensional elasticity
• Metriques de Carnot-Caratheodory et Quasiisometries des Espaces Symetriques de rang un

Related

• Sharp geometric rigidity of isometries on Heisenberg group
• Quasi-isometry rigidity of groups
• Strong C^*-rigidity of the Heisenberg groups
• Coarse differentiation of quasi-isometries II: Rigidity for Sol and\n Lamplighter groups
• Isodiametric sets in the Heisenberg group