Energy minimizing Beltrami fields on Sasakian 3-manifolds

TL;DR

This paper investigates when the Reeb vector field on compact Sasakian 3-manifolds minimizes energy among volume-preserving diffeomorphisms, linking geometric properties with stability and magnetic relaxation phenomena.

Contribution

It characterizes Sasakian manifolds where the Reeb field minimizes energy and analyzes stability under deformations, providing explicit examples including weighted 3-spheres.

Findings

Reeb field is a minimizer when related to the first eigenvalue of curl

Deformations can lead to instability and loss of minimality

Explicit solutions found for weighted 3-spheres in related energy minimization

Abstract

We study on which compact Sasakian 3-manifolds the Reeb field, which is a Beltrami field with eigenvalue 2, is an energy minimizer in its adjoint orbit under the action of volume preserving diffeomorphisms. This minimization property for Beltrami fields is relevant because of its connections with the phenomenon of magnetic relaxation and the hydrodynamic stability of steady Euler flows. We characterize the Sasakian manifolds where the Reeb field is a minimizer in terms of the first positive eigenvalue of the curl operator and show that for $a>a_0$ (a constant that depends on the Sasakian structure) the Reeb field of the $\mathcal{D}$-homothetic deformation of the manifold with constant $a$ (which is still Sasakian) is an unstable critical point of the energy, and hence not even a local minimizer. We also provide some examples of Sasakian manifolds where the Reeb field is a minimizer, highlighting the case of the weighted 3-spheres, on which another minimization problem (for the quartic Skyrme-Faddeev energy) is shown to admit exact solutions.