On Villat’s Kernels and BMD Schwarz Kernels in Komatu-Loewner Equations

The classical Loewner differential equation for simply connected domains is attracting new attention since Oded Schramm launched in 2000 the stochastic Loewner evolution (SLE) based on it. The Loewner equation itself has been extended to various canonical domains of multiple connectivity after the works by Y. Komatu in 1943 and 1950, but the Komatu-Loewner (K-L) equations have been derived rigorously only in the left derivative sense. In a recent work, Z.-Q. Chen, M. Fukushima and S. Rhode prove that the K-L equation for the standard slit domain is a genuine ODE by using a probabilistic method together with a PDE method, and that the right hand side of the equation admits an expression in terms of the complex Poisson kernel of the Brownian motion with darning (BMD). In the present paper, K-L equations for the annulus and circularly slit annili are investigated. For the annulus, we establish a K-L equation as a genuine ODE possessing a normalized Villat’s kernel on its right hand side by using a variant of the Carathéodory convergence theorem for annuli indicated by Komatu. This method is also used to obtain the same K-L equation in the right derivative sense on annulus for a more general family of growing hulls that satisfies a specific right continuity condition usually adopted in the SLE theory. Villat’s kernel is then identified with a BMD Schwarz kernel for the annulus. Finally we derive K-L equations for circularly slit annuli in terms of their normalized BMD Schwarz kernels, but only in the left derivative sense when at least one circular slit is present.