Energy Minimizing Beltrami Fields on Sasakian 3-Manifolds

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 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.

Existence and characterisation of magnetic energy minimisers on oriented, compact Riemannian 3-manifolds with boundary in arbitrary helicity classes

“…The minimisation problem (MP2) was also considered in the context of compact, Sasakian manifolds (without boundary). For instance in [21] the authors investigate the question whether or not the associated Reeb vector field, which is an eigenfield of curl, is always an energy minimiser of (MP2). As is shown [21,Theorem 2] this is the case if and only if the smallest positive eigenvalue of the curl operator is +2 and it is further shown [21, Theorem 1] that the Reeb vector field can in fact be an unstable critical point of the energy.…”

“…Thus the result of Theorem 2.2 holds more generally true for all critical points of the energy functional rather than solely for energy minimisers. It is also worth stating that according to [21,Remark 2] all solutions of the stationary, incompressible Euler equations with constant pressure function on a closed 3-manifold of negative sectional curvature are local energy minimisers of (MP2), i.e. each divergence-free vector field X on such a manifold, satisfying ∇ X X = 0 , where the left hand side denotes the covariant derivative with respect to the Levi-Civita connection, is a local energy minimiser.…”

Geodesic and Conformally Reeb Vector Fields on Flat 3-Manifolds

“…In other words, our notion of induced contact structures seems to sit somewhere in between weakly and strongly compatible contact structures. While the '1/4pinched contact sphere theorem' is known to be false for weakly compatible metrics in general [25], there are no counterexamples yet in the case of induced contact structures.…”

Magnetic Jacobi Fields in 3-Dimensional Sasakian Space Forms

“…Remark 4.1. On an oriented Riemannian 3-manifold (M, g, dv g ), a vector field X is said to be a Beltrami field if it satisfies div X = 0 and curl X = λX for some smooth function λ [30].…”