The **Riemann Mapping Theorem** is a fundamental result in complex analysis, which states:

**Theorem:**
Let \( U \) be a non-empty simply connected open subset of the complex plane \( \mathbb{C} \), which is not the entire plane. Then there exists a bijective holomorphic (conformal) map \( f \) from \( U \) onto the open unit disk \( D = \{ z \in \mathbb{C} : |z| < 1 \} \). Furthermore, this map is unique up to composition with a Möbius transformation of the form
\[
T(z) = e^{i\theta} \frac{z - a}{1 - \overline{a} z}
\]
for some \( a \in D \) and \( \theta \in \mathbb{R} \).

### Proof Outline:

#### Step 1: Uniformization and Normalization
Let \( U \) be a non-empty simply connected open subset of \( \mathbb{C} \), which is not \( \mathbb{C} \) itself. We seek a holomorphic map \( f: U \to D \) such that \( f(U) = D \), and \( f \) is injective.

By the **Carathéodory-Osgood theorem** (an extension of the **Weierstrass approximation theorem**), we know that a holomorphic map can be found if we normalize the problem. First, pick a point \( z_0 \in U \). Then, by the **Schwarz lemma**, we can normalize the map by requiring \( f(z_0) = 0 \) and \( f'(z_0) > 0 \). This normalization ensures uniqueness up to Möbius transformations.

#### Step 2: Existence of the Map
The existence of such a conformal map relies on a deeper principle: **Dirichlet’s principle** or the **method of extremal length**. This principle provides a way to solve certain extremal problems, and in this context, it can be used to construct the desired holomorphic map.

Without going into full detail (which requires advanced variational methods and potential theory), the idea is to minimize a certain functional, which gives a harmonic function whose conjugate is used to construct the desired conformal map.

#### Step 3: Uniqueness of the Map
Once we have established the existence of the conformal map, we use the normalization condition \( f(z_0) = 0 \) and \( f'(z_0) > 0 \) to ensure uniqueness. This normalization fixes the rotation and scaling, but it leaves some freedom for Möbius transformations. Specifically, the set of all such holomorphic maps is unique up to the transformations of the form:
\[
T(z) = e^{i\theta} \frac{z - a}{1 - \overline{a} z}
\]
where \( a \in D \) and \( \theta \in \mathbb{R} \).

Thus, any two conformal maps from \( U \) to \( D \) must differ by such a Möbius transformation.

#### Step 4: Regularity and Boundary Behavior
Finally, the map \( f \) is holomorphic, and by the **Schauder estimates** (from elliptic regularity theory), we can show that the map is smooth (infinitely differentiable) within \( U \). Furthermore, under appropriate conditions, the map can extend continuously to the boundary of \( U \), as per the **Carathéodory extension theorem**.

Thus, the theorem is proven.

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### Do you want more details or have any questions?

### Related Questions:
1. What is the method of extremal length in conformal mapping?
2. How does Dirichlet’s principle help in constructing holomorphic functions?
3. Can you explain the Carathéodory extension theorem in more detail?
4. How does the Schwarz lemma contribute to the uniqueness part of the Riemann Mapping Theorem?
5. What are the generalizations of the Riemann Mapping Theorem to higher dimensions?

### Tip:
Understanding the Schwarz lemma is crucial for grasping why the Riemann Mapping Theorem's normalization condition ensures uniqueness.

Explore the proof of the Riemann Mapping Theorem, a key result in complex analysis. Learn about conformal mapping, holomorphic functions, and critical theorems such as the Carathéodory-Osgood Theorem, the Schwarz Lemma, and Dirichlet’s Principle. Suitable for advanced undergraduate students.

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