# 微积分网课代修|函数代写Function theory代考|ISC5473 A Geometric View 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|A Geometric View of the Schwarz Lemma

In 1938, Lars Ahlfors [AHL1] caused a sensation by proving that the Schwarz lemma is really an inequality about the curvatures of Riemannian metrics. In the present section we will give an exposition of Ahlfors’s ideas. Afterward we can provide some applications.

We shall go into considerable detail in the present discussion, so that the reader has ample motivation and context. Certainly it should be plain that there is definite resonance with the material on the Poincaré and Bergman metrics in Chapter $1 .$

## 微积分网课代修|偏微分方程代写Partial Differential Equation代考|Geometric Ideas

In classical analysis a metric is a device for measuring distance. If $X$ is a set, then a metric $\lambda$ for $X$ is a function
$$\lambda: X \times X \longrightarrow \mathbb{R}$$
satisfying, for all $x, y, z \in X$,

(1) $\lambda(x, y)=\lambda(y, x)$
(2) $\lambda(x, y) \geq 0$ and $\lambda(x, y)=0$ iff $x=y$;
(3) $\lambda(x, y) \leq \lambda(x, z)+\gamma(z, y)$.
The trouble with a metric defined in this generality is that it does not interact well with calculus. What sort of interaction might we wish to see?

Given two points $P, Q \in X$, one would like to consider the curve of least length connecting $P$ to $Q$. Any reasonable construction of such a curve leads to a differential equation, and thus we require that our metric lend itself to differentiation. Yet another consideration is curvature: in the classical setting curvature is measured by the rate of change of the normal vector field. The concepts of normal and rate of change lead inexorably to differentiation. Thus we shall now take a different approach to the concept of “metric.” The reader will see definite parallels here with our treatment of “metric” in Chapter 1 on invariant geometry.

## 微积分网课代修|偏微分方程代写Partial Differential Equation代考|A Geometric View of the Schwarz Lemma

1938 年，Lars Ahlfors [AHL1] 通过证明施瓦茨引理确实是关于黎曼度量曲率的不等式 而引起轰动。在本节中，我们将阐述 Ahlfors 的思想。之后我们可以提供一些应用程 序。

## 微积分网课代修|偏微分方程代写Partial Differential Equation代 考|Geometric Ideas

$$\lambda: X \times X \longrightarrow \mathbb{R}$$

(1) $\lambda(x, y)=\lambda(y, x)$
(2) $\lambda(x, y) \geq 0$ 和 $\lambda(x, y)=0$ 当且当 $x=y$;
(3) $\lambda(x, y) \leq \lambda(x, z)+\gamma(z, y)$.