Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantitative tightness for three-dimensional contact manifolds: a sub-Riemannian approach

Andrei A. Agrachev, Stefano Baranzini, Eugenio Bellini, Luca Rizzi

TL;DR

This paper develops a sub-Riemannian framework to quantify the maximal tight neighbourhood around Reeb orbits in 3D contact manifolds by introducing contact Jacobi curves and analyzing their Schwarzian derivatives and canonical curvatures. It proves sharp lower bounds for the tightness radius in terms of Schwarzian-data and non-sharp, curvature-based bounds, and highlights model cases where the Schwarzian approach is sharp. A Cartan–Hadamard-type theorem for K-contact SR manifolds is established, linking curvature, tightness, and topological properties. The results are compared with existing Riemannian approaches, showing the new method often yields sharper, model-sharp estimates and broad applicability to K-contact and left-invariant structures, with concrete examples on Heisenberg, SU(2), and SL$(2)$.

Abstract

Through the use of sub-Riemannian metrics we provide quantitative estimates for the maximal tight neighbourhood of a Reeb orbit on a three-dimensional contact manifold. Under appropriate geometric conditions we show how to construct closed curves which are boundaries of overtwisted disks. We introduce the concept of \emph{contact} Jacobi curve, and prove lower bounds of the so-called tightness radius (from a Reeb orbit) in terms of Schwarzian derivative bounds. We compare these results with the corresponding ones from [Etnyre, Komendarczyk, Massot - Invent. Math. 2012 and Trans. Amer. Math. Soc. 2016], and we show that our estimates are sharp for classical model structures. We also prove similar, but non-sharp, estimates in terms of sub-Riemannian canonical curvature bounds. We apply our results to K-contact sub-Riemannian manifolds. In this setting, we prove a contact analogue of the celebrated Cartan--Hadamard theorem.

Quantitative tightness for three-dimensional contact manifolds: a sub-Riemannian approach

TL;DR

This paper develops a sub-Riemannian framework to quantify the maximal tight neighbourhood around Reeb orbits in 3D contact manifolds by introducing contact Jacobi curves and analyzing their Schwarzian derivatives and canonical curvatures. It proves sharp lower bounds for the tightness radius in terms of Schwarzian-data and non-sharp, curvature-based bounds, and highlights model cases where the Schwarzian approach is sharp. A Cartan–Hadamard-type theorem for K-contact SR manifolds is established, linking curvature, tightness, and topological properties. The results are compared with existing Riemannian approaches, showing the new method often yields sharper, model-sharp estimates and broad applicability to K-contact and left-invariant structures, with concrete examples on Heisenberg, SU(2), and SL.

Abstract

Through the use of sub-Riemannian metrics we provide quantitative estimates for the maximal tight neighbourhood of a Reeb orbit on a three-dimensional contact manifold. Under appropriate geometric conditions we show how to construct closed curves which are boundaries of overtwisted disks. We introduce the concept of \emph{contact} Jacobi curve, and prove lower bounds of the so-called tightness radius (from a Reeb orbit) in terms of Schwarzian derivative bounds. We compare these results with the corresponding ones from [Etnyre, Komendarczyk, Massot - Invent. Math. 2012 and Trans. Amer. Math. Soc. 2016], and we show that our estimates are sharp for classical model structures. We also prove similar, but non-sharp, estimates in terms of sub-Riemannian canonical curvature bounds. We apply our results to K-contact sub-Riemannian manifolds. In this setting, we prove a contact analogue of the celebrated Cartan--Hadamard theorem.
Paper Structure (34 sections, 53 theorems, 360 equations)

This paper contains 34 sections, 53 theorems, 360 equations.

Table of Contents

  1. Introduction
  2. Tightness radius from a Reeb orbit
  3. Contact Jacobi curves
  4. Tightness radius estimates
  5. Tightness radius estimates via Schwarzian derivative
  6. Tightness radius estimates via canonical curvature
  7. Cartan--Hadamard theorem for K-contact structures
  8. Comparison with the state of the art
  9. Structure of the paper
  10. Acknowledgements
  11. Preliminaries
  12. Three-dimensional contact manifolds
  13. Contact sub-Riemannian geometry
  14. Exponential coordinates
  15. Cylindrical coordinates on the annihilator bundle
  16. ...and 19 more sections

Key Result

Theorem 1

Let $(M,\omega, g)$ be a complete three-di-men-sional contact sub-Riemannian manifold and let $\Gamma$ be an embedded piece of Reeb orbit with $r_{\mathrm{inj}}(\Gamma)>0$. Then, letting the following estimates hold: Moreover, if $r_o^+(\Gamma) < r_{\mathrm{inj}}(\Gamma)$, then for any $q\in \Gamma$, the set is an overtwisted disk, and thus $(M,\omega)$ is an overtwisted contact manifold.

Theorems & Definitions (132)

  • Definition : Tightness radius
  • Definition : Contact Jacobi curve
  • Theorem 1: Geometric tightness radius estimates
  • Theorem 2: Tightness radius estimates with Schwarzian derivative bounded above
  • Theorem 3: Tightness radius estimates with curvature bounded above
  • Theorem 4: Contact Cartan--Hadamard
  • Corollary 5
  • Theorem 1.1: QDarboux
  • Remark 1.2
  • Lemma 1.3
  • ...and 122 more