Overtwisted energy-minimizing curl eigenfields

This paper concerns topological and geometric properties of energy-minimizing solutions to the steady Euler equations for a fluid on a three-dimensional manifold. Specifically, we consider energy-minimizing divergence-free eigenfields of the curl operator in dimension three from the perspective of contact topology. We give a negative answer to a question of Etnyre and the first author by constructing curl eigenfields which minimize L2 energy on their co-adjoint orbit, yet are orthogonal to an overtwisted contact structure. We then show progress towards the conjecture that K-contact structures on Seifert fibred manifolds always define tight minimizers.

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