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Title:  On Existence and Convergence of SLE in Multiply Connected Domains 
Author(s):  Kou, Ming 
Doctoral Committee Chair(s):  Robert O. Bauer 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
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 KomatuLoewner equation is Lipschitz. This is crucial in proving the existence theorems of the KomatuLoewner equations for SLE in multiply connected domains. A now proof of the existence theorem for the chordal KomatuLoewner 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. 
Issue Date:  2007 
Type:  Text 
Language:  English 
Description:  81 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 2007. 
URI:  http://hdl.handle.net/2142/86895 
Other Identifier(s):  (MiAaPQ)AAI3301172 
Date Available in IDEALS:  20150928 
Date Deposited:  2007 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois