Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotics of Riemannian Lie groups with nilpotency step 2

Enrico Le Donne, Luca Nalon, Sebastiano Nicolussi Golo, Seung-Yeon Ryoo

Abstract

We derive sharp estimates comparing asymptotic Riemannian or sub-Riemannian metrics in 2-step nilpotent Lie groups. For each metric, we construct a Carnot metric whose square remains at bounded distance from the square of the original metric. In particular, we deduce the analogue of a conjectire by Burago-Margulis: every 2-step nilpotent Riemannian Lie group is at bounded distance from its asymptotic cone. As a consequence, we obtain a refined estimate of the error term in the asymptotic expansion of the volume of the (sub-)Riemannian metric balls. To achive this, we develop a novel technique to efficiently perturb rectifiable curves modifying their endpoints in a prescribed vertical direction.

Asymptotics of Riemannian Lie groups with nilpotency step 2

Abstract

We derive sharp estimates comparing asymptotic Riemannian or sub-Riemannian metrics in 2-step nilpotent Lie groups. For each metric, we construct a Carnot metric whose square remains at bounded distance from the square of the original metric. In particular, we deduce the analogue of a conjectire by Burago-Margulis: every 2-step nilpotent Riemannian Lie group is at bounded distance from its asymptotic cone. As a consequence, we obtain a refined estimate of the error term in the asymptotic expansion of the volume of the (sub-)Riemannian metric balls. To achive this, we develop a novel technique to efficiently perturb rectifiable curves modifying their endpoints in a prescribed vertical direction.

Paper Structure

This paper contains 17 sections, 26 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sub-Riemannian metrics on Lie groups
  4. Nilpotent Lie groups and asymptotic metrics
  5. Models for sub-Riemannian 2-step nilpotent Lie groups
  6. Energy of curves
  7. Complexification of real Lie algebras
  8. Main results
  9. Vertical variations
  10. Proof of theorems
  11. Consequences, examples, and sharpness of results
  12. Rate of convergence to the asymptotic cone
  13. The non-validity for 3-step Riemannian metrics
  14. Volume growth of metric balls
  15. The non-validity for Finsler metrics in the Heisenberg group
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $G$ be a simply connected, $2$-step nilpotent Lie group equipped with a left-invariant sub-Riemannian metric $d$. Denote by $d_\infty$ the canonical asymptotic metric on $G$. Then, there exists $C > 0$ such that In particular

Theorems & Definitions (60)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1: Canonical asymptotic metric
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • ...and 50 more