K"ahler 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 incompressiblility leads to equations of Monge--Amp\`ere type for the stream function, when the Laplacian of the pressure is known. In two dimensions a K\"ahler geometry is described, which is associated with the Monge--Amp\`ere problem. This K\"ahler structure is then generalised to `two-and-a-half dimensional' flows, of which Burgers' vortex is one example. In three dimensions, we show how a generalized Calabi--Yau structure emerges in a special case.