Geometric Function Theory Notes

Geometric Function Theory is a branch of mathematics that deals with the geometric properties of analytic functions, especially those functions defined on the complex plane.

Conformal mappings are transformations that preserve angles locally. In Geometric Function Theory, the study of conformal mappings is central. These mappings are often used to map one region onto another while preserving angles.

**Schwarz Lemma:**

The Schwarz lemma provides information about the size and location of the image of the unit disk under a holomorphic function. It is a fundamental tool in the study of conformal mappings.

**Riemann Mapping Theorem:**

This theorem states that any simply connected proper subdomain of the complex plane is conformally equivalent to the unit disk. It has significant implications for the study of conformal mappings.

**Harmonic Measure:**

Harmonic measure is a measure associated with the boundary behavior of harmonic functions. It plays a crucial role in understanding the behavior of harmonic functions and their connections to conformal mappings.

**Quasiconformal Mappings:**

Quasiconformal mappings are generalizations of conformal mappings that relax the condition of angle preservation. They are used to study the distortion properties of mappings between domains.

**Boundary Behavior of Holomorphic Functions:**

Understanding the behavior of holomorphic functions near their boundary is a key aspect of Geometric Function Theory. This includes topics like boundary regularity and the relation between the behavior of a function and its boundary values.

**Carathéodory’s Theorem:**

Carathéodory’s theorem characterizes the boundary behavior of conformal mappings. It provides conditions under which a homeomorphism on the boundary extends to a homeomorphism of the closure.

**Harmonic Measure and Potential Theory:**

Geometric Function Theory often involves the study of harmonic measure and its connections to potential theory. This includes topics like the Dirichlet problem and the Poisson kernel.

**Loewner’s Theorem:**

Loewner’s theorem is concerned with the parametrization of univalent (injective) functions. It characterizes the growth of such functions and their boundary behavior.

**Hyperbolic Geometry and Poincaré Metric:**

The study of hyperbolic geometry, including the Poincaré metric, is an integral part of Geometric Function Theory. It provides a framework for understanding the geometry of non-Euclidean spaces.

**Schramm-Loewner Evolution (SLE):**

SLE is a mathematical theory that describes the random curves that arise in certain physical and mathematical processes. It has connections to conformal invariance.

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