Carl Ringqvist: The Loewner Equation: An introduction and the winding of its trace

In the early 1920's, Karl Löwner (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 and second part gives an introduction to present theory. 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. The proof of such a bound by methods of S. Rohde, H. Tran and M. Zinsmeister is reproduced. These methods are then used for investigating similar bounds for the argument of the same function; a quantity with geometrical significance as well as applications within the treatment of the famous SLE-curves.

During the seminar, we outline the ideas and main results of this investigation. The most notable result, that the upper limit for the Holder-norm for which our desired bound is possible is no larger than 22‾√, is derived in detail.