Contact structures and Beltrami fields on the torus and the sphere

TL;DR

This paper constructs explicit examples of contact structures on the 3-sphere and 3-torus using curl eigenfields, challenging existing theorems and establishing rigidity results for tight contact structures.

Contribution

It provides new explicit contact structures with weakly compatible metrics and disproves a conjecture about the contact sphere theorem, also proving a rigidity result for tight contact forms on the sphere.

Findings

Constructed explicit tight and overtwisted contact structures on the sphere and torus.

Showed the contact sphere theorem does not hold for weakly compatible metrics.

Proved rigidity of tight contact structures on the 3-sphere with round metric.

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.