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Quantitative versions of Pansu Asymptotic Theorem and of Mitchell Tangent Theorem

Enrico Le Donne, Sebastiano Nicolussi Golo, Andrea Tettamanti

TL;DR

This work provides quantitative refinements of two foundational geometric limits for geodesic Lie groups: the Pansu asymptotic cone in nilpotent, sub-Finsler settings and the Mitchell tangent cone in general sub-Finsler settings. By introducing asymptotic and tangent gradings, along with constants α_infty, α_0 and β, the authors derive explicit power-law rates for |d(p,q)−d_infty(p,q)| and |d(p,q)−d_0(p,q)| on compacts, improving prior sublinear bounds. The methodology hinges on one-parameter families of contracted and dilated metrics, Grönwall-type lemmas adapted to each setting, and the notion of Carnot quotient ideals to localize the comparison to Carnot factors; this yields uniform convergence results and explicit GH convergence rates. The results illuminate how choices of gradings affect convergence speed and provide sharper bounds in natural constructions (e.g., products with Carnot factors), with clear implications for the study of sub-Finsler geometry and its infinitesimal limits.

Abstract

We quantitatively study the speed of convergence of geodesic Lie groups to their metric limits. For nilpotent geodesic Lie groups, we give estimates on the difference of the original metrics and the asymptotic metrics, while for general geodesic Lie groups, we give similar estimates for the difference of the original metrics and the tangent metrics. In both settings, our results sharpen existing bounds in the literature.

Quantitative versions of Pansu Asymptotic Theorem and of Mitchell Tangent Theorem

TL;DR

This work provides quantitative refinements of two foundational geometric limits for geodesic Lie groups: the Pansu asymptotic cone in nilpotent, sub-Finsler settings and the Mitchell tangent cone in general sub-Finsler settings. By introducing asymptotic and tangent gradings, along with constants α_infty, α_0 and β, the authors derive explicit power-law rates for |d(p,q)−d_infty(p,q)| and |d(p,q)−d_0(p,q)| on compacts, improving prior sublinear bounds. The methodology hinges on one-parameter families of contracted and dilated metrics, Grönwall-type lemmas adapted to each setting, and the notion of Carnot quotient ideals to localize the comparison to Carnot factors; this yields uniform convergence results and explicit GH convergence rates. The results illuminate how choices of gradings affect convergence speed and provide sharper bounds in natural constructions (e.g., products with Carnot factors), with clear implications for the study of sub-Finsler geometry and its infinitesimal limits.

Abstract

We quantitatively study the speed of convergence of geodesic Lie groups to their metric limits. For nilpotent geodesic Lie groups, we give estimates on the difference of the original metrics and the asymptotic metrics, while for general geodesic Lie groups, we give similar estimates for the difference of the original metrics and the tangent metrics. In both settings, our results sharpen existing bounds in the literature.
Paper Structure (21 sections, 35 theorems, 229 equations)

This paper contains 21 sections, 35 theorems, 229 equations.

Table of Contents

  1. Introduction
  2. Sub-Finsler Lie groups
  3. Sub-Finsler metrics on Lie groups
  4. Baker--Campbell--Hausdorff formula
  5. Gradings of vector spaces and Lie algebras
  6. Large and small scale geometry of sub-Finsler groups
  7. Asymptotic Carnot group and Pansu Asymptotic Theorem
  8. Osculating Carnot group and Mitchell Tangent Theorem
  9. One-parameter families of sub-Finsler Lie group structures
  10. Contracted metrics on simply connected nilpotent sub-Finsler Lie groups
  11. Dilated metrics on sub-Finsler Lie groups
  12. Carnot quotient ideals
  13. Grönwall Lemma type results
  14. Grönwall Lemma for Pansu Asymptotic Theorem
  15. Grönwall Lemma for Mitchell Tangent Theorem
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $(G,d)$ be a geodesic Lie group. Assume $G$ is simply connected and nilpotent. After fixing an asymptotic grading, consider the associated Pansu limit metric $d_\infty$ on $G$ and the constants $\alpha_\infty$ and $\beta$ defined in def_alpha_inf and def_beta, respectively. Then, for some $C>0$,

Theorems & Definitions (84)

  • Theorem 1.1: Quantitative Pansu
  • Theorem 1.2: Quantitative Mitchell
  • Definition 2.1: Asymptotic grading
  • Definition 2.2: Tangent grading
  • Theorem 2.3: Ball-Box Theorem for Carnot groups Don25
  • Proposition 2.4: Properties of linear gradings
  • proof
  • Definition 3.1: Hausdorff approximating sequence
  • Definition 3.2: Gromov-Hausdorff limit
  • Proposition 3.3: Criterion for GH convergence
  • ...and 74 more