Conformal Mappings and Dispersionless Toda Hierarchy

Abstract

Let ${\mathfrak{D}}$ be the space consists of pairs (f, g), where f is a univalent function on the unit disc with f(0) = 0, g is a univalent function on the exterior of the unit disc with g(∞) = ∞ and f′(0)g′(∞) = 1. In this article, we define the time variables ${t_n, n\in \mathbb{Z}}$, on ${\mathfrak{D}}$ which are holomorphic with respect to the natural complex structure on ${\mathfrak{D}}$ and can serve as local complex coordinates for ${\mathfrak{D}}$ . We show that the evolutions of the pair (f, g) with respect to these time coordinates are governed by the dispersionless Toda hierarchy flows. An explicit tau function is constructed for the dispersionless Toda hierarchy. By restricting ${\mathfrak{D}}$ to the subspace Σ consists of pairs where ${f(w)=1/\overline{g(1/\bar{w})}}$, we obtain the integrable hierarchy of conformal mappings considered by Wiegmann and Zabrodin [31]. Since every C ¹ homeomorphism γ of the unit circle corresponds uniquely to an element (f, g) of ${\mathfrak{D}}$ under the conformal welding ${\gamma=g^{-1}\circ f}$, the space Homeo _C (S ¹) can be naturally identified as a subspace of ${\mathfrak{D}}$ characterized by f(S ¹) = g(S ¹). We show that we can naturally define complexified vector fields ${\partial_n, n\in \mathbb{Z}}$ on Homeo _C (S ¹) so that the evolutions of (f, g) on Homeo _C (S ¹) with respect to ∂ _n satisfy the dispersionless Toda hierarchy. Finally, we show that there is a similar integrable structure for the Riemann mappings (f ⁻¹, g ⁻¹). Moreover, in the latter case, the time variables are Fourier coefficients of γ and 1/γ ⁻¹.