In this work, the author studies contact structures via adapted Riemannian metrics. The main observation is a relation between characteristic surfaces of contact structures and zero sets of solutions to certain subelliptic partial differential equations. This relation makes it possible to derive, under a symmetry assumption, lower bounds for the volume of the manifold supporting overtwisted or virtually overtwisted contact structures arising from a certain class of invariant curl eigenfields. Further, it has implications in the energy relaxation of this special class of fluid flows. Specifically, the author provides a counterexample to the conjecture of Etnyre and Ghrist posed in their work on hydrodynamics of contact structures.

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