I'm learning about tight vs. overtwisted contact structures in contact geometry. I understand that we care about the existence/nonexistence of overtwisted disks in a contact structure in part because the distinction has proved useful (e.g., in classification).


But since contact geometry has a lot of applications to physics, I'm curious: Are there any big *physical* reasons for why we care about this distinction?


(reposted from math.SE)



Physical motivation for tight/overtwisted dichotomy
In the physics of fluids, a reason for caring about tightness of the contact structure is the idea/conjecture that overtwisted discs raise the energy of the fluid.


The velocity field of an inviscid, incompressible fluid flow on a Riemannian manifold corresponds to a contact 1-form in dimension three.$^\ast$ In this context one can investigate how the topological tight/overtwisted dichotomy for contact structures relates to the physical properties of the fluid like its energy. *Is the energy minimizing flow necessarily a tight contact structure?* Ghrist and Komendarczyk, in [Overtwisted energy-minimizing curl eigenfields](https://arxiv.org/abs/math/0411319), show that the answer is "no" in general, but "yes" under additional symmetry conditions.


$^\ast$ For an introduction and overview of this correspondence, see [On the contact topology and geometry of ideal fluids](https://www.math.upenn.edu/~ghrist/preprints/fluidshandbook.pdf).


