9.3 Proof of the Riemann mapping theorem

The Riemann mapping theorem is a cornerstone of complex analysis, establishing the existence of conformal mappings between simply connected domains. It provides a powerful tool for understanding complex functions and their properties, allowing us to study arbitrary domains by mapping them to well-understood ones.

The proof of this theorem involves constructing a sequence of holomorphic functions that converge to the desired mapping. It relies on key results like Montel's theorem, the argument principle, and Hurwitz's theorem. The proof demonstrates the flexibility of conformal mappings and their importance in complex analysis.

Riemann mapping theorem

• Fundamental result in complex analysis that establishes the existence of conformal mappings between simply connected domains
• Plays a crucial role in the study of complex functions and their properties
• Provides a powerful tool for understanding the geometry and topology of complex domains

Statement of theorem

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• If $D$ is a simply connected domain in the complex plane $\mathbb{C}$ that is not the entire plane, then there exists a unique conformal mapping $f$ from $D$ onto the unit disk $\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$
• The mapping $f$ is a one-to-one holomorphic function that preserves angles and local geometry
• The uniqueness of $f$ is up to composition with an automorphism of the unit disk (a Möbius transformation)

Significance in complex analysis

• Demonstrates the remarkable flexibility and richness of conformal mappings in the complex plane
• Allows for the study of complex functions on arbitrary simply connected domains by reducing them to the well-understood unit disk
• Provides a bridge between complex analysis and other areas of mathematics, such as geometry, topology, and dynamical systems

Proof overview

• The proof of the Riemann mapping theorem involves several key steps and ideas from complex analysis
• Relies on a combination of classical results, such as Montel's theorem, the argument principle, and Hurwitz's theorem
• Constructively builds the desired conformal mapping using tools like the Schwarz-Christoffel formula and the properties of holomorphic functions

Key steps and ideas

• Construct a sequence of holomorphic functions on the domain $D$ that converges uniformly to a mapping onto the unit disk or upper half-plane
• Use the properties of normal families (Montel's theorem) to ensure the existence of a convergent subsequence
• Apply the argument principle to show that the limit function is univalent (one-to-one) on $D$
• Invoke Hurwitz's theorem to prove that the limit function is non-vanishing and preserves the orientation of angles
• Establish the conformality of the mapping by verifying its injectivity, surjectivity, and the existence of a holomorphic inverse
• Prove the uniqueness of the mapping up to composition with automorphisms of the unit disk using the properties of Möbius transformations

Preliminary results

• The proof of the Riemann mapping theorem relies on several classical results from complex analysis
• These results provide the necessary tools and insights to construct and analyze the desired conformal mapping

Montel's theorem

• States that a family of holomorphic functions on a domain $D$ is normal if it is uniformly bounded on compact subsets of $D$
• Ensures the existence of a uniformly convergent subsequence of holomorphic functions
• Plays a crucial role in the construction of the conformal mapping by allowing the extraction of a convergent limit function

Argument principle

• Relates the number of zeros and poles of a meromorphic function inside a closed contour to the change in the argument of the function along the contour
• Used to prove that the limit function obtained from Montel's theorem is univalent (one-to-one) on the domain $D$
• Helps establish the injectivity of the conformal mapping

Hurwitz's theorem

• States that if a sequence of non-vanishing holomorphic functions converges uniformly on compact subsets of a domain, then the limit function is either identically zero or never vanishes
• Ensures that the limit function obtained from Montel's theorem preserves the orientation of angles and does not vanish on $D$
• Contributes to proving the conformality of the mapping

Constructing the mapping

• The proof of the Riemann mapping theorem involves constructing a conformal mapping from the given simply connected domain $D$ to a canonical domain, such as the unit disk or the upper half-plane
• Different methods can be employed to build the mapping, depending on the specific properties of the domain and the desired target domain

Mapping to the unit disk

• One approach is to construct a sequence of holomorphic functions that map $D$ onto expanding subsets of the unit disk
• The functions are chosen to satisfy certain normalization conditions, such as fixing the image of a specific point in $D$ and having a positive derivative at that point
• Montel's theorem ensures the existence of a uniformly convergent subsequence, whose limit function maps $D$ onto the entire unit disk

Mapping to the upper half-plane

• Alternatively, the mapping can be constructed by targeting the upper half-plane $\mathbb{H} = \{z \in \mathbb{C} : \text{Im}(z) > 0\}$
• The construction proceeds similarly to the unit disk case, with appropriate modifications to the normalization conditions
• The resulting limit function maps $D$ conformally onto the upper half-plane

Schwarz-Christoffel formula

• In some cases, the conformal mapping can be explicitly constructed using the Schwarz-Christoffel formula
• This formula provides a method for mapping the upper half-plane or the unit disk onto polygonal domains
• By composing the Schwarz-Christoffel mapping with the mapping from $D$ to the upper half-plane or unit disk, a conformal mapping from $D$ to a polygonal domain can be obtained

Establishing conformality

• To prove that the constructed mapping is indeed a conformal mapping, several key properties need to be established
• These properties ensure that the mapping preserves angles, local geometry, and has a holomorphic inverse

Injectivity of the mapping

• The mapping $f$ constructed in the proof must be shown to be injective (one-to-one) on the domain $D$
• This is typically achieved using the argument principle, which relates the number of zeros of $f - w$ inside $D$ to the change in the argument of $f - w$ along the boundary of $D$
• By carefully analyzing the argument change, it can be proven that $f - w$ has exactly one zero in $D$ for each $w$ in the target domain, implying the injectivity of $f$

