Conformal Maps and Geometry

The following sections are included:

• Dedication
• Preface
• Acknowledgements
• Contents

All parts of analysis have a significant geometric component and this is even more so in the case of complex analysis. Unfortunately, this connection is not usually covered by standard complex analysis courses. Probably the only result that might appear in a basic course is the geometrical form of the argument principle, which relates the number of zeroes inside a contour to the winding number of its image…

In this chapter, we discuss a class of results that form a foundation for the rest of the book. We are interested in conformal classification of planar domains, that is, given two domains, can we find a conformal map from one domain onto the other?…

In this chapter, we discuss general properties of univalent functions. We pay special attention to universal boundary estimates and connections between analytic and geometric properties.

In this chapter, we discuss various quantities that do not change under conformal transformations or change in a very simple, predictable, and controllable way.

We have seen examples of conformal invariants. The first class of examples stemmed from the fact that harmonic functions are conformally invariant. This implied that solutions to the Dirichlet boundary problem are also conformally invariant. In particular, this gives the conformal invariance of Green's function and harmonic measures. These are important examples of functions that are conformally invariant. There are also some quantities that are conformally invariant. We have already encountered two examples of this type: two rectangles or two annuli are conformally equivalent if and only if they have the same 'shape'. More generally, given a conformal rectangle or a doubly connected domain, they can be mapped onto a proper rectangle or an annulus, respectively. The 'shape' of these uniformizing domains is a conformal invariant. This approach works with virtually any definition of the shape. In the case of rectangles, the canonical notion is the ratio of side lengths and in the case of annuli it is the conformal modulus. In this chapter, we develop a general and versatile approach to conformal invariants. In particular, we see that the 'shape' of a rectangle and an annulus are particular instances of the same notion.

Loewner evolution is a classical subject first introduced in 1923 by Charles Loewner in his famous paper, “Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I”. This work was motivated by the extremal problems of function theory and showed a way to describe an evolution of slit maps. The first application of this technique was the proof of the n = 3 case of the Bieberbach conjecture. Bieberbach, who was the editor of Mathematische Annalen, the journal where Loewner published his paper, was so sure that this technique was sufficiently powerful that Loewner would be able to give the complete proof very soon that he added ‘Part I’ to the title. But Loewner never returned to the subject and to some extent the Loewner evolution had faded away until interest was revived in 1985 when de Branges (1985) used it to finally prove the Bieberbach conjecture. After the initial excitement which revived interest in the Loewner evolution, it went into decline one more time, but interest was renewed when it became one of the key ingredients in the seminal paper of Oded Schramm (2000). In this paper, he introduced a stochastic version of the Loewner evolution which is called SLE. Originally, SLE meant stochastic Loewner evolution, but in recent years, people have started to use Schramm–Loewner evolution instead. This stochastic version turned out to be an extremely valuable tool in the study of critical models of statistical physics and led to a very rapid progress in the mathematically rigorous understanding of these models. Interested readers can find more information about SLE in books by Lawler (2005) and Kemppainen (2017). This book also contains an introduction to the theory of Loewner evolution written from a more probabilistic perspective.

The following sections are included:

• Bibliography
• Index