Loewner equations and reductions of dispersionless hierarchies
The equations of Loewner type can be derived in two very different contexts: one of them is complex analysis and the theory of parametric conformal maps and the other one is the theory of integrable systems. In this paper we compare the both approaches. After recalling the derivation of Löwner equations based on complex analysis we review one- and multi-variable reductions of dispersionless integrable hierarhies (dKP, dBKP, dToda, and dDKP). The one-vaiable reductions are described by solutions of different versions of Loewner equation: chordal (rational) for dKP, quadrant for dBKP, radial (trigonometric) for dToda and elliptic for DKP. We also discuss multi-variable reductions which are given by a system of Loewner equations supplemented by a system of partial differential equations of hydrodynamic type. The solvability of the hydrodynamic type system can be proved by means of the generalized hodograph method.