Conformal weldings and Dispersionless Toda hierarchy

TL;DR

This paper demonstrates that the evolution of conformal maps associated with circle homeomorphisms can be described by the dispersionless Toda hierarchy, revealing deep integrable structures in conformal mapping theory.

Contribution

It establishes a connection between conformal map evolutions and the dispersionless Toda hierarchy, extending previous integrable structure results to a broader class of solutions.

Findings

Conformal map evolutions follow the dispersionless Toda hierarchy.

Relations to integrable structures of conformal maps are clarified.

An extended hierarchy encompassing previous solutions is proposed.

Abstract

Given a $C^1$ homeomorphism of the unit circle $\gamma$, let $f$ and $g$ be respectively the normalized conformal maps from the unit disc and its exterior so that $\gamma= g^{-1}\circ f$ on the unit circle. In this article, we show that by suitably defined time variables, the evolutions of the pairs $(g, f)$ and $(g^{-1}, f^{-1})$ can be described by an infinite set of nonlinear partial differential equations known as dispersionless Toda hierarchy. Relations to the integrable structure of conformal maps first studied by Wiegmann and Zabrodin \cite{WZ} are discussed. An extension of the hierarchy which contains both our solution and the solution of \cite{WZ} is defined.