# 微积分网课代修|函数代写Function theory代考|ISC5473 Ahlfors’s Version of the Schwarz Lemma

• 单变量微积分
• 多变量微积分
• 傅里叶级数
• 黎曼积分
• ODE
• 微分学

## 微积分网课代修|偏微分方程代写Partial Differential Equation代考|Ahlfors’s Version of the Schwarz Lemma

If $U \subseteq \mathbb{C}$ is a planar domain and $\rho$ is a metric on $U$, then the curvature of the metric $\rho$ at a point $z \in U$ is defined to be
$$\kappa_{U, \rho}(z)=\kappa(z) \equiv \frac{-\Delta \log \rho(z)}{\rho(z)^{2}} .$$
(Here zeros of $\rho(z)$ will result in singularities of the curvature function- $\kappa$ is undefined at such points.)

Since $\rho$ is twice continuously differentiable, this definition makes sense. It assigns to each $z \in U$ a numerical quantity. The most important preliminary fact about $\kappa$ is its conformal invariance:

Proposition 2.4.1. Let $U_{1}$ and $U_{2}$ be planar domains and $h: U_{1} \rightarrow U_{2} a$ conformal map (in particular, $h^{\prime}$ never vanishes). If $\rho$ is a metric on $U_{2}$, then
$$\kappa_{U_{1}, h^{} \rho}(z)=\kappa_{U_{2}, \rho}(h(z)), \quad \forall z \in U_{1} .$$ Proof. We need to calculate: \begin{aligned} \kappa_{U_{1}, h^{} \rho}(z) & \equiv \frac{-\Delta \log \left[\rho(h(z)) \cdot\left|h^{\prime}(z)\right|\right]}{\left[\rho(h(z)) \cdot \mid h^{\prime}(z)\right]^{2}} \ &=\frac{-\Delta \log [\rho(h(z))]-\Delta\left[\log \left(\left|h^{\prime}(z)\right|\right)\right]}{\left[\rho(h(z)) \cdot\left|h^{\prime}(z)\right|\right]^{2}} . \end{aligned}

## 微积分网课代修|偏微分方程代写Partial Differential Equation代考|Liouville and Picard Theorems

It turns out that curvature gives criteria for when there do or do not exist nonconstant holomorphic functions from a domain $U_{1}$ to a domain $U_{2}$. The most basic result along these lines is as follows

Theorem 2.5.1. Let $U \subseteq \mathbb{C}$ be an open set equipped with a metric $\sigma(z)$ having the property that its curvature $\kappa_{\sigma}(z)$ satisfies
$$\kappa_{\sigma}(z) \leq-B<0$$ for some positive constant $B$ and for all $z \in U$. Then any holomorphic function $$f: \mathbb{C} \rightarrow U$$ must be constant. Proof. For $\alpha>0$ we consider the mapping
$$f: D(0, \alpha) \rightarrow U$$

## 微积分网课代修|偏微分方程代写Partial Differential Equation代 考|Ahlfors’s Version of the Schwarz Lemma

$$\kappa_{U, \rho}(z)=\kappa(z) \equiv \frac{-\Delta \log \rho(z)}{\rho(z)^{2}} .$$
（这里的零 $\rho(z)$ 将导致曲率函数的奇异性- $\kappa$ 在这些点上是末定义的。）

$$\kappa_{U_{1}, h \rho}(z)=\kappa_{U_{2}, \rho}(h(z)), \quad \forall z \in U_{1} .$$

$$\kappa_{U_{1}, h \rho}(z) \equiv \frac{-\Delta \log \left[\rho(h(z)) \cdot\left|h^{\prime}(z)\right|\right]}{\left[\rho(h(z)) \cdot \mid h^{\prime}(z)\right]^{2}}=\frac{-\Delta \log [\rho(h(z))]-\Delta\left[\log \left(\left|h^{\prime}(z)\right|\right)\right]}{\left[\rho(h(z)) \cdot\left|h^{\prime}(z)\right|\right]^{2}} .$$

## 微积分网课代修|偏微分方程代写Partial Differential Equation代 考|Liouville and Picard Theorems

$$\kappa_{\sigma}(z) \leq-B<0$$ 对于一些正常数 $B$ 并为所有人 $z \in U$. 那么任何全纯函数 $$f: \mathbb{C} \rightarrow U$$ 必须是恒定的。证明。为了 $\alpha>0$ 我们考虑映射
$$f: D(0, \alpha) \rightarrow U$$