Dispersionless version of the constrained Toda hierarchy and symmetric radial L\"owner equation

Abstract

We study the dispersionless version of the recently introduced constrained Toda hierarchy. Like the Toda lattice itself, it admits three equivalent formulations: the formulation in terms of Lax equations, the formulation of the Zakharov-Shabat type and the formulation through the generating equation for the dispersionless limit of logarithm of the tau-function. We show that the dispersionless constrained Toda hierarchy describes conformal maps of reflection-symmetric planar domains to the exterior of the unit disc. We also find finite-dimensional reductions of the hierarchy and show that they are characterized by a differential equation of the L\"owner type which we call the symmetric radial L\"owner equation. It is also shown that solutions to the symmetric radial L\"owner equation are conformal maps of the exterior of the unit circle with two symmetric slits to the exterior of the unit circle.