Integrable Hamiltonian structure of conformal maps

Abstract:
A survey of recent studies on the integrable structures of the complex analysis is presented. In particular we consider a conformal map of a domain on the complex plane to a disc of the unit radius. We characterize the domain by means of its harmonic moments and the Schwarz function.

It is possible to define such a Poisson structure that the functions $z(w)$, $\overline{z}(w^{-1})$ appeared to be a conjugate variables with respect to this structure. Here $z(w)$ is a univalent conformal map of the exterior of the unit disc to the exterior of the domain. Deformations of the functions $z(w)$, $\overline{z}(w^{-1})$ with respect to the moments appeared to be the Hamiltonian equations where the Hamiltonians are partial derivatives of the Schwarz function potential with respect to the moments. These equations  are known as the Lax-Sato equations.

Some applications of this theory are discussed.