Surjectivity of the mapping

• In addition to injectivity, the mapping $f$ must also be surjective onto the target domain (unit disk or upper half-plane)
• Surjectivity is often established by showing that the image of $D$ under $f$ is both open and closed in the target domain, and then invoking the connectedness of the target domain
• The openness of the image follows from the open mapping theorem for holomorphic functions, while the closedness is a consequence of the compactness of the closure of $D$ and the continuity of $f$

Holomorphic inverse function

• To complete the proof of conformality, the existence of a holomorphic inverse function for $f$ needs to be demonstrated
• The inverse function theorem for holomorphic functions states that if $f$ is holomorphic and has a non-zero derivative at a point, then $f$ is locally invertible near that point, and the inverse is also holomorphic
• By showing that $f'$ is non-vanishing on $D$ (using Hurwitz's theorem or the properties of the constructed mapping), the existence of a global holomorphic inverse for $f$ can be established, confirming its conformality

Uniqueness of the mapping

• The Riemann mapping theorem asserts that the conformal mapping $f$ from $D$ to the unit disk is unique up to composition with an automorphism of the unit disk
• This uniqueness property is a consequence of the symmetries and rigidity of the unit disk

Automorphisms of the unit disk

• The automorphisms of the unit disk are the Möbius transformations that map the unit disk onto itself
• These transformations form a three-parameter family and can be represented as $\varphi(z) = e^{i\theta} \frac{z - a}{1 - \bar{a}z}$, where $|a| < 1$ and $\theta \in [0, 2\pi)$
• Composing a conformal mapping $f$ with an automorphism of the unit disk yields another conformal mapping from $D$ to the unit disk

Normalization conditions

• To prove the uniqueness of the conformal mapping, additional normalization conditions are imposed on $f$
• Common normalization conditions include specifying the image of a particular point in $D$ and the direction of the derivative at that point
• For example, if $f(z_0) = 0$ and $f'(z_0) > 0$ for some $z_0 \in D$, then any other conformal mapping $g$ from $D$ to the unit disk satisfying these conditions must coincide with $f$
• The uniqueness of $f$ up to automorphisms of the unit disk follows from the fact that any two conformal mappings satisfying the normalization conditions differ by a rotation of the unit disk

Consequences and applications

• The Riemann mapping theorem has far-reaching consequences and applications in complex analysis and related fields
• It provides a powerful tool for studying the properties of complex functions and the geometry of complex domains

Uniformization of simply connected domains

• The theorem implies that any simply connected domain in the complex plane (other than the entire plane itself) can be conformally mapped onto a "simple" canonical domain, such as the unit disk or the upper half-plane
• This process is known as uniformization and allows for the study of complex functions on arbitrary simply connected domains by reducing them to well-understood canonical domains
• Uniformization simplifies the analysis of geometric and analytic properties of functions and facilitates the solution of various problems in complex analysis

Conformal equivalence

• Two domains in the complex plane are said to be conformally equivalent if there exists a conformal mapping between them
• The Riemann mapping theorem establishes the conformal equivalence of any two simply connected domains (other than the entire plane)
• This equivalence relation partitions the class of simply connected domains into conformal equivalence classes, each represented by a canonical domain (unit disk or upper half-plane)
• Conformal equivalence allows for the transfer of properties and results between different domains, providing a unified framework for studying complex functions

Riemann surfaces

• The concept of conformal equivalence can be extended to more general surfaces, leading to the theory of Riemann surfaces
• A Riemann surface is a one-dimensional complex manifold, locally modeled on the complex plane or the extended complex plane (Riemann sphere)
• The uniformization theorem, a generalization of the Riemann mapping theorem, states that every simply connected Riemann surface is conformally equivalent to one of three canonical surfaces: the complex plane, the unit disk, or the Riemann sphere
• Riemann surfaces play a fundamental role in the study of algebraic curves, holomorphic functions, and the geometry of complex manifolds

Historical context

• The Riemann mapping theorem is named after the German mathematician Bernhard Riemann, who first stated and proved the theorem in his dissertation in 1851
• Riemann's proof was a significant milestone in the development of complex analysis and laid the foundation for the theory of conformal mappings

Riemann's original proof

• Riemann's original proof of the theorem relied on the Dirichlet principle, which asserts the existence of a harmonic function with prescribed boundary values on a domain
• He used the Dirichlet principle to construct a harmonic function on the given simply connected domain and then used its harmonic conjugate to define the conformal mapping
• However, the Dirichlet principle was later found to be flawed, and Riemann's proof was considered incomplete

Subsequent simplifications and generalizations

• In the years following Riemann's work, several mathematicians sought to provide rigorous proofs of the Riemann mapping theorem and to extend its scope
• Notable contributions were made by Carathéodory, Koebe, and Poincaré, among others
• Carathéodory gave a simplified proof using normal families and the Montel's theorem, which became the basis for many modern proofs of the theorem
• Koebe and Poincaré independently proved the uniformization theorem, generalizing the Riemann mapping theorem to arbitrary simply connected Riemann surfaces
• The Riemann mapping theorem continues to be an active area of research, with ongoing work on its generalizations, applications, and connections to other branches of mathematics