Euler flows and singular geometric structures

Tichler proved (Tischler D. 1970 Topology 9, 153-154. (doi:10.1016/0040-9383(70)90037-6)) that a manifold admitting a smooth non-vanishing and closed oneform fibres over a circle. More generally, a manifold admitting k-independent closed one-form fibres over a torus Tk. In this article, we explain a version of this construction for manifolds with boundary using the techniques of b-calculus (Melrose R. 1993 The Atiyah Patodi Singer index theorem. Research Notes in Mathematics. Wellesley, MA: A. K. Peters; Guillemin V, Miranda E, Pires AR. 2014 Adv. Math. (N. Y.) 264, 864-896. (doi:10.1016/j.aim.2014.07.032)). We explore new applications of this idea to fluid dynamics and more concretely in the study of stationary solutions of the Euler equations. In the study of Euler flows on manifolds, two dichotomic situations appear. For the first one, in which the Bernoulli function is not constant, we provide a new proof of Arnold's structure theorem and describe b-symplectic structures on some of the singular sets of the Bernoulli function. When the Bernoulli function is constant, a correspondence between contact structures with singularities (Miranda E, Oms C. 2018 Contact structures with singularities. https://arxiv.org/abs/1806.05638) and what we call b-Beltrami fields is established, thus mimicking the classical correspondence between Beltrami fields and contact structures (see for instance Etnyre J, Ghrist R. 2000 Trans. Am. Math. Soc. 352, 5781-5794. (doi:10.1090/S0002-9947-00-02651-9)). These results provide a new technique to analyse the geometry of steady fluid flows on non-compact manifolds with cylindrical ends.
Robert Cardona is supported by FPI-BGSMath doctoral grant. Eva Miranda is supported by the CatalanInstitution for Research and Advanced Studies via an ICREA Academia Prize 2016 and partially supported bythe grants reference number MTM2015-69135-P (MINECO/FEDER) and reference number 2017SGR932 (AGAUR).Daniel Peralta-Salas is supported by the ERC Starting Grant 335079, the MTM grant 2016-76702-P, and partiallysupported by the ICMAT–Severo Ochoa grant SEV-2015-0554. This material is based upon work supported by theNational Science Foundation under Grant No. DMS-1440140 while Eva Miranda was in residence at the MathematicalSciences Research Institute in Berkeley, California, during the Fall 2018 semester.