Skip to content

Miniworkshop: Geometry and Dynamics of Singular Structures

When
Oct 01, 2020 from 04:00 PM to 07:30 PM (Europe/Madrid / UTC200)
Add event to calendar
iCal

The miniworkshop Geometry and Dynamics of singular structures will feature talks by experts on Geometry and Dynamical systems with a special focus on their interaction.

This workshop is organized in occasion of the thesis defenses of Arnau Planas and Cédric Oms.

When? October 1st starting at 3pm

Where? Online (details to be posted)

The speakers will be

Summaries of the talks

Daniel Peralta-Salas

Title: Contact structures and Beltrami fields on the torus and the sphere

Abstract: A Beltrami field on a Riemannian 3-manifold (M,g) is a vector-valued eigenfunction of the curl operator. Beltrami fields have found far reaching applications in several areas of physics, such as fluid mechanics or magnetohydrodynamics. In the early 2000's, Etnyre and Ghrist developed a surprising connection with contact geometry: any nonvanishing Beltrami field engenders a contact structure and, conversely, a Reeb field of a contact form is Beltrami for some metric. This result turns out to be related to the emerging area of contact Riemannian geometry, where the geometric properties of the metrics that are (weakly) compatible with a contact form are studied. The connection is that the Riemannian metric g on M is weakly compatible with the contact structure engendered by a nonvanishing Beltrami field. A major result in this direction is the contact sphere theorem proved by Etnyre, Komendarczyk and Massot in 2012: a compatible metric with appropriately pinched sectional curvature implies that the associated contact structure is tight. This left open the problem of whether an analogous pinching result holds for weakly compatible metrics; in particular, is it possible to classify the contact structures engendered by Beltrami fields on the round 3-sphere or the flat 3-torus? Can they be overtwisted?
In this talk I will report on some recent progress in this direction obtained in collaboration with Radu Slobodeanu. I will show that there exist overtwisted contact structures on the (round) 3-sphere and the (flat) 3-torus for which the ambient metric is weakly compatible, thus implying that the contact sphere theorem of Etnyre, Komendarczyk and Massot does not hold for weakly compatible metrics. Our proofs are based on the construction of S1-invariant nonvanishing curl eigenfields on S3 and T3 using suitable families of Jacobi polynomials and eigenfunctions of the Laplacian with contractible nodal sets, respectively.