Zero Set Structure of Real Analytic Beltrami Fields

Journal of Geometric Analysis, vol. 31, pp. 9928–9950

Abstract

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 $${\mathbb {R}}^3$$ R3 , 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 $$+1$$ +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.

Authors 1

1. Wadim Gerner corresponding Aachen Lehrstuhl für Angewandte Analysis

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen University, Lehrstuhl für Angewandte Analysis, Turmstraße 46, 52064, Aachen, Germany

   RWTH Aachen University, Lehrstuhl für Angewandte Analysis, Aachen, Germany