Path geometries, Fourier modes and holomorphic curves

A path geometry on a surface M prescribes an immersed path for every
direction in each tangent space of M. Fixing a Riemannian metric g on M, a
path geometry can be encoded in terms of a real-valued function f on the
unit tangent bundle of (M,g). Requiring the paths to agree with the
geodesics of some connection is equivalent to the condition that the
vertical Fourier expansion of f contains only terms of degree 1 and 3. I
will relate the vanishing of these Fourier modes to (pseudo-)holomorphic
curves and discuss related PDE problems. Joint with Gabriel Paternain.