In this paper we construct L€ owner chains which enable us to derive quasiconformal extension criteria for typical classes of univalnet functions. This method also provides us explicit quasiconformal extensions.

In this note we consider a multi-slit Loewner equation with constant coefficients that describes the growth of multiple SLE curves connecting N points on R to infinity within the upper halfplane. For every N ∈ N, this equation provides a measure valued process t → {α N,t }, and we are interested in the limit behaviour as N → ∞. We prove tightness of the sequence {α N,t } N ∈N under certain assumptions and address some further problems.

Recently del Monaco and Schleißinger addressed an interesting problem whether one can take the limit of multiple Schramm-Loewner evolution (SLE) as the number of slits N goes to infinity. When the N slits grow from points on the real line R in a simultaneous way and go to infinity within the upper half plane H, an ordinary differential equation describing time evolution of the conformal map g t (z) was derived in the N → ∞ limit, which is coupled with a complex Burgers equation in the inviscid limit. It is well known that the complex Burgers equation governs the hydrodynamic limit of the Dyson model defined on R studied in random matrix theory, and when all particles start from the origin, the solution of this Burgers equation is given by the Stieltjes transformation of the measure which follows a time-dependent version of Wigner's semicircle law. In the present paper, first we study the hydrodynamic limit of the multiple SLE in the case that all slits start from the origin. We show that the time-dependent version of Wigner's semicircle law determines the time evolution of the SLE hull, K t ⊂ H ∪ R , in this hydrodynamic limit. Next we consider the situation such that a half number of the slits start from a > 0 and another half of slits start from −a < 0, and determine the multiple SLE in the hydrodynamic limit. After reporting these exact solutions, we will discuss the universal long-term behavior of the multiple SLE and its hull K t in the hydrodynamic limit.

In 1972, Becker [J. Reine Angew. Math. 255 (1972, 23-43] discovered a construction of quasiconformal extensions making use of the classical radial Loewner chains. In this paper we develop a chordal analogue of Becker's construction. As an application, we establish new sufficient conditions for quasiconformal extendibility of holomorphic functions and give a simplified proof of one well-known result by Becker and Pommerenke for functions in the half-plane [J. Reine Angew. Math. 354 (1984), 74-94].

In this paper we confirm that several crucial theorems due to Pommerenke and Becker for the theory of Löwner chains work well without normalization on the complex-valued first coefficient. As applications of those considerations, some new univalent and quasiconformal extension criteria are given in the last section.