Kahler Geometry and the Navier-Stokes Equations

We study the Navier-Stokes and Euler equations of incompressible hydrodynamics in two and three spatial dimensions and show how the constraint of incompress-iblility leads to equations of Monge{Ampere type for the stream function, when the Laplacian of the pressure is known. In two dimensions a Kahler geometry is described, which is associated with the Monge{Ampere problem. This Kahler struc-ture is then generalised to `two-and-a-half dimensional ' ows, of which Burgers' vortex is one example. In three dimensions, we show how a generalized Calabi{Yau structure emerges in a special case