Math 829F: Riemann Surfaces |
| GP 1. | Work out the details of the construction of the complex structures on the torus via Riemannian metrics induced by the Euclidean metric on \(\mathbb{R}^3\). Construct the Teichmüller space \(\mathcal{T}(\mathbb{T}^2)\ = \mathcal{T}_1\) from this perspective. Compare and contrast these constructions with the ones that we studied in [JJ]. |
| GP 2. | Show that the Fricke coordinate map \(\mathcal{F}_g :\mathcal{T}_g \to \mathbb{R}^{6g-6}\) is injective, hence a bijection onto its image. Discuss how this induces a topology and a complex structure on \(\mathcal{T}_g\). |
| GP 3. | Summarize the proof of Teichmüller's theorem: The Teichmüller space \(\mathcal{T}_g\) of a closed Riemann surface \(S\) of genus \(g\) is equivalent to the space \(\mathcal{Q}(S)\) of holomorphic quadratic differentials on \(S\). |
| GP 4. | Prove the Uniformization Theorem: Every simply-connected Riemann surface is biholomorphically equivalent to one and only one of \(S^2\), \(\mathbb{C}\), or \(H\). |
| GP 5. | Summarize the proof of the existence theorem for Beltrami coefficients: For every Beltrami coefficient \(\mu \in B(\mathbb{C})_1,\) there exists a homeomorphism \(f\) of \(\mathbb{C}^* = \mathbb{C} \cup \{\infty\}\) onto \(\mathbb{C}^*\) that is a quasi-conformal mapping of \(\mathbb{C}\) with complex dilation \(\mu\). Moveover, \(f\) is uniquely determined by the normalization conditions: \[ f(0) = 0,\ \ f(1) = 1,\ \ \text{and}\ \ f(\infty) = \infty. \] Discuss the applications of this theorem. |
| GP 6. | Discuss the different compactifications of Teichmüller space in as much detail as you see fit. In particular, discuss the importance of Thurston's compactification in relation to the mapping class group. |
| GP 7. | Let \(\Omega \subset \mathbb{R}^d\) be a domain (i.e., a connected open set). A function \(\Gamma(x,y)\) on \(\bar{\Omega}\) is called a Green's function for \(\Omega\) if \begin{eqnarray*} \Delta_x \Gamma(x,y) & = & 0 \ \ \ \text{for}\ x \neq y, \\ \Gamma(x,y) & = & 0\ \ \ \text{for}\ x \in \partial \Omega,\ y \in \Omega \end{eqnarray*} and \(\Gamma(x - y) - G(x - y)\) is bounded, where \(G\) is the function defined in Lemma 3.4.1. If \(y\) is fixed, then one says that \(\Gamma(x,y)\) is the Green's function of \(\Omega\) with singularity at \(y \in \Omega\). Show that a Green's function for \(\Omega\) (if it exists) is uniquely determined by the above requirements. What is the Green's function \(f\) of a ball \(B(y,R) \subset \mathbb{R}^d?\) |
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