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Canonical frames in contact 3-manifolds and applications

Brayan Ferreira, Marcelo Miranda, Alejandro Vicente

Abstract

We study contact 3-manifolds $Y$ with a special global frame inspired by Cartan's structure equations. This frame is dual to a generalized Finsler structure defined by Bryant. We present some examples and rigidity results on the class of manifolds whose frame satisfies certain natural conditions on a scalar function $K\colon Y\to \mathbb{R}$, related to the frame. This function realizes the curvature when $Y$ is the unit tangent bundle with respect to a metric on a surface. As applications, we obtain sharp estimates for the action of a Reeb orbit in terms of this scalar function, under the assumption that the frame satisfies specific conditions. In particular, we recover a classical upper bound on the systole of positively curved metrics on $S^2$ due to Toponogov.

Canonical frames in contact 3-manifolds and applications

Abstract

We study contact 3-manifolds with a special global frame inspired by Cartan's structure equations. This frame is dual to a generalized Finsler structure defined by Bryant. We present some examples and rigidity results on the class of manifolds whose frame satisfies certain natural conditions on a scalar function , related to the frame. This function realizes the curvature when is the unit tangent bundle with respect to a metric on a surface. As applications, we obtain sharp estimates for the action of a Reeb orbit in terms of this scalar function, under the assumption that the frame satisfies specific conditions. In particular, we recover a classical upper bound on the systole of positively curved metrics on due to Toponogov.

Paper Structure

This paper contains 16 sections, 14 theorems, 135 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Existence and classification results
  3. Landsberg contact manifolds and unit tangent bundles
  4. Applications
  5. Examples and properties of contact canonical frames
  6. Examples
  7. Properties
  8. On contact canonical frames with constant $K$
  9. The case $K\equiv 0$
  10. The case $K\equiv 1$
  11. The case $K\equiv -1$
  12. Applications
  13. Proof of Theorem \ref{['thm:sturmcompar']}
  14. Estimating action via CZ index
  15. Conley--Zehnder index
  16. ...and 1 more sections

Key Result

Theorem 1.3

Let $(Y,\lambda)$ be a contact 3-manifold and $\{R,X_1, X_2\}$ a contact canonical frame. If $K\equiv 0$, then

Figures (1)

  • Figure 1: Frame $(R,X_1,X_2)$ for $\mathbb{R}^3$ in blue, red and green, respectively.

Theorems & Definitions (37)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 27 more