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# 数学代写|黎曼曲面代写Riemann surface代考|MAT00111M

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## 数学代写|黎曼曲面代写Riemann surface代考|Quotients Under Group Actions

Definition 4. Let $\Delta$ be a domain in $\mathbb{C}$. A group $G: \Delta \rightarrow \Delta$ of holomorphic transformations acts discontinuously on $\Delta$ if for any $P \in \Delta$ there exists a neighborhood $V \ni P$ such that
$$g V \cap V=\emptyset, \quad \forall g \in G, \quad g \neq I .$$
The quotient space $\Delta / G$ is defined by the equivalence relation
$$P \sim P^{\prime} \Leftrightarrow \exists g \in G: P^{\prime}=g P .$$
By the natural projection $\pi: \Delta \rightarrow \Delta / G$ every point is mapped to its equivalence class. Every point $P \in \Delta$ has a neighborhood $V$ satisfying (1.10). Then $U=\pi(V)$ is open and $\pi_{\left.\right|_V}: V \rightarrow U$ is a homeomorphism. Its inversion $z: U \rightarrow V \subset \Delta \subset \mathbb{C}$ is a local parameter. One can cover $\Delta / G$ by domains of this type. The transition functions are the corresponding group elements $g$; therefore they are holomorphic.
Theorem 2. $\Delta / G$ is a Riemann surface.

Tori
Let us consider the case $\Delta=\mathbb{C}$ and the group $G$ generated by two translations
$$z \rightarrow z+w, \quad z \rightarrow z+w^{\prime},$$
where $w, w^{\prime} \in \mathbb{C}$ are two non-parallel vectors, $\operatorname{Im} w^{\prime} / w \neq 0$, see Fig. 1.2. The group $G$ is commutative and consists of the elements
$$g_{n, m}(z)=z+n w+m w^{\prime}, \quad n, m \in \mathbb{Z} .$$
The factor $\mathbb{C} / G$ has a nice geometrical realization as the parallelogram
$$T=\left{z \in \mathbb{C} \mid z=a w+b w^{\prime}, a, b \in[0,1)\right} .$$
There are no $G$-equivalent points in $T$ and on the other hand every point in $\mathbb{C}$ is equivalent to some point in $T$. Since the edges of the parallelogram $T$ are $G$-equivalent $z \sim z+w, z \sim z+w^{\prime}, \mathcal{R}$ is a compact Riemann surface, which is topologically a torus. We discuss this case in more detail in Sect. 1.5.5.

## 数学代写|黎曼曲面代写Riemann surface代考|Polyhedral Surfaces as Riemann Surfaces

One can build a Riemann surface gluing together pieces of the complex plane $\mathbb{C}$.

Consider a finite set of disjoint polygons $F_i$ and identify isometrically pairs of edges in such a way that the result is a compact oriented polyhedral surface $\mathcal{P}$. A polyhedron in 3-dimensional Euclidean space is an example of such a surface.
Theorem 3. The polyhedral surface $\mathcal{P}$ is a Riemann surface.

In order to define a complex structure on a polyhedral surface let us distinguish three kinds of points (see Fig. 1.3):

1. Inner points of triangles
2. Inner points of edges
3. Vertices
One can map isometrically the corresponding polygon $F_i$ (or pairs of neighboring polygons) into $\mathbb{C}$. This provides local parameters at the points of the first and the second kind. Let $P$ be a vertex and $F_i, \ldots, F_n$ the sequence of successive polygons with this vertex (see the point (iii) above). Denote by $\theta_i$ the angle of $F_i$ at $P$. Then define
$$\gamma=\frac{2 \pi}{\sum_{i=1}^n \theta_i} .$$
Consider a suitably small ball neighborhood of $\mathrm{P}$, which is the union $U^r=$ $\cup_i F_i^r$, where $F_i^r=\left{Q \in F_i|| Q-P \mid<r\right}$. Each $F_i^r$ is a sector with angle $\theta_i$ at $P$. We map it as above into $\mathbb{C}$ with $P$ mapped to the origin and then apply $z \mapsto z^\gamma$, which produces a sector with the angle $\gamma \theta_i$. The mappings corresponding to different polygons $F_i$ can be adjusted to provide a homeomorphism of $U^r$ onto a disc in $\mathbb{C}$. All transition functions of the constructed charts are holomorphic since they are compositions of maps of the form $z \mapsto a z+b$ and $z \mapsto z^\gamma$ (away from the origin).

It turns out that any compact Riemann surface can be recovered from some polyhedral surface [Bos].

## 数学代写|黎曼曲面代写Riemann surface代考|Quotients Under Group Actions

$$g V \cap V=\emptyset, \quad \forall g \in G, \quad g \neq I .$$

$$P \sim P^{\prime} \Leftrightarrow \exists g \in G: P^{\prime}=g P .$$

Tori

$$z \rightarrow z+w, \quad z \rightarrow z+w^{\prime},$$

$$g_{n, m}(z)=z+n w+m w^{\prime}, \quad n, m \in \mathbb{Z} .$$

$$T=\left{z \in \mathbb{C} \mid z=a w+b w^{\prime}, a, b \in[0,1)\right} .$$

## 数学代写|黎曼曲面代写Riemann surface代考|Polyhedral Surfaces as Riemann Surfaces

$$\gamma=\frac{2 \pi}{\sum_{i=1}^n \theta_i} .$$

## MATLAB代写

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