On Existence and Convergence of SLE in Multiply Connected Domains

Abstract

Using tools from complex analysis, this thesis extends some arguments given by Garabedian in proving Hadamard's variation formula of the Green's function to show that the vector field associated with the chordal and bilateral Komatu-Loewner equation is Lipschitz. This is crucial in proving the existence theorems of the Komatu-Loewner equations for SLE in multiply connected domains. A now proof of the existence theorem for the chordal Komatu-Loewner equation, as an example of the general method, is also given. Next, this thesis discusses the convergence of bilateral SLE to radial SLE in multiply connected domains in a special case: the convergence of annulus bilateral SLE to radial SLE. The convergence is described in terms of the driving function, which is a random motion on the boundary of the domain. As the inner circular hole of a domain where a bilateral SLE curve grows shrinks to a point, its driving function goes to a limit, which is a natural driving function of the corresponding radial SLE, up to a time change. For general standard (multiply connected) domains, a comparison theorem of bilateral SLE and radial SLE is given and proved: if the radius of the inner circular hole is small enough (bilateral case), the natural driving function for the radial SLE associated to a radial standard domain is close to the natural driving function for the bilateral SLE associated to a bilateral standard domain, which is the radial domain except that it has the circular hole. If kappa = 6, the comparison theorem holds without taking the limit of the size of the inner circular hole and implies a locality property for SLE in multiply connected domains.