Abstract
In this paper, we prove a classification theorem for the zero sets of real analytic Beltrami fields. Namely, we show that the zero set of a real analytic Beltrami field on a real analytic, connected 3-manifold without boundary is either empty after removing its isolated points or can be written as a countable, locally finite union of differentiably embedded, connected 1-dimensional submanifolds with (possibly empty) boundary and tame knots. Further, we consider the question of how complicated these tame knots can possibly be. To this end, we prove that on the standard (open) solid toroidal annulus in {mathbb {R}}^3, there exist for any pair (p, q) of positive, coprime integers countable infinitely many distinct real analytic metrics such that for each such metric, there exists a real analytic Beltrami field, corresponding to the eigenvalue +1 of the curl operator, whose zero set is precisely given by a standard (p, q)-torus knot. The metrics and the corresponding Beltrami fields are constructed explicitly and can be written down in Cartesian coordinates by means of elementary functions alone.
Highlights
Such vector fields appear naturally in physics and have been widely studied in mathematics. They appear as stationary magnetic fields of the equations of ideal magnetohydrodynamics, and in particular in astrophysics, in the case of constant pressure and a resting plasma [3, Chapter III §1.A]
They appear as stationary solutions of the incompressible Euler equations for an appropriate pressure function [3, Chapter II §1.A]
From a variational point of view, Beltrami fields are closely related to the helicity constraint magnetic energy minimisation, see [2,5,34], while Beltrami fields with a non-constant proportionality function were studied for instance in [14] and [28]
Summary
The zero sets, referred to as nodal sets, of Laplacian eigenfunctions were thoroughly studied for example in [18,21,30] and [31]. A priori, the zero set of a Beltrami field might be 2-dimensional. We will show that these features are more generally true for any real analytic Beltrami field defined on an abstract manifold without boundary. Our result states that, after removing the isolated points, the remaining nodal set is either empty or a countable, locally finite union of analytically embedded 1-manifolds with (possibly non-present) C1-endpoints, see Definition 2.1, and tame knots. Our approach differs from the approach in [8], since we do not assume any symmetry, but instead rely on results from semianalytic geometry [7,17,22], most notably the curve selection lemma [23, §2], [27, Lemma 3.1], [24, Lemma 6.6]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
