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- W614399296 abstract "In the early 1920's Karl Lowner (later Charles Loewner) introduced a simple differential equation that encodes domains in the complex plane changing continuously in time t into a real-valued function of t. This thesis centers around this equation, called the Loewner Equation, and has three separate parts. The first part proves it is satisfied by conformal maps taking the complement of a simple curve in the upper half plane to the upper half plane. The second part proves the existence and uniqueness of a conformal solution to the Loewner Equation with continuous driving function. Although by no means new results, the intention is nevertheless to provide them from a fresh point of view by means of compactness, rigor and variations of selected partial proofs.The third part tentatively explores new domains. It starts with treating the existence of a generating curve for the domain of the Loewner Equation with Holder-1/2 continuous driving function of norm less than 4. In establishing the existence of such a curve, finding a bound for the absolute value of the Loewner function's derivative is crucial. We reproduce the proof of the existence of such a bound by methods of S. Rohde, H. Tran and M. Zinsmeister, and note that these methods seem suitable for investigating similar bounds for the argument of the same function for driving-norm less than 2√2. We present a result for norm less than √2, but otherwise reach the conclusion that the methods considered are unable to produce the desired bound for norms in the interval [√2, 2√2). The explicit traces and maps of logarithmic spirals are calculated showing that the correct upper limit for the norm regarding the existence of a non-trivial bound to the argument is no larger than 2√2." @default.
- W614399296 created "2016-06-24" @default.
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- W614399296 date "2015-01-01" @default.
- W614399296 modified "2023-09-24" @default.
- W614399296 title "The Loewner Equation: An introduction and the winding of its trace" @default.
- W614399296 hasPublicationYear "2015" @default.
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