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Contact structures and Beltrami fields on the torus and the sphere

Daniel Peralta-Salas, Radu Slobodeanu

Abstract

We present new explicit tight and overtwisted contact structures on the (round) 3-sphere and the (flat) 3-torus for which the ambient metric is weakly compatible. Our proofs are based on the construction of nonvanishing curl eigenfields using suitable families of Jacobi or trigonometric polynomials. As a consequence, we show that the contact sphere theorem of Etnyre, Komendarczyk and Massot (2012) does not hold for weakly compatible metric as it was conjectured. We also establish a geometric rigidity for tight contact structures by showing that any contact form on the 3-sphere admitting a compatible metric that is the round one is isometric, up to a constant factor, to the standard (tight) contact form.

Contact structures and Beltrami fields on the torus and the sphere

Abstract

We present new explicit tight and overtwisted contact structures on the (round) 3-sphere and the (flat) 3-torus for which the ambient metric is weakly compatible. Our proofs are based on the construction of nonvanishing curl eigenfields using suitable families of Jacobi or trigonometric polynomials. As a consequence, we show that the contact sphere theorem of Etnyre, Komendarczyk and Massot (2012) does not hold for weakly compatible metric as it was conjectured. We also establish a geometric rigidity for tight contact structures by showing that any contact form on the 3-sphere admitting a compatible metric that is the round one is isometric, up to a constant factor, to the standard (tight) contact form.

Paper Structure

This paper contains 13 sections, 19 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminary results
  3. Basics of contact geometry
  4. Beltrami fields and contact structures
  5. Eigenfunctions of the Laplacian on $\mathbb T^2$ and $\mathbb S^2$
  6. Overtwisted Beltrami fields on $\mathbb{S}^3$: proof of Theorem \ref{['maint']}
  7. Beltrami fields with constant norm on $\mathbb{S}^3$: proof of Proposition \ref{['maint2']}
  8. Nonvanishing Beltrami fields on $\mathbb{T}^3$: proof of Theorem \ref{['maint3']}
  9. Examples and final remarks
  10. Existence of overtwisted curl eigenfields on $\mathbb S^3$
  11. Other examples in $\mathbb S^3$ and $\mathbb T^3$
  12. An application to planar open book decompositions
  13. Nonvanishing Beltrami fields with nonconstant proportionality factor

Key Result

Theorem 1

Let $(M,\alpha)$ be a closed contact 3-manifold. If there exists a compatible metric $g$ with pinched sectional curvature then the contact structure is tight and the manifold is covered by a $3$-sphere.

Theorems & Definitions (43)

  • Definition 1
  • Theorem 1: Etnyre, Komendarczyk and Massot, 2012
  • Theorem 2
  • Corollary 1
  • Proposition 1
  • Theorem 3
  • Theorem 4: Gray stability
  • Theorem 5: Eliashberg classification Eli
  • Proposition 2
  • Remark 1
  • ...and 33 more