# C4.8 Complex Analysis: Conformal Maps and Geometry - Material for the year 2019-2020

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The only necessary prerequisite is a basic complex analysis course: analytic functions, Taylor series, contour integration, Cauchy theorems, residues, maximum modulus, Liouville's theorem. Some basic results that might not be covered by a basic course (such as argument principle and the Rouche theorem) will be given in the introductory chapter of the lecture notes.

16 lectures

### Assessment type:

- Written Examination

The theory of conformal maps has a very long and rich history. Its foundations were laid out by Riemann in mid XIX century and it was one of the central areas of the Complex Analysis since then.

The aim of the course is to teach the principal techniques and methods of analytic and geometric function theory. We start by proving the Riemann mapping theorem which claims that all non-trivial simply connected domains are conformal images of the unit disc. This provides a connection between planar domains and 1-to-1, or univalent, analytic functions in the unit disc.

In this course we will study the Riemann mapping theorem and general theory of univalent functions, study how geometric properties of domains are related to analytic properties of functions. We will also study conformal invariants: quantities that do not change under conformal transformations. We will see that many of these invariants have a simple geometric interpretation and could be used to obtain numerous beautiful results.

Students will have been introduced to ideas and techniques of geometric function theory that play important role and have a lot of applications in other areas of analysis. In particular, they will learn the proof of the Riemann mapping theorem and the concept of conformal invariants.

We will cover the following topics:

- The Riemann mapping theorem. The main goal will be to prove Riemann's theorem which tells us that any non-trivial simply-connected domain can be conformally mapped onto the unit disc. This will be the key result for the entire course since it will allow us to connect the geometry of a domain with the analytical properties of a conformal map which sends this domain to the unit disc. Within this section we will discuss
- Möbius transformations and the Schwarz lemma
- Normal families, the Montel and Hurwitz theorems
- Proof of Riemann mapping theorem
- Constructive uniformization: Christoffel-Schwarz mappings and iterative methods
- Boundary correspondence, accessible points and prime ends
- Uniformization of multiply-connected domains (without proof)
- Applications: the Dirichlet boundary problem
- Basic theory of univalent functions. In this part we will study the universal estimates that are valid for all conformal maps on the unit disc.
- Area theorem and coefficient estimates
- The Koebe $1/4$ theorem, distortion theorems
- Conformal invariants: extremal length and its applications
- Conformal invariants
- Conformal invariance of Green's function and harmonic measure
- Extremal length, its conformal invariance and applications

There is a number of books that cover many of these topic. I would suggest

- L. Ahlfors,
*Complex analysis*. This is a very good advanced textbook on Complex analysis. If you are a bit rusty on the basic complex analysis, then you might find everything you need (and a bit more) in Chapters 1-4. We will cover some of the material from chapters 5-6. - L. Ahlfors,
*Conformal Invariants*. We will cover some topics from Chapters 1-6. - Ch. Pommerenke,
*Univalent functions*. This book is mostly for further reading. We will discuss some of the results that are covered in Chapters 1,5,6, and 10. - Ch. Pommerenke,
*Boundary behaviour of conformal maps*. This is an updated version of the previous book. We will be interested in Chapters 1,4, and 8. - P. Duren,
*Univalent functions*. This is an excellent book about general theory of the univalent functions. We are mostly interested in the first three chapters. - G. Goluzin,
*Geometric Theory of Functions of a Complex Variable*. This book contains vast amount of information about the geometric function theory. We will cover some of the results from the first four chapters.